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Morfismos issue for December 2013

VOLUMEN 17NÚMERO 2

JULIO A DICIEMBRE 2013 ISSN: 1870-6525

Chief Editors - Editores Generales

• Isidoro Gitler • zelaznoGsuseJ

Associate Editors - Editores Asociados

• Ruy Fabila • zednanreHleamsI• amreL-zednanreHomisenO • Hector Jasso Fuentes

• Sadok Kallel • Miguel Maldonado• Carlos Pacheco • Enrique Ramırez de Arellano

• Enrique Reyes • Dai Tamaki• Enrique Torres Giese

Apoyo Tecnico

• zehcnaSadnarAanairdA • Irving Josue Flores Romero• zelaznoGsetneuFoinotnAocraM • oczorOzednanreHramO

• Roxana Martınez • Laura Valencia

Morfismos noicceridalneelbinopsidatse http://www.morfismos.cinvestav.mx.Para mayores informes dirigirse al telefono +52 (55) 5747-3871. Toda corres-

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:noicceridalaocinortcele morfismos@math.cinvestav.mx.

VOLUMEN 17NÚMERO 2

JULIO A DICIEMBRE 2013ISSN: 1870-6525

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del Instituto Politecnico Nacional (Cinvestav), a traves del DepartamentoordePnaS.loC,8052.oNlanoicaNocincetiloPotutitsnI.vA.sacitametaMed,00837475-55.leT,.F.D,06370.P.C,oredaM.AovatsuGnoicageleD,ocnetacaZ

www.cinvestav.mx, morfismos@math.cinvestav.mx, Editores Generales: Drs.sohcereDedavreseR.sorraBonipsEzelaznoGsuseJyreltiGorodisI

No. 04-2012-011011542900-102, ISSN: 1870-6525, ambos otorgados por elInstituto Nacional del Derecho de Autor. Certificado de Licitud de TıtuloNo. 14729, Certificado de Licitud de Contenido No. 12302, ambos otorga-

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Cinvestav, Avenida Instituto Politecnico Nacional 2508, Colonia San PedronerimirpmiedonimretesoremunetsE.F.D,ocixeM,06370.P.C,ocnetacaZ

febrero de 2014 con un tiraje de 50 ejemplares.

Las opiniones expresadas por los autores no necesariamente reflejan la.noicacilbupaledserotidesoledarutsop

-nocsoledlaicrapolatotnoiccudorperaladibihorpetnematcirtseadeuQlednoicazirotuaaiverpnis,noicacilbupaledsenegamiesodinet Cinvestav.

Information for Authors

The Editorial Board of Morfismos calls for papers on mathematics and related areas tobe submitted for publication in this journal under the following guidelines:

• Manuscripts should fit in one of the following three categories: (a) papers covering thegraduate work of a student, (b) contributed papers, and (c) invited papers by leadingscientists. Each paper published in Morfismos will be posted with an indication ofwhich of these three categories the paper belongs to.

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Informacion para Autores

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• Se consideraran tres tipos de trabajos: (a) artıculos derivados de tesis de grado dealta calidad, (b) artıculos por contribucion y (c) artıculos por invitacion escritos porlıderes en sus respectivas areas. En todo artıculo publicado en Morfismos se indicarael tipo de trabajo del que se trate de acuerdo a esta clasificacion.

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Editorial Guidelines

Morfismos is the journal of the Mathematics Department of Cinvestav. Oneof its main objectives is to give advanced students a forum to publish their earlymathematical writings and to build skills in communicating mathematics.

Publication of papers is not restricted to students of Cinvestav; we want to en-courage students in Mexico and abroad to submit papers. Mathematics researchreports or summaries of bachelor, master and Ph.D. theses of high quality will beconsidered for publication, as well as contributed and invited papers by researchers.All submitted papers should be original, either in the results or in the methods.The Editors will assign as referees well-established mathematicians, and the accep-tance/rejection decision will be taken by the Editorial Board on the basis of thereferee reports.

Authors of Morfismos will be able to choose to transfer copy rights of theirworks to Morfismos. In that case, the corresponding papers cannot be consideredor sent for publication in any other printed or electronic media. Only those papersfor which Morfismos is granted copyright will be subject to revision in internationaldata bases such as the American Mathematical Society’s Mathematical Reviews, andthe European Mathematical Society’s Zentralblatt MATH.

Morfismos

Lineamientos Editoriales

Morfismos, revista semestral del Departamento de Matematicas del Cinvestav,tiene entre sus principales objetivos el ofrecer a los estudiantes mas adelantadosun foro para publicar sus primeros trabajos matematicos, a fin de que desarrollenhabilidades adecuadas para la comunicacion y escritura de resultados matematicos.

La publicacion de trabajos no esta restringida a estudiantes del Cinvestav; de-seamos fomentar la participacion de estudiantes en Mexico y en el extranjero, asıcomo de investigadores mediante artıculos por contribucion y por invitacion. Losreportes de investigacion matematica o resumenes de tesis de licenciatura, maestrıao doctorado de alta calidad pueden ser publicados en Morfismos. Los artıculos apublicarse seran originales, ya sea en los resultados o en los metodos. Para juzgaresto, el Consejo Editorial designara revisores de reconocido prestigio en el orbe in-ternacional. La aceptacion de los artıculos propuestos sera decidida por el ConsejoEditorial con base a los reportes recibidos.

Los autores que ası lo deseen podran optar por ceder a Morfismos los derechos depublicacion y distribucion de sus trabajos. En tal caso, dichos artıculos no podranser publicados en ninguna otra revista ni medio impreso o electronico. Morfismossolicitara que tales artıculos sean revisados en bases de datos internacionales como loson el Mathematical Reviews, de la American Mathematical Society, y el ZentralblattMATH, de la European Mathematical Society.

Morfismos

Special MIMS Proceedings Issue

This issue is devoted to the proceedings of the conference “Op-erads and Configuration Spaces” that took place at the Mediter-ranean Institute for the Mathematical Sciences (MIMS), Cite desSciences, in Tunis capital city, June 18–22, 2012. This conferencewas part of the launch of MIMS in the region. Plenary speak-ers gave a series of lectures that were attended by students andyoung researchers from Tunisia and Algeria. The MIMS thanksChristophe Cazanave, Jeffrey Giansiracusa, Paolo Salvatore, InesSaihi, Ismar Volic, Benjamin Walter, and all participants for mak-ing this a successful first conference. It also thanks Oscar-RandalWilliams for his special contribution.

Este numero esta dedicado a las memorias de la conferencia“Operads and Configuration Spaces” realizada en el MediterraneanInstitute for Mathematical Sciences (MIMS), Cite des Sciences, enla ciudad de Tunez, del 18 al 22 de junio de 2012. La conferenciafue parte de las actividades inaugurales del MIMS en la region.Los ponentes plenarios dieron una serie de conferencias a las queasistieron estudiantes e investigadores jovenes de Tunez y Argelia.El MIMS agradece a Christophe Cazanave, Jeffrey Giansiracusa,Paolo Salvatore, Ines Saihi, Ismar Volic, Benjamin Walter y todoslos participantes por hacer de esta primera conferencia un exito.Tambien agradece a Oscar-Randal Williams por su contribucionespecial.

Contents - Contenido

Configuration space integrals and the topology of knot and link spaces

Ismar Volic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Topological chiral homology and configuration spaces of spheres

Oscar Randal-Williams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Cooperads as symmetric sequences

Benjamin Walter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Moduli spaces and modular operads

Jeffrey Giansiracusa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Morfismos, Vol. 17, No. 2, 2013, pp. 1–56

Configuration space integrals and the topologyof knot and link spaces

Ismar Volic 1

Abstract

This article surveys the use of configuration space integrals in thestudy of the topology of knot and link spaces. The main focus isthe exposition of how these integrals produce finite type invari-ants of classical knots and links. More generally, we also explainthe construction of a chain map, given by configuration spaceintegrals, between a certain diagram complex and the deRhamcomplex of the space of knots in dimension four or more. A gen-eralization to spaces of links, homotopy links, and braids is alsotreated, as are connections to Milnor invariants, manifold calculusof functors, and the rational formality of the little balls operads.

2010 Mathematics Subject Classification: 57Q45, 57M27, 81Q30, 57-R40.Keywords and phrases: configuration space integrals, Bott-Taubes inte-grals, knots, links, homotopy links, braids, finite type invariants, Vas-siliev invariants, Milnor invariants, chord diagrams, weight systems,manifold calculus, embedding calculus, little balls operad, rational for-mality of configuration spaces.

Contents2noitcudortnI15repapehtfonoitazinagrO1.16seiranimilerP2

2.1 Differential forms and integration along the fiber 69stonkgnolfoecapS2.2

1The author was supported by the National Science Foundation grant DMS1205786.

1

2 Ismar Volic

2.3 Finite type invariants 102.4 Configuration spaces and their compactification 17

3 Configuration space integrals and finite type knotinvariants 203.1 Motivation: The linking number 203.2 “Self-linking” for knots 223.3 Finite type two knot invariant 233.4 Finite type k knot invariants 30

4 Generalization to Kn, n > 3 335 Further generalizations and applications 38

5.1 Spaces of links 385.2 Manifold calculus of functors and finite typeinvariants 435.3 Formality of the little balls operad 48

References 52

1 Introduction

Configuration space integrals are fascinating objects that lie at the inter-section of physics, combinatorics, topology, and geometry. Since theirinception over twenty years ago, they have emerged as an important toolin the study of the topology of spaces of embeddings and in particularof spaces of knots and links.

The beginnings of configuration space integrals can be traced backto Guadagnini, Martellini, and Mintchev [19] and Bar-Natan [4] whosework was inspired by Chern-Simons theory. The more topological pointof view was introduced by Bott and Taubes [9]; configuration spaceintegrals are because of this sometimes even called Bott-Taubes inte-grals in the literature (more on Bott and Taubes’ work can be foundin Section 3.3). The point of this early work was to use configurationspace integrals to construct a knot invariant in the spirit of the classicallinking number of a two-component link. This invariant turned out tobe of finite type (finite type invariants are reviewed in Section 2.3) andD. Thurston [51] generalized it to construct all finite type invariants.We will explain D. Thurston’s result in Section 3.4, but the idea is asfollows:

Given a trivalent diagram Γ (see Section 2.3), one can construct abundle

π : Conf[p, q;K3,Rn] −→ K3,

Configuration space integrals and knots 3

where K3 is the space of knots in R3. Here p and q are the numbers ofcertain kinds of vertices in Γ and Conf[p, q;K3,Rn] is a pullback spaceconstructed from an evaluation map and a projection map. The fiber ofπ over a knot K ∈ K3 is the compactified configuration space of p + qpoints in R3, first p of which are constrained to lie on K. The edges ofΓ also give a prescription for pulling back a product of volume formson S2 to Conf[p, q;K3,Rn]. The resulting form can then be integratedalong the fiber, or pushed forward, to K3. The dimensions work outso that this is a 0-form and, after adding the pushforwards over alltrivalent diagrams of a certain type, this form is in fact closed, i.e. it isan invariant. Thurston then proves that this is a finite type invariantand that this procedure gives all finite type invariants.

The next generalization was carried out by Cattaneo, Cotta-Ramu-sino, and Longoni [12]. Namely, let Kn, n > 3, be the space of knots inRn. The main result of [12] is that there is a cochain map

(1) Dn −→ Ω∗(Kn)

between a certain diagram complex Dn generalizing trivalent diagramsand the deRham complex of Kn. The map is given by exactly the sameintegration procedure as Thurston’s, except the degree of the form thatis produced on Kn is no longer zero. Specializing to classical knots(where there is no longer a cochain map due to the so-called “anoma-lous face”; see Section 3.4) and degree zero, one recovers the work ofThurston. Cattaneo, Cotta-Ramusino, and Longoni have used the map(1) to show that spaces of knots have cohomology in arbitrarily highdegrees in [13] by studying certain algebraic structures on Dn that cor-respond to those in the cohomology ring of Kn. Longoni also proved in[33] that some of these classes arise from non-trivalent diagrams.

Even though configuration space integrals were in all of the afore-mentioned work constructed for ordinary closed knots, it has in recentyears become clear that the variant for long knots is also useful. Be-cause some of the applications we describe here have a slight preferencefor the long version, this is the space we will work with. The differencebetween the closed and the long version is minimal from the perspectiveof this paper, as explained at the beginning of Section 2.2.

More recently, configuration space integrals have been generalizedto (long) links, homotopy links, and braids [30, 57], and this work issummarized in Section 5.1. One nice feature of this generalization is

4 Ismar Volic

that it provides the connection to Milnor invariants. This is becauseconfiguration space integrals give finite type invariants of homotopylinks, and, since Milnor invariants are finite type, this immediately givesintegral expressions for these classical invariants.

We also describe two more surprising applications of configurationspace integrals. Namely, one can use manifold calculus of functors toplace finite type invariants in a more homotopy-theoretic setting as de-scribed in Section 5.2. Functor calculus also combines with the formalityof the little n-discs operad to give a description of the rational homol-ogy of Kn, n > 3. Configuration space integrals play a central role heresince they are at the heart of the proof of operad formality. Some detailsabout this are provided in Section 5.3.

In order to keep the focus of this paper on knot and links and keepits length to a manageable size, we will regrettably only point the readerto three other topics that are growing in promise and popularity. Thefirst is the work of Sakai [44] and its expansion by Sakai and Watan-abe [49] on long planes, namely embeddings of Rk in Rn fixed outsidea compact set. These authors use configuration space integrals to pro-duce nontrivial cohomology classes of this space with certain conditionson k and n. This work generalizes classes produced by others [14, 58]and complements recent work by Arone and Turchin [2] who show, us-ing homotopy-theoretic methods, that the homology of Emb(Rk,Rn) isgiven by a certain graph complex for n ≥ 2k+2. Sakai has further usedconfiguration space integrals to produce a cohomology class of K3 indegree one that is related to the Casson invariant [43] and has given anew interpretation of the Haefliger invariant for Emb(Rk,Rn) for somek and n [44]. In an interesting bridge between two different points ofview on spaces of knots, Sakai has in [44] also combined the configura-tion space integrals with Budney’s action of the little discs operad onKn [10].

The other interesting development is the recent work of Koytcheff[29] who develops a homotopy-theoretic replacement of configurationspace integrals. He uses the Pontryagin-Thom construction to “pushforward” forms from Conf[p, q;Kn,Rn] to Kn. The advantage of thisapproach is that is works over any coefficients, unlike ordinary configu-ration space integration, which takes values in R. A better understand-ing of how Koytcheff’s construction relates to the original configurationspace integrals is still needed.

Configuration space integrals and knots 5

The third topic is the role configuration space integrals have re-cently played in the construction of asymptotic finite type invariants ofdivergence-free vector fields [25]. The approach in this work is to applyconfiguration space integrals to trajectories of a vector field. In thisway, generalizations of some familiar asymptotic vector field invariantslike asymptotic linking number, helicity, and the asymptotic signaturecan be derived.

Lastly, some notes on the style and expositional choices we havemade in this paper are in order. We will assume an informal tone, espe-cially at times when writing down something precisely would require usto introduce cumbersome notation. To quote from a friend and coauthorBrian Munson [40], “we will frequently omit arguments which would dis-tract us from our attempts at being lighthearted”. Whenever this is thecase, a reference to the place where the details appear will be supplied.In particular, most of the proofs we present here have been worked outin detail elsewhere, and if we feel that the original source is sufficient,we will simply give a sketch of the proof and provide ample referencesfor further reading. It is also worth pointing out that many open prob-lems are stated througout and our ultimate hope is that, upon lookingat this paper, the reader will be motivated to tackle some of them.

1.1 Organization of the paper

We begin by recall some of the necessary background in Section 2. Weonly give the basics but furnish abundant references for further reading.In particular, we review integration along the fiber in Section 2.1 andpay special attention to integration for infinite-dimensional manifoldsand manifolds with corners. In Section 2.2 we define the space of longknots and state some observations about it. A review of finite typeinvariants is provided in Section 2.3; they will play a central role later.This section also includes a discussion of chord diagrams and trivalentdiagrams. Finally in Section 2.4, we talk about configuration spaces andtheir Fulton-MacPherson compactification. These are the spaces overwhich our integration will take place.

Section 3 is devoted to the construction of finite type invariants viaconfiguration space integrals. The motivating notion of the linking num-ber is recalled in Section 3.1, and that leads to the failed constructionof the “self-linking” number in Section 3.2 and its improvement to thesimplest finite type (Casson) invariant in Section 3.3. This section is at

6 Ismar Volic

the heart of the paper since it gives all the necessary ideas for all of theconstructions encountered from then on. Finally in Section 3.4 we con-struct all finite type knot invariants via configuration space integrals.

Section 4 is dedicated to the description of the cochain map (1)and includes the definition of the cochain complex Dn. We also discusshow this generalizes D. Thurston’s construction that yields finite typeinvariants.

Finally in Section 5, we give brief accounts of some other features,generalizations, and applications of configuration space integrals. Moreprecisely, in Section 5.1, we generalize the constructions we will haveseen for knots to links, homotopy links, and braids; in Section 5.2,we explore the connections between manifold calculus of functors andconfiguration space integrals; and in Section 5.3, we explain how config-uration space integrals are used in the proof of the formality of the littlen-discs operad and how this leads to information about the homologyof spaces of knots.

2 Preliminaries

2.1 Differential forms and integration along the fiber

The strategy we will employ in this paper is to produce differentialforms on spaces of knots and links via configuration space integrals.Since introductory literature on differential forms is abundant (see forexample [8]), we will not recall their definition here. We also assumethe reader is familiar with integration of forms over manifolds.

We will, however, recall some terminology that will be used through-out: Given a smooth oriented manifold M , one has the deRham cochaincomplex Ω∗(M) of differential forms :

0 −→ Ω0(M)d−→ Ω1(M)

d−→ Ω2(M) −→ · · ·

where Ωk(M) is the space of smooth k-forms on M . The differential dis the exterior derivative. A form α ∈ Ωk(M) is closed if dα = 0 andexact if α = dβ for some β ∈ Ωk−1(M). The kth deRham cohomologygroup of M , Hk(M), is defined the usual way as the kernel of d modulothe image of d, i.e. the space of closed forms modulo the subspace ofexact forms. All the cohomology we consider will be over R.

Configuration space integrals and knots 7

The wedge product, or exterior product, of differential forms givesΩ∗(M) the structure of an algebra, called the deRham, or exterior alge-bra of M .

According to the deRham Theorem, deRham cohomology is isomor-phic to the ordinary singular cohomology. In particular, H0(M) is thespace of functionals on connected components of M , i.e. the space ofinvariants of M . The bulk of this paper is concerned with invariants ofknots and links.

The complex Ω∗(M) can be defined for manifolds with boundary bysimply restricting the form to the boundary. Locally, we take restrictionsof forms on open subsets of Rk to Rk−1 × R+. Further, one can definedifferential forms on manifolds with corners (an n-dimensional manifoldwith corners is locally modeled on Rk

+ × Rn−k, 0 ≤ k ≤ n; see [23]for a nice introduction to these spaces) in exactly the same fashion byrestricting forms to the boundary components, or strata of M .

The complex Ω∗(M) can also be defined for infinite-dimensionalmanifolds such as the spaces of knots and links we will consider here.One usually considers the forms on the vector space on which M is lo-cally modeled and then patches them together into forms on all of M .When M satisfies conditions such as paracompactness, this “patched-together” complex again computes the ordinary cohomology of M . An-other way to think about forms on an infinite-dimensional manifold Mis via the diffeological point of view which considers forms on open setsmapping into M . For more details, see [30, Section 2.2] which givesfurther references.

Given a smooth fiber bundle π : E → B whose fibers are compactoriented k-dimensional manifolds, there is a map

(2) π∗ : Ωn(E) −→ Ωn−k(B)

called the pushforward or integration along the fiber. The idea is to de-fine the form on B pointwise by integrating over each fiber of π. Namely,since π is a bundle, each point b ∈ B has a k-dimensional neighborhoodUb such that π−1(b) ∼= B×Ub. Then a form α ∈ Ωn(E) can be restrictedto this fiber, and the coordinates on Ub can be “integrated away”. Theresult is a form on B whose dimension is that of the original form butreduced by the dimension of the fiber. The idea is to then patch thesevalues together into a form on B. Thus the map (2) can be described

8 Ismar Volic

by

(3) α !−→!b !−→

"

π−1(b)α

#.

In terms of evaluation on cochains, π∗α can be thought of as an(n− k)-form on B who value on a k-chain X is

"

Xπ∗α =

"

π−1(X)α.

For introductions to the pushforward, see [8, 21].

Remark 2.1.1. The assumption that the fibers of π be compact canbe dropped, but then forms with compact support should be used toguarantee the convergence of the integral.

To check if π∗α is a closed form, it suffices to integrate dα, i.e. wehave [8, Proposition 6.14.1]

dπ∗α = π∗dα.

The situation changes when the fiber is a manifold with boundary orwith corners, as will be the case for us. Then by Stokes’ Theorem, dπ∗αhas another term. Namely, we have

(4) dπ∗α = π∗dα+ (∂π)∗α

where (∂π)∗α is the integral of the restriction of α to the codimensionone boundary of the fiber. This can be seen by an argument similarto the proof of the ordinary Stokes’ Theorem. For Stokes’ Theorem formanifolds with corners, see, for example [31, Chapter 10].

In the situations we will encounter here, α will be a closed form, inwhich case dα = 0. Thus the first term in the above formula vanishes,so that we get

(5) dπ∗α = (∂π)∗α.

Our setup will combine various situations described above – we will havea smooth bundle π : E → B of infinite-dimensional spaces with fibersthat are finite-dimensional compact manifolds with corners.

Configuration space integrals and knots 9

2.2 Space of long knots

We will be working with long knots and links (links will be definedin Section 5.1), which are easier to work with than ordinary closedknots and links in many situations. For example, long knots are H-spaces via the operation of stacking, or concatenation, which gives their(co)homology groups more structure. Also, the applications of manifoldcalculus of functors to these spaces (Section 5.2) prefer the long model.Working with long knots is not much different than working with ordi-nary closed ones since the theory of long knots in Rn is the same as thetheory of based knots in Sn. From the point of view of configurationspace integrals, the only difference is that, for the long version, we willhave to consider certain faces at infinity (see Remark 2.4.4).

Before we define long knots, we remind the reader that a smoothmap f : M → N between smooth manifolds M and N is

• an immersion if the derivative of f is everywhere injective;

• an embedding if it is an immersion and a homeomorphism onto itsimage.

Now let Mapc(R,Rn) be the space of smooth maps of R to Rn, n ≥ 3,which outside the standard interval I (or any compact set, really) agreeswith the map

R −→ Rn

t $−→ (t, 0, 0, ..., 0).

Give Mapc(R,Rn) the C∞ topology (see, for example, [22, Chapter2]).

Definition 2.2.1. Define the space of long (or string) knots Kn ⊂Mapc(R,Rn) as the subset of maps K ∈ Mapc(R,Rn) that are embed-dings, endowed with the subspace topology.

A related space that we will occassionally have use for is the spaceof long immersions Immc(R,Rn) defined the same way as the space oflong knots except its points are immersions. Note that Kn is a subspaceof Immc(R,Rn).

A homotopy in Kn (or any other space of embeddings) is called anisotopy. A homotopy in Immc(R,Rn) is a regular homotopy.

10 Ismar Volic

K ∈ Kn

Figure 1: An example of a knot in Rn.

An example of a long knot is given in Figure 1. Note that we haveconfused the map K with its image in Rn. We will do this routinelyand it will not cause any issues.

From now on, the adjective “long” will be dropped; should we needto talk about closed knots, we will say this explicitly.

The space Kn is a smooth inifinite-dimensional paracompact mani-fold [30, Section 2.2], which means that, as explained in Section 2.1, wecan consider differential forms on it and study their deRham cohomol-ogy.

Classical knot theory (n = 3) is mainly concerned with computing

• H0(K3), which is generated (over R; recall that in this paper, thecoefficient ring is always R) by knot types, i.e. by isotopy classesof knots; and

• H0(K3), the set of knot invariants, namely locally constant (R-valued) functions on K3. These are therefore precisely the func-tions that take the same value on isotopic knots.

The question of computation of knot invariants will be of particularinterest to us (see Section 3).

However, higher (co)homology of Kn is also interesting, even forn > 3. Of course, in this case there is no knotting or linking (by asimple general position argument), so H0 and H0 are trivial, but onecan then ask about H>0 and H>0. It turns out that our configurationspace integrals contain much information about cohomology in variousdegrees and for all n > 3 (see Section 4).

2.3 Finite type invariants

An interesting set of knot invariants that our configuration space inte-grals will produce are finite type, or Vassiliev invariants. These invari-

Configuration space integrals and knots 11

ants are conjectured to separate knots, i.e. to form a complete set ofinvariants. To explain, there is no known invariant or a set of invariants(that is reasonably computable) with the following property:

Given two non-isotopic knots, there is an invariant in thisset that takes on different values on these two knots.

The conjecture that finite type invariants form such a set of invari-ants has been open for some twenty years. There is some evidence thatthis might be true since finite type invariants do separate homotopylinks [20] and braids [6, 24] (see Section 5.1 for the definitions of thesespaces).

Since finite type invariants will feature prominently in Section 3, wegive a brief overview here. In addition to the separation conjecture,these invariants have received much attention because of their connec-tion to physics (they arise from Chern-Simons Theory), Lie algebras,three-manifold topology, etc. The literature on finite type invariants isabundant, but a good start is [5, 15].

Suppose V is a knot invariant, so V ∈ H0(K3). Consider the spaceof singular links, which is the subspace of Immc(R,Rn) consisting ofimmersions that are embeddings except for a finite number of double-point self-intersections at which the two derivatives (coming from trav-eling through the singularity along two different pieces of the knot) arelinearly independent.

Each singularity can be locally “resolved” in two natural ways (up toisotopy), with one strand pushed off the other in one of two directions.A k-singular knot (a knot with k self-intersections) can thus be resolvedinto 2k ordinary embedded knots. We can then define V on singularknots as the sum of the values of V on those resolutions, with signs asprescribed in Figure 2. The expression given in that picture is calledthe Vassiliev skein relation and it depicts the situation locally around asingularity. The rest of the knot is the same for all three pictures. Theknot should be oriented (by, say, the natural orientation of R); otherwisethe two resolutions can be rotated into one another.

Definition 2.3.1. A knot invariant V is finite type k (or Vassiliev oftype k) if it vanishes on singular knots with k + 1 self-intersections.

Example 2.3.2. There is only one (up to constant multiple) type 0invariant, since such an invariant takes the same value on two knots

12 Ismar Volic

!− V

"!= V

" !V"

Figure 2: Vassiliev skein relation.

that only differ by a crossing change. Since all knots are related bycrossing changes, this invariant must take the same value on all knots.It is not hard to see that there is also only one type 1 invariant.

Example 2.3.3. The coefficients of the Conway, Jones, HOMFLY, andKauffman polynomials are all finite type invariants [4, 7].

Let Vk be the real vector space generated by all finite type k invari-ants and let

V =#

k

Vk.

This space is filtered; it is immediate from the definitions that Vk ⊂Vk+1.

One of the most interesting features of finite type invariants is thata value of a type k invariant V on a k-singular knot only depends onthe placement of the singularities and not on the immersion itself. Thisis due to a simple observation that, if a crossing of a k-singular knot ischanged, the difference of the evaluation of V on the two knots (beforeand after the switch) is the value of V on a (k+1)-singular knot by theVassiliev skein relation. But V is type k so the latter value is zero, andhence V “does not see” the crossing change. Since one can get from anysingular knot to any other singular knot that has the singularities in thesame place (“same place” in the sense that for both knots, there are 2kpoints on R that are partitioned in pairs the same way; these pairs willmake up the k singularities upon the immersion of R in R3), V in facttakes the same value on all k singular knots with the same singularitypattern.

The notion of what it means for singularities to be in the “sameplace” warrants more explanation and leads to the beautiful and richconnections between finite type invariants and the combinatorics ofchord diagrams as follows.

Configuration space integrals and knots 13

Definition 2.3.4. A chord diagram of degree k is a connected graph(one should think of a 1-dimensional cell complex) consisting of an ori-ented line segment and 2k labeled vertices marked on it (considered upto orientation-preserving diffeomorphism of the segment). The graphalso contains k oriented chords pairing off the vertices (so each vertexis connected to exactly one other vertex by a chord).

We will refer to labels and orientations as decorations and, whenthere is no danger of confusion, we will sometimes draw diagrams with-out them.

Examples of chord diagrams are given in Figure 3. The reader mightwish for a more proper combinatorial (rather than descriptive) definitionof a chord diagram, and such a definition can be found in [30, Section3.1] (where the definition is for the case of trivalent diagrams which wewill encounter below, but it specializes to chord diagrams as the latterare a special case of the former).

61 2 3 4 1 2 3 4 5

Figure 3: Examples of chord diagrams. The left one is of degree 2 andthe right one is of degree 3. We will always assume the segment isoriented from left to right.

Definition 2.3.5. Define CDk to be the real vector space generated bychord diagrams of degree k modulo the relations

1. If Γ contains more than one chord connecting two vertices, thenΓ = 0;

2. If a diagram Γ differs from Γ′ by a relabeling of vertices or orien-tations of chords, then

Γ− (−1)σΓ′ = 0

where σ is the sum of the order of the permutation of the labelsand the number of chords with different orientation;

3. If Γ contains a chord connecting two consecutive vertices, thenΓ = 0 (this is the one-term, or 1T relation);

14 Ismar Volic

a

−

−=

i j j i

k l l kj j

l la a

a

Figure 4: The four-term (4T) relation. Chord orientations have beenomitted, but they should be the same for all four pictures.

4. If four diagrams differ only in portions as pictured in Figure 4,then their sum is 0 (this is the four-term, or 4T relation).

We can now also define the graded vector space

CD =!

k

CDk.

Let CWk = Hom(CDk,R), the dual of CDk. This is called the space ofweight systems of degree k, and is by definition the space of functionalson chord diagrams that vanish on 1T and 4T relations.

We can now define a function

f : Vk −→ CWk

V −→"

W : CDk −→ RΓ #−→ V (KΓ)

#

where KΓ is any k-singular knot with singularities as prescribed byΓ. By this we mean that there are 2k points on R labeled the sameway as in Γ, and if x and y are points for which there exists a chordin Γ, then KΓ(x) = KΓ(y). The map f is well-defined because of theobservation that type k invariant does not depend on the immersionwhen evaluated on a k-singular knot.

The reason that the image of f is indeed in CWk, i.e. the reason thatthe function W we defined above vanishes on the 1T and 4T relations isnot hard to see (in fact, 1T and 4T relations are part of the definition ofCDk precisely because W vanishes on them): 1T relation corresponds tothe singular knot essentially having a loop at the singularity; resolvingthose in two ways results in two isotopic knots on which V has to take thesame value (since it is an invariant). Thus the difference of those values

Configuration space integrals and knots 15

is zero, but by the skein relation, V is then zero on a knot containingsuch a singularity. The vanishing on the 4T relation arises from the factthat passing a strand around a singularity of a (k − 1)-singular knotintroduces four k-singular knots, and the 4T relation reflects the factthat at the end one gets back to the same (k − 1)-singular knot.

It is also immediate from the definitions that the kernel of f isprecisely type k − 1 invariants, so that we have an injection (which wewill denote by the same letter f)

(6) f : Vk/Vk−1 !→ CWk.

The following theorem is usually referred to as the FundamentalTheorem of Finite Type Invariants, and is due to Kontsevich [27].

Theorem 2.3.6. The map f from equation (6) is an isomorphism.

Kontsevich proves this remarkable theorem by constructing the in-verse to f , a map defined by integration that is now known as theKontsevich Integral. There are now several proof of this theorem, andthe one relevant to us gives the inverse of f in terms of configurationspace integrals. See Remark 3.4.2 for details.

Lastly we describe an alternative space of diagrams that will bebetter suited for our purposes.

Definition 2.3.7. A trivalent diagram of degree k is a connected graphconsisting of an oriented line segment (considered up to orientation-preserving diffeomorphism) and 2k labeled vertices of two types: seg-ment vertices, lying on the segment, and free vertices, lying off thesegment. The graph also contains some number of oriented chords con-necting segment vertices and some number of oriented edges connectingtwo free vertices or a free vertex and a segment vertex. Each vertex istrivalent, with the segment adding two to the count of the valence of asegment vertex.

Note that chord diagrams as described in Definition 2.3.4 are alsotrivalent diagrams. Examples of trivalent diagrams that are not chorddiagrams are given in Figure 5.

As mentioned before, a more combinatorial definition of trivalentdiagrams is given in [30, Section 3.1].

Analogously to Definition 2.3.5, we have

16 Ismar Volic

8

1

4 5

321 2 3 4 5 6

7

Figure 5: Examples of trivalent diagrams (that are not chord diagrams).The left one is of degree 4 and the right one is of degree 3.

Definition 2.3.8. Define T Dk to be the real vector space generated bytrivalent diagrams of degree k modulo the relations

1. If Γ contains more than one edge connecting two vertices, thenΓ = 0;

2. If a diagram Γ differs from Γ′ by a relabeling of vertices or orien-tations of chords or edges, then

Γ− (−1)σΓ′ = 0

where σ is the sum of the order of the permutation of the labelsand the number of chords and edges with different orientation;

3. If Γ contains a chord connecting two consecutive segment vertices,then Γ = 0 (this is same 1T relation from before);

4. If three diagrams differ only in portions as pictured in Figure 6,then their sum is 0 (these are called the STU and IHX relation,respectively.).

Finally let

T D =!

k

T Dk.

Remark 2.3.9. The IHX relations actually follows from the STU re-lation [5, Figure 9], but it is important enough that it is usually left inthe definition of the space of trivalent diagrams (it gives a connectionbetween finite type invariants and Lie algebras).

Theorem 2.3.10 ([5], Theorem 6). There is an isomorphism

CDk∼= T Dk.

Configuration space integrals and knots 17

i

= −

I H X

−

jijji

= −

jiijiS UT

j

Figure 6: The STU and the IHX relations.

The isomorphism is given by using the STU relation repeatedly toremove all free vertices and turn a trivalent diagram into a chord dia-gram. The relationship between the STU and the 4T relation is thatthe latter is the difference of two STU relations applied to two differentedges of the “tripod” diagram in Figure 11.

One can now again define the space of weight systems of degree kfor trivalent diagrams as the space of functionals on T Dk that vanish onthe 1T, STU, and IHX relations. We will denote these by T Wk. FromTheorem 2.3.10, we then have

(7) CWk∼= T Wk.

2.4 Configuration spaces and their compactifications

At the heart of the construction of our invariants and other cohomologyclasses of spaces of knots are configuration spaces and their compactifi-cations.

Definition 2.4.1. The configuration space of p points in a manifold Mis the space

Conf(p,M) = (x1, x2, ..., xp) ∈ Mp : xi = xj for i = j

18 Ismar Volic

Thus Conf(p,M) is Mp with all the diagonals, i.e. the fat diagonal,taken out.

We take Conf(0,M) to be a point. Since configuration spaces ofp points on R or on S1 have p! components, we take Conf(p,R) andConf(p, S1) to mean the component where the points x1, ..., xp appear inlinear order (i.e. x1 < x2 < · · · < xp on R or the points are encounteredin that order as the circle is traversed counterclockwise starting withx1).

Example 2.4.2. It is not hard to see that Conf(1,Rn) = Rn andConf(2,Rn) ≃ Sn−1 (where by “≃” we mean homotopy equivalence).The latter equivalence is given by the Gauss map which gives the direc-tion between the two points:

φ : Conf(2,Rn) −→ Sn−1(8)

(x1, x2) $−→x1 − x2|x1 − x2|

.

We will want to integrate over Conf(p,Rn), but this is an openmanifold and our integral hence may not converge. The “correct” com-pactification to take, in the sense that it has the same homotopy typeas Conf(p,Rn) is due to [3, 16], and is defined as follows.

Recall that that, given a submanifold N of a manifold M , the blowupof M along N , Bl(M,N), is obtained by replacing N by the unit normalbundle of N in M .

Definition 2.4.3. Define the Fulton-MacPherson compactification ofConf(k,M), denoted by Conf[k,M ], to be the closure of the image ofthe inclusion

Conf(p,M) −→ Mp ×!

S⊂1,...,p, |S|≥2

Bl(MS ,∆S),

where MS is the product of the copies of M indexed by S and ∆S isthe (thin) diagonal in MS , i.e. the set of points (x, ..., x) in MS .

Here are some properties of Conf[k,M ] (details and proofs can befound in [45]):

• Conf[k,M ] is homotopy equivalent to Conf(k,M);

• Conf[k,M ] is a manifold with corners, compact when M is com-pact;

Configuration space integrals and knots 19

• The boundary of Conf[k,M ] is characterized by points collidingwith directions and relative rates of collisions kept track of. Inother words, three points colliding at the same time gives a dif-ferent point in the boundary than two colliding, then the thirdjoining them;

• The stratification of the boundary is given by stages of collisionsof points, so if three points collide at the same time, the resultinglimiting configuration lies in a codimension one stratum. If twocome together and then the third joins them, this gives a pointin a codimension two stratum since the collision happened in twostages. In general, a k-stage collision gives a point in a codimen-sion k stratum.

• In particular, codimension one boundary of Conf[k,M ] is givenby points colliding at the same time. This will be important inSection 4, since integration along codimension one boundary isrequired for Stokes’ Theorem.

• Collisions can be efficiently encoded by different parenthesizations,e.g. the situations described two items ago are parenthesized as(x1x2x3) and ((x1x2)x3). Since parenthesizations are related totrees, the stratification of Conf[k,M ] can thus also be encoded bytrees and this leads to various connections to the theory of operads(we will not need this here);

• Conf[k,R] is the associahedron, a classical object from homotopytheory;

Additional discussion of the stratification of Conf[p,M ] can be foundin [30, Section 4.1].

Remark 2.4.4. Since we will consider configurations on long knots, andthese live in Rn, we will think of Conf[p,Rn] as the subspace of Conf[p+1, Sn] where the last point is fixed at the north pole. Consequently,we will have to consider additional strata given by points escaping toinfinity, which corresponds to points colliding with the north pole.

Remark 2.4.5. All the properties of the compactifications we men-tioned hold equally well in the case where some, but not all, diagonalsare blown up. One can think of constructing the compactification byblowing up the diagonals one at a time, and the order in which we blow

20 Ismar Volic

up does not matter. Upon each blowup, one ends up with a mani-fold with corners. This “partial blowup” will be needed in the proof ofProposition 3.3.1 (see also Remark 3.3.2).

3 Configuration space integrals and finite typeknot invariants

3.1 Motivation: The linking number

Let Mapc(R !R,R3) be the space of smooth maps which are fixed out-side some compact set as in the setup leading to Definition 2.2.1 (seeDefinition 5.1.1 for details) and define the space of long (or string) linkswith two components, L3

2, as the subspace of Mapc(R ! R,R3) given byembeddings.

Now recall the definition of the configuration space from Defini-tion 2.4.1 and the Gauss map φ from Example 2.4.2. Consider the map

Φ : R× R× L32

ev−→ Conf(2,R3)φ−→ S2

(x1, x2, L = (K1,K2)) %−→ (K1(x1),K2(x2)) %−→K2(x2)−K1(x1)

|K2(x2)−K1(x1)|

So ev is the evaluation map which picks off two points in R3, one oneach of the strands in the image of a link L ∈ L3

2, and φ records thedirection between them. The picture of Φ is given in Figure 7.

K1(x1)K2

K2(x2)

ΦK1

Figure 7: The setup for the computation of the linking number.

Also consider the projection map

π : R× R× L32 −→ L3

2

which is a trivial bundle and we can thus perform integration along thefiber on it as described in Section 2.1.

Configuration space integrals and knots 21

Putting these maps together, we have a diagram

R× R× L32

Φ !!

π""

S2

L32

which, on the complex of deRham cochains (differential forms), gives adiagram

Ω∗(R× R× L32)

π∗""

Ω∗(S2)Φ∗##

Ω∗−2(L32)

Here Φ∗ is the usual pullback and π∗ is the integration along the fiber.

We will now produce a form on L32 by starting with a particular form

on S2. So let symS2 ∈ Ω2(S2) be the unit symetric volume form on S2,i.e.

symS2 =x dydz − y dxdz + z dxdy

4π(x2 + y2 + z2)3/2.

This is the form that integrates to 1 over S2 and is rotation-invariant.

Let α = Φ∗(symS2).

Definition 3.1.1. The linking number of the link L = (K1,K2) is

Lk(K1,K2) = π∗α =

!

R×Rα.

The expression in this definition is the famous Gauss integral. Sinceboth the form symS2 and the fiber R × R are two dimensional, theresulting form is 0-dimensional, i.e. it is a function that assigns a numberto each two-component link. The first remarkable thing is that this formis actually closed, so that the linking number is an element of H0(L3

2),an invariant. The second remarkable thing is that the linking numberis actually an integer because it essentially computes the degree of theGauss map. Another way to think about this is that the linking numbercounts the number of times one strand of L goes over the other in aprojection of the link, with signs.

Remark 3.1.2. The reader should not be bothered by the fact that thedomain of integration is not compact. As will be shown in the proof ofProposition 3.3.1, the integral along faces at infinity vanishes.

22 Ismar Volic

3.2 “Self-linking” for knots

One could now try to adapt the procedure that produced the linkingnumber to a single knot in hope of producing some kind of a “self-linking” knot invariant. The picture describing this is given in Figure8.

K(x1)

K

Φ

K(x2)

Figure 8: The setup for the attempted computation of a self-linkingnumber.

Since the domain of the knot is R, we now take Conf(2,R) ratherthan R× R. Thus the corresponding diagram is

Conf(2,R)×K3 Φ !!

π""

S2

K3

The first issue is that the integration over the fiber Conf(2,R) maynot converge since this space is open. One potential fix is to use theFulton-MacPherson compactification from Definition 2.4.3 and replaceConf(2,R) by Conf[2,R]. This indeed takes care of the convergenceissue, and we now have a 0-form whose value on a knot K ∈ K3 is

(9) A(K) =

!

Conf[2,R]

"K(x1)−K(x2)

|K(x1)−K(x2)|

#∗symS2 =

!

Conf[2,R]

φ∗symS2

(The reason we are denoting this by A(K) is that it will have somethingto do with the so-caled “anomalous correction” in Section 3.4.)

Checking if this form is closed comes down to checking that (5) issatisfied (symS2 is closed so the first term of (4) goes away). In otherwords, we need to check that the restriction of the above integral to theface where two points on R collide vanishes. However, there is no reasonfor this to be true.

Configuration space integrals and knots 23

More precisely, in the stratum where points x1 and x2 in Conf[2,R]collide, which we denote by ∂x1=x2 Conf[2,R], the Gauss map becomesthe normalized derivative

(10)K ′(x1)

|K ′(x1)|.

Pulling back symS2 via this map to ∂x1=x2 Conf[2,R] × K3 and inte-grating over ∂x1=x2 Conf[2,R], which is now 1-dimensional, produces a1-form which is the boundary of π∗α:

dπ∗α(K) =

!

∂x1=x2 Conf[2,R]

"K ′(x1)

|K ′(x1)|

#∗symS2 .

Since this integral is not necessarily zero, π∗α fails to be invariant.

One resolution to this problem is to look for another term whichwill cancel the contribution of dπ∗α. As it turns out, that correction isgiven by the framing number [38].

The strategy for much of what is to come is precisely what we havejust seen: We will set up generalizations of this “self-linking” integrationand then correct them with other terms if they fail to give an invariant.

Remark 3.2.1. The integral π∗α described here is related to the famil-iar writhe. One way to say why our integral fails to be an invariant isthat the writhe is not an invariant – it fails on the Type I Reidemeistermove.

3.3 A finite type two knot invariant

It turns out that the next interesting case generalizing the “self-linking”integral from Section 3.2 is that of four points and two directions aspictured in Figure 9.

The two maps Φ now have subscripts to indicate which points arebeing paired off (the variant where the two maps are Φ12 and Φ34 doesnot yield anything interesting essentially because of the 1T relation fromDefinition 2.3.5). Diagrammatically, the setup can be encoded by thechord diagram . This diagram tells us how many points we areevaluating a knot on and which points are being paired off. This is all

24 Ismar Volic

Φ13

K

Φ24

K(x2)

K(x4)K(x1)

K(x3)

Figure 9: Toward a generalization of the self-linking number.

the information that is needed to set up the maps

Conf[4,R]×K3 Φ=Φ13×Φ24 !!

π""

S2 × S2

K3

Let sym2S2 be the product of two volume forms on S2 × S2 and let

α = Φ∗(sym2S2).

Since α and Conf[4,R], the fiber of π, are both 4-dimensional, integra-tion along the fiber of π thus yields a 0-form π∗α which we will denoteby

(IK3) ∈ Ω0(K3).

By construction, the value of this form on a knot K ∈ K3 is

(IK3) (K) =

!

π−1(K)=Conf[4,R]

α

The question now is if this form is an element of H0(K3). This amountsto checking if it is closed. Since symS2 is closed, this question reducesby (5) to checking whether the restriction of π∗α to the codimensionone boundary vanishes:

d(IK3) (K) =

!

∂ Conf[4,R]

α|∂?= 0

There is one such boundary integral for each stratum of ∂ Conf[4,R],and we want the sum of these integrals to vanish. We will considervarious types of faces:

Configuration space integrals and knots 25

• prinicipal faces, where exactly two points collide;

• hidden faces, where more than two points collide;

• the anomalous face, the hidden face where all points collide (thisface will be important later);

• faces at infinity, where one or more points escape to infinity.

The principal and hidden faces of Conf[4,R3] can be encoded bydiagrams in Figure 10, which are obtained from diagram bycontracting segments between points (this mimics collisions). The loopin the three bottom diagrams corresponds to the derivative map sincethis is exactly the setup that leads to equation (10). In other words,for each loop, the map with which we pull back the volume form is thederivative.

2=3=4

1 2 3=4

1

1=2 4 41 2=3

41=2=3 1=2=3=4

3

Figure 10: Diagrams encoding collisions of points.

Proposition 3.3.1. The restrictions of (IK3) to hidden faces andfaces at infinity vanish.

Proof. Since there are no maps keeping track of directions between var-ious pairs of points on the knot, for example between K(x1) and K(x2)or K(x2) and K(x3), the blowup along those diagonals did not need tobe performed. As a result, the stratum where K(x1) = K(x2) = K(x3)is in fact codimension two (three points moving on a one-dimensionalmanifold became one point) and does not contribute to the integral.The same is true for the stratum where K(x2) = K(x3) = K(x4). Thisis an instance of what is sometimes refered to as the disconnected stra-tum. The details of why integrals over such a stratum vanish are givenin [56, Proposition 4.1]. This argument for vanishing does not work forprincipal faces since, when two points collide, this gives a codimensionone face regardless of whether that diagonal was blown up or not.

26 Ismar Volic

For the anomalous face, we have the integral

!

∂A Conf[4,R]

"K ′(x1)

|K ′(x1)|× K ′(x1)

|K ′(x1)|

#∗sym2

S2 .

The map ΦA (the extension of Φ to the anomalous face) then factors as

∂AConf[4,R]×K3 ΦA !!

""

S2 × S2

S2

##

The pullback thus factors through S2 and, since we are pulling back a4-form to a 2-dimensional manifold, the form must be zero.

Now suppose a point, say x1, goes to infinity. The map Φ13 isconstant on this stratum so that the extension of Φ to this stratumagain factors through S2. If more than one point goes to infinity, thefactorization is through a point since both maps are constant.

Remark 3.3.2. It is somewhat strange that one of the vanishing argu-ments in the above proof required us to essentially go back and changethe space we integrate over. Fortunately, in light of Remark 2.4.5, thisis not such a big imposition. The reason this happened is that, whenconstructing the space Conf[4,R], we only paid attention to how manypoints there were on the diagram and not how they were pairedoff. The version of the construction where all the diagram informationis taken into account from the beginning is necessary for constructingintegrals for homotopy links as will be described in Section 5.1.

There is however no reason for the integrals corresponding to theprincipal faces (top three diagrams in Figure 10) to vanish. One wayaround this is to look for another space to integrate over which has thesame three faces and subtract the integrals. This difference will thenbe an invariant. To find this space, we again turn to diagrams. Thediagram that fits what we need is given in Figure 11 since, when wecontract edges to get 4 = 1, 4 = 2, and 4 = 3, the result is same threerelevant pictures as before (up to relabeling and orientation of edges).

The picture suggests that we want a space of four configurationpoints in R3, three of which lie on a knot, and we want to keep track ofthree directions between the points on the knot and the one off the knot

Configuration space integrals and knots 27

4

1 2 3

Figure 11: Diagram whose edge contractions give top three diagrams ofFigure 10.

K(x3)K

x4K(x2)

Φ24 Φ34

K(x1)Φ14

Figure 12: The situation schematically given by the diagram from Fig-ure 11.

(since the diagram has those three edges). In other words, we want thesituation from Figure 12.

To make this precise, consider the pullback space

(11) Conf[3, 1;K3,R3] !!

""

Conf[4,R3]

proj""

Conf[3,R]×K3 eval !! Conf[3,R3]

where eval is the evaluation map and proj the projection onto the firstthere points of a configuration. There is now an evident map

π′ : Conf[3, 1;K3,R3] −→ K3

whose fiber over K ∈ K3 is precisely the configuration space of fourpoints, three of which are constrained to lie on K.

Proposition 3.3.3 ([9]). The map π′ is a smooth bundle whose fiber isa finite-dimensional smooth manifold with corners.

This allows us to perform integration along the fiber of π′. So let

Φ = Φ14 × Φ24 × Φ34 : Conf[3, 1;K3,R3] −→ (S2)3

28 Ismar Volic

be the map giving the three directions as in Figure 12. The relevantmaps are thus

Conf[3, 1;K,R3] Φ !!

π′

""

(S2)3

KAs before, let α′ = Φ∗(sym3

S2). This form can be integrated along thefiber Conf[3, 1;K,R3] over K. Notice that both the form and the fiberare now 6-dimensional, so integration gives a form

(IK3) ∈ Ω0(K3).

The value of this form on K ∈ K3 is thus

(IK3) (K) =

!

(π′)−1(K)=Conf[3,1;,K,R3]

α′

We now have the analog of Proposition 3.3.1:

Proposition 3.3.4. The restrictions of (IK3) to the hidden facesand faces at infinity vanish. The same is true for the two principalfaces given by the collisions of two points on the knot.

Proof. The same arguments as in Proposition 3.3.1 work here, althoughan alternative is possible: For any of the hidden faces or the two prin-cipal faces from the statement of the proposition, at least two of themaps will be the same. For example, if K(x1) = K(x2), then Φ14 = Φ24

and Φ hence factors through a space of strictly lower dimension thanthe dimension of the form.

A little more care is needed for faces at infinity. If some, but not allpoints go to infinity (this includes x4 going to infinity in any direction),the same argument as in Proposition 3.3.1 holds. If all points go toinfinity, then it can be argued that, yet again, the map Φ factors througha space of lower dimension, but this has to be done a little more carefully;see proof of [30, Proposition 4.31] for details.

We then have

Theorem 3.3.5. The map

K −→ R

K $−→"(IK3) (K)− (IK3) (K)

#

Configuration space integrals and knots 29

is a knot invariant, i.e. an element of H0(K3). Further, it is a finitetype two invariant.

This theorem was proved in [19] and in [4]. Since the set up therewas for closed knots, the diagrams were based on circles, not segments,and one hence had to include some factors in the above formula havingto do with the automorphisms of those diagrams. We will encounterautomorphism factors like these in Theorem 3.4.1.

Recall the discussion of finite type invariants from Section 2.3. Theinvariant from Theorem 3.3.5 turns out to be the unique finite type twoinvariant which takes the value of zero on the unknot and one on thetrefoil. It is also equal to the coefficient of the quadratic term of theConway polynomial, and it is equal to the Arf invariant when reducedmod 2. In addition, it appears in the surgery formula for the Cassoninvariant of homology spheres and is thus also known as the Casson knotinvariant. A treatment of this invariant from the Casson point of viewcan be found in [42].

Proof of Theorem 3.3.5. In light of Propositions 3.3.1 and 3.3.4, theonly thing to show is that the integrals along the principal faces match.This is clear essentially from the pictures (since the diagram picturesrepresenting those collisions are the same), except that the collisions ofK(xi), i = 1, 2, 3, with x4 produce an extra map on each face (extensionof Φi4 to that face) that is not present in the first integral. Since K(x4)can approach K(xi) from any direction, the extension of Φi4 sweeps outa sphere, and this is independent of the other two maps. By Fubini’sTheorem, we then have for, say, the case i = 1,

!

∂K(x1)=x4Conf[3,1;K,R3]

(Φ14 × Φ24 × Φ34)∗sym3

S2

=

!

S2

(Φ14)∗symS2 ·

!

∂K(x1)=x4Conf[3,1;K,R3]

(Φ24 × Φ34)∗sym2

S2

=

!

Conf[3,R]

(Φ12 × Φ13)∗sym2

S2

The last line is obtained by observing that the first integral in the pre-vious line is 1 (since symS2 is a unit volume form) and by rewritingthe domain ∂K(x1)=x4

Conf[3, 1;K,R3] as Conf[3,R] (and relabeling the

30 Ismar Volic

points). The result is precisely the integral of one of the principal faces of(IK3) . The remaining two principal faces can similarly be matchedup. Some care should be taken with signs, and we leave it to the readerto check those, keeping in mind that relabeling the vertices of a diagrammay introduce a sign in the integral (this corresponds to permuting thecopies of R and R3 and, since the dimensions of these spaces are odd,this in turn preserves or reverses the orientation of the fiber dependingon the sign of the permutation). Changing orientations of chords ofedges also might affect the sign (this corresponds to composing withthe antipodal map to S2 which changes the sign of the pullback form).

To show that this is a finite type two invariant is not difficult. Thekey is that the resolutions of the three singularities can be chosen assmall as desired. Then the integration domain can be broken up intosubsets on which the difference of the integrals between the two resolu-tions is zero. Details can be found in [56, Section 5].

3.4 Finite type k knot invariants

Recall the space of trivalent diagrams from Definition 2.3.8. The twodiagrams appearing in Theorem 3.3.5 are the two (up to decorations) el-ements of T D2. However, the recipe for integration that these diagramsgave us in the previous section generalizes to any diagram. Namely, anydiagram Γ ∈ T Dk with p segement vertices and q free vertices gives aprescription for constructing a pullback as in (11):

(12) Conf[p, q;K3,R3] !!

""

Conf[p+ q,R3]

proj""

Conf[p,R]×K3 eval !! Conf[p,R3]

There is again a bundle

π : Conf[p, q;K3,R3] −→ K3

whose fibers are manifolds with corners. We also have a map

Φ : Conf[p, q;K3,R3] −→ (S2)e

where

Configuration space integrals and knots 31

• Φ is the product of the Gauss maps between pairs of configurationpoints corresponding to the edges of Γ, and

• e is the number of chords and edges of Γ.

Let α = Φ∗(symeS2). It is not hard to see that, because of the

trivalence condition on T Dk, the dimension of the fiber of π is 2e, as isthe dimension of α. Thus we get a 0-form π∗α, or, in the notation ofSection 3.3, a form

(IK3)Γ ∈ Ω0(K3),

whose value on K ∈ K3 is

(IK3)Γ(K) =

!

π−1(K)=Conf[p,q;K,R3]

α.

Now recall from discussion preceeding (7) that T Wk is the spaceof weight systems of degree k, i.e. functionals on T Dk. Also recall the“self-linking” integral from equation (9). Finally let T Bk be a basis of di-agrams for T Dk (this is finite and canonical up to sign) and let |Aut(Γ)|be the number of automorphisms of Γ (these are automorphisms thatfix the segment, regarded up to labels and edge orientations).

Theorem 3.4.1 ([51]). For each W ∈ T Wk, the map

K3 −→ R(13)

K $−→"

Γ∈T Bk

#W (Γ)

|Aut(Γ)|(IK3)Γ − µΓA(K)

$,

where µΓ is a real number that only depends on Γ, is a finite type kinvariant. Furthermore, the assignment W $→ V ∈ Vk gives an isomor-phism

I0K3 : T Wk −→ Vk/Vk−1

(where the map (13) is followed by the quotient map Vk → Vk/Vk−1).

Proof. The argument here is essentially the same as in Theorem 3.3.5but is complicated by the increased number of cases. Once again, tostart, one should observe that the relations in Definition 2.3.8 are com-patible with the sign changes that occur in the integral if copies of R orR3 in the bundle Conf[p, q;K3,R3] are switched (the orientation of thisspace would potentially change and so would the sign of the integral)or if a Gauss map is replaced by its antipode.

32 Ismar Volic

To prove that the integrals along hidden faces vanish, one considersvarious types of faces as in Propositions 3.3.1 and 3.3.4. If the pointsthat are colliding form a “disconnected stratum” in the sense that theset of vertices and edges of the corresponding part of Γ forms two subsetssuch that no chord of edge connects a vertex of one subset to a vertexof the other, we revise the definition of Conf[p, q;K3,R3] and turn thisstratum into one of codimension greater than one. This takes care ofmost hidden faces [56, Section 4.2]. The remaining ones are disposedof by symmetry arguments due to Kontsevich [26] (see also [56, Section4.3]) which show that some of the integrals are equal to their negativesand thus vanish. Another reference for the vanishing along hidden facesis [12, Theorem A.6] (the authors of that paper consider closed knotsbut this does not change the arguments).

The vanishing of the integrals along faces at infinity goes exactly thesame way as in Proposition 3.3.1; the map Conf[p, q;K3,R3] → (S2)e

always factors through a space of lower dimension. More details can befound in [56, Section 4.5].

Lastly, the vanishing of principal faces occurs due to the STU andIHX relations. The STU case is provided in Figure 13 (we have omittedthe labels on diagrams and signs to simplify the picture).

=!W ( )±W ( )±W ( )

"(IK3) (K) = 0

d!W ( )(IK3) (K)±W ( )(IK3) (K)±W ( )(IK3) (K)

"

= W ( )d(IK3) (K)±W ( )d(IK3) (K)±W ( )d(IK3) (K)

= W ( )(IK3) (K)±W ( )(IK3) (K)±W ( )(IK3) (K)

Figure 13: Cancellation due to the STU relation

Similar cancellation occurs with principal faces resulting from colli-sion of free vertices, where one now uses the IHX relation. The contri-butions from all principal faces thus cancel.

The one boundary integral that is not know to vanish (but is con-jectured to; it is known that it does in a few low degree cases) is thatof the anomalous face corresponding to all points colliding. It turns outthat this integral is some multiple µΓ of the self-linking integral A(K),

Configuration space integrals and knots 33

and hence the term µΓA(K) is subtracted so that we get an invariant.For further details, see [56, Section 4.6].

To show that this invariant is finite type k and that we get an iso-morphism T Wk → Vk/Vk−1 is tedious but straightforward. The pointis that, as in the proof of Theorem 3.3.5, the resolutions of the k + 1singularities can be chosen to differ in arbitratily small neighborhoodsand so the integrals from the sum of (13) cancel out. For this to happen,the domain of integration is subdivided and the integrals end up pairingoff and canceling on appropriate neighborhoods. For details on how thisleads to the conclusion that the invariant is finite type, see [56, Lemma5.4]. Finally, it is easy to show that the map I0K3 , when composed withthe isomorphism (7), is an inverse to the map f from equation (6), andthis gives the desired isomorphism [56, Theorem 5.3].

Remark 3.4.2. In combination with (7), Theorem 3.4.1 thus gives analternative proof of Kontsevich’s Theorem 2.3.6.

Remark 3.4.3. The reason we put a superscript “0” on the map I0K3

is that this is really just the degree zero manifestation of a chain mapdescribed in Section 4.

Remark 3.4.4. In Theorem 3.3.5, there is one weight system for thedegree two case and it takes one the values 1 and −1 for the two relevantdiagrams. In addition, the anomalous correction in that case vanishes,so Theorem 3.4.1 is indeed a generalization of Theorem 3.3.5.

Theorem 3.4.1 thus gives a way to construct all finite type invariants.In addition, configuration space integrals provide an important link be-tween Chern-Simons Theory (where the first instances of such integralsoccur), topology, and combinatorics. Unfortunately, computations withthese integrals are difficult and only a handful have been performed.

4 Generalization to Kn, n > 3

Just as there was no reason to stop at diagrams with four vertices,there is no reason to stop at trivalent diagrams. One might as well takediagrams that are at least trivalent, such as the one from Figure 14 (lessthan trivalent turns out not to give anything useful).

To make this precise, we generalize Definition 2.3.7 as follows:

34 Ismar Volic

Figure 14: A more general diagram (without decorations).

Definition 4.1. For n ≥ 3, define a diagram to be a connected graphconsisting of an oriented line segment (considered up to orientation-preserving diffeomorphism) and some number of vertices of two types:segment vertices, lying on the segment, and free vertices, lying off thesegment. The graph also contains some number of

• chords connecting distinct segment vertices;

• loops connecting segment vertices to themselves; and

• edges connecting two free vertices or a free vertex and a segmentvertex.

Each vertex is at least trivalent, with the segment adding two to thecount of the valence of a segment vertex. In addition,

• if n is odd, all vertices are labeled, and edges, loops, and chordsare oriented;

• if n is even, external vertices, edges, loops, and chords are labeled.

We also identify arcs, which are parts of the segment between suc-cessive segment vertices.

Definition 4.2. For each n ≥ 3, let Dn be the real vector space gener-ated by diagrams from Definition 4.1 modulo the relations

1. If Γ contains more than one edge connecting two vertices, thenΓ = 0;

2. If n is odd and if a diagram Γ differs from Γ′ by a relabeling ofvertices or orientations of loops, chord, and edges, then

Γ− (−1)σΓ′ = 0

where σ is the sum of the sign of the permutation of vertex la-bels and the number or loops, chords, and edges with differentorientation.

Configuration space integrals and knots 35

3. If n is even and if a diagram Γ differs from Γ′ by a relabeling ofsegment vertices or loops, chord, and edges, then

Γ = (−1)σΓ′,

where σ is the sum of the signs of these permutations.

Note that T Dk is the quotient of the subspace of Dn generated bytrivalent diagrams with 2k vertices.

Definition 4.3. Define the degree of Γ ∈ Dn to be

deg(Γ) = 2(# edges)− 3(# free vertices)− (# segment vertices).

Thus if Γ is a trivalent diagram, deg(Γ) = 0.

Coboundary δ is given on each diagram by contracting edges andsegments (but not chords or loops since this does not represent a collisionof points). Namely, let e be an edge or an arc of Γ and let Γ/e be Γwith e contracted. Then

δ(Γ) =!

edges and arcs e of Γ

ϵ(Γ)Γ/e,

where ϵ(Γ) is a sign determined by

• if n is odd and e connects vertex i to vertex j, ϵ(Γ) = (−1)j ifj > i and ϵ(Γ) = (−1)i+1 if j < i;

• if n is even and e is an arc connecting vertex i to vertex j, thenϵ(Γ) is computed as above, and if e is an edge, then ϵ(Γ) = (−1)s,where s = (label of e) + (# segment vertices) + 1.

On Dn, δ is the linear extension of this. An example (without dec-orations and hence modulo signs) is given in Figure 15.

Theorem 4.4 ([12], Theorem 4.2). For n ≥ 3, (Dn, δ) is a cochaincomplex.

Proof. It is a straightforward combinatorial exercise to see that δ raisesdegree by 1 and that δ2 = 0.

Returning to configuration space integrals, for each Γ ∈ Dn andK ∈ Kn, we can still define an integral as before. Vertices of Γ tell

36 Ismar Volic

δ!

(last two are zero)

"= ±

± ±

±

Figure 15: An example of a coboundary.

us what pullback bundle Conf[p, q;Kn,Rn] to construct, i.e. how manypoints to have on and off the knot. The only difference is that the map

Φ : Conf[p, q;Kn,Rn] −→ (Sn−1)(# loops, chords, and edges of Γ)

is now a product of Gauss maps for each edge and chord of Γ as well asthe derivative map for each loop of Γ. Further, none of what was donein Section 3.4 requires n = 3. More precisely, we still get a form, forn ≥ 3,

(IKn)Γ ∈ Ωd(Kn),

given by integration along the fiber of the bundle

π : Conf[p, q;Kn,Rn] −→ Kn.

The degree of the form is no longer necessarily zero but of degree equalto

(n− 1)(# loops, chords, and edges of Γ)

−n(# free vertices of Γ)

−(# segment vertices of Γ)

and its value on K ∈ Kn is as before

(IKn)Γ(K) =

!

π−1(K)=Conf[p,q;K,Rn]

α.

Theorem 4.5 ([12], Theorem 4.4). For n > 3, configuration spaceintegrals give a cochain map

IKn : (Dn, δ) −→ (Ω∗(Kn), d).

Configuration space integrals and knots 37

(Here d is the ordinary deRham differential.)

Proof. To prove this, one first observes that, just as in Theorem 3.4.1,the relations in Dn correspond precisely to what happens on the in-tegration side if points or maps are permuted or if Gauss maps arecomposed with the antipodal map. In fact, those relations in Dn aredefined precisely because of what happens on the integration side.

Once this is established, it is necessary to show that, for each Γ, theintegrals along the hidden faces and faces at infinity vanish, and thisgoes exactly the same way as in Theorem 3.4.1. The integrals alongprincipal faces correspond precisely to contractions of edges and arcs,so that the map commutes with the differential.

Remark 4.6. There is an algebra structure on Dn given by the shuffleproduct that is compatible with the wedge product of forms [13]. Thismakes IKn a map of algebras as well.

By evaluating IKn on certain diagrams, Cattaneo, Cotta-Ramusino,and Longoni [12] also prove

Corollary 4.7. Given any i > 0, the knot space Kn, n > 3, has non-trivial cohomology in dimension greater than i.

Complex Dn is known to have the same homology as the Kn, so itcontains a lot of information about the topology of long knots. However,we do not know that this map induces an isomorphism. More precisely,we have

Conjecture 4.8. The map IKn is a quasi-isomorphism.

Even though we do not have Theorem 4.5 for n = 3, the constructionis compatible with what we already did in the case of classical knots K3.Namely, for n = 3, one does not get a cochain map in all degrees becauseof the anomalous face. But in degree zero, it turns out that

H0(D3) = T D

(up to certain diagram automorphism factors; see [30, Section 3.4]). Inother words, the kernel of δ in degree zero is defined by imposing the1T, STU, and IHX relations.

Thus, after correcting by the anomalous correction and after identi-fying T D with its dual, the space of weight systems T W (the dualizationgymnastics is described in [30, Section 3.4]), we get a map

(H0(D3))∗ = T W −→ H0(K).

38 Ismar Volic

But this is precisely the map I0K3 from Theorem 3.4.1 and we alreadyknow that the image of this map is the finite type knot invariants.

5 Further generalizations and applications

In this section we give brief overviews of other contexts in which con-figuration space integrals have appeared in recent years.

5.1 Spaces of links

Configuration space integrals can also be defined for spaces of long links,homotopy links, and braids. (The reader should keep in mind that it isactually quite surprising that they can be defined for homotopy links.)The results stated here encompass those for knots (by setting m = 1 inLnm).

For m ≥ 1, n ≥ 2, let Mapc("mR,Rn) be the space of smooth mapsof "mR to Rn which, outside of "mI agree with the map "mR → Rn,which is on the ith copy of R given as

t $−→ (t, i, 0, 0, ..., 0).

As in the case of knots (Definition 2.2.1), we can define the spacesof links as subspaces of Mapc("mR,Rn) with the induced topology asfollows.

Definition 5.1.1. Define the space of

• long (or string) links with m strands Lnm ⊂ Mapc("mR,Rn) as the

space of embeddings L : "m R → Rn.

• pure braids on m strands Bnm ⊂ Mapc("mR,Rn) as the space of

embeddings B : "m R → Rn whose derivative in the direction ofthe first coordinate is positive.

• long (or string) homotopy links with m strands Hnm ⊂ Mapc("mR,

Rn) as the space of link maps H : "m R → Rn (smooth maps of"mR in Rn with the images of the copies of R disjoint).

Remark 5.1.2. Another (and in fact, more standard) way to thinkabout Bn

m is as the loop space ΩConf(m,Rn−1).

Configuration space integrals and knots 39

Remark 5.1.3. For technical reasons, it is sometimes better to takestrands that are not parallel outside of In, but this does not changeanything about the theorems described here. For details, see [30, Defi-nition 2.1].

Some observations about these spaces are:

• Bnm ⊂ Ln

m ⊂ Hnm;

• In π0(Hnm), we can pass a strand through itself so this can be

thought of as space of “links without knotting”;

Example of a homotopy link and a braid is given in Figure 16. Notethat, as usual, we have confused the maps H and L with their imagesin Rn.

self-intersection

H ∈ Hn3

L ∈ Bn3 ⊂ Ln

3 ⊂ Hn3

Figure 16: Examples of links. The top picture is a homotopy link, butnot a link (and hence not a braid) because of the self-intersection in thebottom strand.

When we say “link”, we will mean an embedded link. Otherwisewe will say “homotopy link” or “braid”. As with knots, the adjective“long” will be dropped. We will denote components (i.e. strands) of anembedded link by L = (K1,K2, ...,Km).

As before, an isotopy is a homotopy in the space of links or braids,and link homotopy is a path in the space of homotopy links.

As in the case of the space of knots, all of these link spaces aresmooth infinite-dimensional paracompact manifolds so we can considertheir deRham cohomology.

40 Ismar Volic

Finite type invariants of these link spaces can be defined the sameway as for knots (see Section 2.3). Namely, we consider self-intersectionswhich, in the case of links, come from a single strand or two differentstrands (i.e. in the left picture of Figure 2, there are no conditions on thetwo strands making up the singularity). For homotopy links, we onlytake intersections that come from different strands. For braids, this con-dition is automatic since a braid cannot “turn back” to intersect itself.Then a finite type k invariant is defined as an invariant that vanisheson (k + 1) self-intersections. We will denote finite type k invariants oflinks, homotopy links, and braids by LVk, HVk, and BVk, respectively.

As for knots, the question of separation of links by finite type in-variants is still open, but it is known that these invariants separatehomotopy links [20] and braids [6, 24].

We now revisit Section 4 and show how Theorem 4.5 generalizes tolinks. Namely, recall the cochain map

IKn : Dn −→ Ω∗(Kn).

The first order of business is to generalize the diagram complex Dn toa complex LDn

m (which we will use for both links and braids) and asubcomplex HDn

m (which we will use for homotopy links). This gener-alization is simple: LDn

m is defined the same way as Dn except thereare now m segments, as for example in Figure 17.

∈ LDn4

Figure 17: An example of a diagram for links (without decorations).

All the definitions from Section 4 carry over in exactly the sameway and we will not reproduce the details here, especially since they arespelled out in [30, Section 3]. In particular, depending on the parity ofn, the diagrams have to be appropriately decorated. The differential isagain given by contracting arcs and edges.

HDnm is defined by taking diagrams

Configuration space integrals and knots 41

• with no loops, and

• requiring that, if there exists a path between distinct vertices ona given segment, then it must pass through a vertex on anothersegment.

It is a simple combinatorial exercise to show that HDnm is a subcomplex

of LDnm [30, Proposition 3.24].

As expected, in degree zero, complexes Lnm and Hn

m are still definedby imposing the STU and IHX relations, and an extra relation in thecase of Hn

m that diagrams cannot contain closed paths of edges. Inparticular, the spaces of weight systems of degree k for these link spaces,which we will denote by LWk andHWk, consist of functionals vanishingon these relations (with automorphism factors); see [30, Definition 3.35]for details.

As it turns out, the integration is not as easily generalized. Theproblem is that, if we want to produce cohomology classes on Hn

m, thenthe evaluation map from (12) will need to take values in Hn

m, but pointsin this space are not even immersions, let alone embeddings. Hence thetarget of the evaluation map would not be a configuration space butrather some kind of a “partial configuration space” where some pointsare allowed to pass through each other (this is actually a complementof a subspace arrangement, a familiar object from algebraic geometry).But then the projection map would be a map of partial configurationspaces which is far from being a fibration (see [30, Example 4.7]). Thismakes it unlikely that the pullback is a bundle over Hn

m.

A way around this is to patch the integral together from pieces forwhich this difficulty does not occur. This is achieved by breaking up adiagram Γ ∈ LDn

m into its graft components which are essentially thecomponents one would see after the segments and segment vertices areremoved. The second condition defining the subcomplex HDn

m guaran-tees that there will be no more than one segment vertex on each segmentof each graft, and this turns out to remove the issue of the projectionnot being a fibration. Since it would take us too far afield to define thegraft-based pullback bundle precisely, we will refer the reader to [30,Definition 4.16] for details. Suffice it to say here that the constructionessentially takes into account both the vertices and edges of Γ ratherthan just vertices when constructing the pullback bundle (see Remark3.3.2). This procedure is indeed a refinement of the original definition

42 Ismar Volic

of configuration space integrals since it produces the same form on Lnm

as the original definition [30, Proposition 4.24]. The only difference,therefore, is that the graft definition makes it possible to restrict the in-tegration from the complex LDn

m to the subcomplex HDnm and produce

forms on Hnm.

We then have a generalization of Theorem 4.5.

Theorem 5.1.4 ([30], Section 4.5). There are integration maps IL andIH given by configuration space interals and a commutative diagram

HDnm

IH !!! "

""

Ω∗(Hnm)

""LDn

mIL !! Ω∗(Ln

m)

Here IL is a cochain map for n > 3 and IH is a cochain map for n ≥ 3.

Proof. The proof goes exactly the same way as in the case of knots.The only difference is that IH is also a cochain map for n = 3. Thereason for this is that the anomalous face is not an issue for homotopylinks. Namely, the anomaly can only arise when all points on and offthe link collide. But since strands of the link are always disjoint, thisis only possible when all the configuration points on the link are infact on a single strand. In other words, the corresponding diagram Γis concentrated on a single strand. Such a diagram does not occur inHDn

m. (The integral in this case in effect produces a form on the spaceof knots, so that the anomaly can be thought of as a purely knotting,rather than linking, phenomenon.)

Remark 5.1.5. As in the case of knots, there is an algebra structureon LDn

m given by the shuffle product that is compatible with the wedgeproduct of forms [30, Section 3.3.2]. It thus turns out that the maps ILand IH are maps of algebras [30, Proposition 4.29].

For n = 3, we also have a generalization of Theorem 3.4.1.

Theorem 5.1.6 ([30], Theorems 5.6 and 5.8). Configuration space in-tegral maps IL and IH induce isomorphisms

I0L : LWk∼=−→ LVk/LVk−1 ⊂ H0(L3

m)

I0H : HWk∼=−→ LVk/LVk−1 ⊂ H0(H3

m)

Configuration space integrals and knots 43

The isomorphisms are given exactly as in Theorem 3.4.1. In par-ticular, the anomalous correction has to be introduced for the case oflinks.

Lastly, we mention an interesting connection to a class of classicalhomotopy link invariants called Milnor invariants [36]. In brief, theseinvariants live in the lower central series of the link group and essentiallymeasure how far a “longitude” of the link survives in the lower centralseries. It is known that Milnor invariants of long homotopy links arefinite type invariants (and it is important that these are long, ratherthan closed homotopy links) [6, 32]. Theorem 5.1.6 thus immediatelygives us

Corollary 5.1.7. The map IH provides configuration space integral ex-pressions for Milnor invariants of H3

m.

For more details about this corollary, see [30, Section 5.4].

Even though we made no explicit mention of braids in the abovetheorems, everything goes through the same way for this space as well.The complex is still LDn

m but the evaluation now take place on braids,i.e. elements of Bn

m. The integration IB would thus produce forms on Bnm

and all finite type invariants of B3m. However, this is not very satisfying

since we do not yet have a good subcomplex of LDnm or a modification

in the integration procedure that would take into account the definitionof braids. For example, since Bn

m ≃ ΩConf(m,Rn−1), braids can bethought of as “flowing” at the same rate, and the integration hencemight be defined so that it only takes place in “vertical slices” of thebraid. In particular, one should be able to connect configuration spaceintegrals for braids to Kohno’s work on braids and Chen integrals [24].

5.2 Manifold calculus of functors and finite type invari-ants

Configuration space integrals connect in unexpected ways to the theoryof manifold calculus of functors [59, 18]. Before we can state the results,we provide some basic background, but we will assume the reader isfamiliar with the language of categories and functors. For further detailson manifold calculus of functors, the reader might find [40] useful.

For M a smooth manifold, Let Top be the category of topologicalspaces and let

O(M) = category of open subsets of M with inclusions as morphisms.

44 Ismar Volic

Manifold calculus studies functors

F : O(M)op −→ Top

satisfying the conditions:

1. F takes isotopy equivalences to homotopy equivalences, and;

2. For any sequence of open sets U0 ⊂ U1 ⊂ · · · , the canonical mapF (∪iUi)→ holimi F (Ui) is a homotopy equivalence (here “holim”stands for the homotopy limit).

The target category is not limited to topological spaces, but for con-creteness and for our purposes we will stick to that case.

One such functor is the space of embeddings Emb(−, N), where Nis a smooth manifold, since, given an inclusion

O1 !→ O2

of open subsets of M , there is a restriction

Emb(O2, N) −→ Emb(O1, N).

In particular, we can specialize to the space of knots Kn, n ≥ 3, and seewhat manifold calculus has to say about it.

For any functor F : O(M)op → Top, the theory produces a “Taylortower” of approximating functors/fibrations

F (−) −→!T0F (−)← · · ·← TkF (−)← · · ·← T∞F (−)

"

where T∞F (−) is the inverse limit of the tower.

Theorem 5.2.1 ([60]). For F = Emb(−, N) and for 2 dim(M) + 2 ≤dim(N), the Taylor tower converges on (co)homology (for any coeffi-cients), i.e.

H∗(Emb(−, N)) ∼= H∗(T∞ Emb(−, N)).

In particular, evaluating at M gives

H∗(Emb(M,N)) ∼= H∗(T∞ Emb(M,N)).

Remark 5.2.2. For dim(M) + 3 ≤ dim(N), the same convergenceresult is true on homotopy groups [17].

Configuration space integrals and knots 45

Note that when M is 1-dimensional, N has to be at least 4-dimen-sional to guarantee convergence. Hence this says nothing about K3.Nevertheless, the tower can still be constructed in this case and it turnsout to contain a lot of information.

To construct TkKn, n ≥ 3, let I1, ..., Ik+1 be disjoint closed subin-tervals of R and let

∅ = S ⊆ 1, ..., k + 1.

Then letKn

S = Embc(R \!

i∈SIi, Rn),

where Embc as usual stands for the space of “compactly supported”embeddings, namely those that are fixed outside some compact set suchas I.

Thus KnS is a space of “punctured knots”; an example is given in

Figure 18.

Figure 18: An element of Kn1,2,3,4.

These spaces are not very interesting on their own, and are connectedeven for n = 3. However, we have restriction maps Kn

S → KnS∪i given

by “punching another hole”, namely restricting an embedding of R withsome intervals taken out to an embedding of R with one more intervaltaken out. These spaces and maps then form a diagram of knots withholes (such a diagram is sometimes called a punctured cube).

Example 5.2.3. When k = 2, we get

Kn1

!!

""Kn

2##

!!

Kn1,2

!!

Kn3

##

$$

Kn1,3

""Kn

2,3## Kn

1,2,3

46 Ismar Volic

Definition 5.2.4. The kth stage of the Taylor tower for Kn, n ≥ 3, isthe homotopy limit of this punctured cube, i.e.

TkKn = holim∅=S⊆1,..,k+1

KnS .

For the reader not familiar with homotopy limits, it is actually nothard to see what this homotopy limit is: For example, the puncturedcubical diagram from Example 5.2.3 can be redrawn as

Kn1

!! ""Kn

1,3## Kn

1,2,3 Kn1,2

$$

Kn3

%%

## Kn2,3

&&

Kn2

$$

''

Then a point in T2Kn is

• A point in each Kni (once-punctured knot);

• A path in each Kni,j (isotopy of a twice-punctured knot) ;

• A two-parameter path in Kn1,2,3 (two-parameter isotopy of a

thrice-punctured knot); and

• Everything is compatible with the restriction maps. Namely, apath in each Kn

i,j joins the restrictions of the elements of Kni

and Knj to Kn

i,j, and the two-parameter path in Kn1,2,3 is re-

ally a map of a 2-simplex into Kn1,2,3 which, on its edges, is the

restriction of the paths in Kni,j to Kn

1,2,3.

The pattern for TkKn, k = 2, should be clear.

There is a mapKn −→ TkKn

given by punching holes in the knot (the isotopies in the homotopy limitare thus constant).

Remark 5.2.5. It is not hard to see that, for k ≥ 3, Kn is the actualpullback (limit) of the subcubical diagram. So the strategy here is toreplace the limit, which is what we really care about, by the homotopylimit, which is hopefully easier to understand.

Configuration space integrals and knots 47

There is also a map, for all k ≥ 1,

TkKn −→ Tk−1Kn,

since the diagram defining Tk−1Kn is a subdiagram of the one definingTkKn (this map is a fibration; this is a general fact about homotopylimits).

Putting these maps and spaces together, we get the Taylor tower forKn, n ≥ 3:

Kn −→!T0Kn ← · · ·← TkKn ← · · ·← T∞Kn

".

By Theorem 5.2.1, this tower converges on (co)homology for n ≥ 4.

There is a variant of this Taylor tower, the so-called “algebraic Tay-lor tower”, which is a tower of cochain complexs obtained by applyingcochains to each space of punctured knots and then letting T ∗

k (K3) bethe homotopy colimit of the resulting diagram of cochain complexes.

Recall the map I0K3 from Theorem 3.4.1. We then have the followingtheorem, which essentially says that the algebraic Taylor tower classifiesfinite type invariants.

Theorem 5.2.6 ([55], Theorem 1.2). The map I0K3 factors through thealgebraic Taylor tower for K3. Furthermore, we have isomorphisms

T Wk

I0K3

∼=!!

∼=

""

Vk/Vk−1

H0(T ∗2kK3)

∼=##

(and H0(T ∗2kK3) ∼= H0(T ∗

2k+1K3) so all stages are accounted for).

The main ingredient in this proof is the extension of configurationspace integrals to the stages T2kK3 of the space of long knots [54, The-orem 4.5]. The idea of this extension is this: As a configuration pointmoves around a punctured knot (this corresponds to a point moving onthe knot in the usual construction) and approaches a hole, it is madeto “jump”, via a path in the homotopy limit (this is achieved by an ap-propriate partition of unity), to another punctured knot which has thathole filled in, thus preventing the evaluation map from being undefined.

48 Ismar Volic

Theorem 5.2.6 places finite type theory into a more homotopy-theo-retic setting and the hope is that this might give a new strategy forproving the separation conjecture.

Several generalizations of Theorem 5.2.6 should be possible. For ex-ample, it should also be possible to extend the entire chain map fromTheorem 4.5 to the Taylor tower. It should also be possible to show thatthe Taylor multitowers for Ln

m, Hnm, and Bn

m supplied by the multivari-able manifold calculus of functors [41] also admit factorizations of theintegration maps IL, IH, and IB, as well as classify finite type invariantsof these spaces.

Lastly, it seems likely that finite type invariants are all one finds inthe ordinary Taylor tower (and not just its algebraic version). Someprogress toward this goal can be found in [11].

5.3 Formality of the little balls operad

There is another striking connection between the Taylor tower for knotsand configuration space integrals of a slightly different flavor. To ex-plain, we will first modify the space of knots slightly and instead of Kn,n > 3, use the space

Kn = hofiber(Embc(R,Rn) !→ Immc(R,Rn)),

where hofiber stands for the homotopy fiber. This is the space of “em-beddings modulo immersions” and a point in it is a long knot alongwith a path, i.e. a regular homotopy, to the long unknot (since this is anatural basepoint in the space of immersions) through compactly sup-ported immersions. This space is easier to work with, but it is not verydifferent from Kn: the above inclusion is nullhomotopic [46, Proposition5.17] so that we have a homotopy equivalence

Kn ≃ Kn × Ω2Sn−1.

The difference between long knots and its version modulo immersionsis thus well-undersrtood, especially rationally.

Now let Bn = Bn(p)p≥0 be the little n-discs operad (i.e. little ballsin Rn), where Bn(p) is the space of configurations of p closed n-discswith disjoint interiors contained in the unit disk of Rn. The little n-discs operad is an important object in homotopy theory, and a goodintroduction for the reader who is not familiar with it, or with operads

Configuration space integrals and knots 49

in general, is [39]. Taking the chains and the homology of Bn gives twooperads of chain complexes, C∗(Bn;R) and H∗(Bn;R) (where the latteris a chain complex with zero differential). Then we have the followingformality theorem.

Theorem 5.3.1 (Kontsevich [28]; Tamarkin for n = 2 [50]). For n ≥ 2,there exists a chain of weak equivalences of operads of chain complexes

C∗(Bn;R)≃←− !Dn ≃−→ H∗(Bn;R),

where !Dn is a certain diagram complex. In other words, Bn is (stably)formal over R.

For details about the proof of Theorem 5.3.1, the reader should con-sult [35]. The reason this theorem is relevant here is that !Dn is a diagramcomplex (it is in fact a commutative differential graded algebra coop-erad) that is essentially the complex Dn we encountered in Section 4.

The main difference is that loops are not allowed in !Dn. In addition,the map

!Dn −→ C∗(Bn;R)is given by configuration space integrals and is the same as the mapIKn we saw before, with one important difference that the bundle weintegrate over is different. To explain briefly, Bn can be regarded as acollection of configuration spaces Conf[p,Rn]. For a diagram Γ ∈ !Dn

with p segment vertices and q free vertices, consider the projection

π : Conf[p+ q,Rn] −→ Conf[p,Rn].

This is a bundle in a suitable sense; see [35, Section 5.9]. The edges of Γagain determine some Gauss maps Conf[p+ q,Rn]→ Sn−1, so that theproduct of volume forms can be pulled back to Conf[p+ q,Rn] and thenpushed forward to Conf[p,Rn]. This part is completely analogous towhat we have seen in Section 4. The bulk of the proof of Theorem 5.3.1then consist of showing that the map !Dn → Ω∗(Conf[p,Rn]) is an equiv-

alence (we are liberally passing between cooperad !Dn and its dual, aswell as chains and cochains on configuration spaces). This again comesdown to vanishing arguments, which are the same as in Theorem 4.5.

The map!Dn −→ H∗(Bn;R)

is easy, with some obvious diagrams sent to the generators of the (co)ho-mology of configuration spaces and the rest to zero.

50 Ismar Volic

Theorem 5.3.1 has been used in a variety of situations, such asMcClure-Smith’s proof of the Deligne Conjecture and Tamarkin’s proofof Kontsevich’s deformation quantization theorem. For us, the impor-tance is in that it gives information about the rational homology of Kn,n > 3.

To explain, first observe that the construction of the Taylor towerfor Kn from Section 5.2 can be carried out in exactly the same way forKn. Then, by retracting the arcs of a punctured knot, we get

KnS ≃ Conf(|S|− 1,Rn).

If we had used Kn, we would also have copies of spheres keeping trackof tangential data. In the Kn version, they are not present since thespace of immersions, which carries this data, has been removed.

The restriction maps “add a point”, as in Figure 19.

↓

x2x1

x3

x1

x3

x2x′2

Figure 19: Restriction of punctured knots.

This structure yields a homology spectral sequence that can be as-sociated to the Taylor tower for Kn, n ≥ 3. It starts with

E1−p,q = Hq(Conf(p,Rn)),

and, for n ≥ 4, converges to H∗(T∞Kn). By Theorem 5.2.1, this spectralsequence hence converges to H∗(Kn).

Remark 5.3.2. This is the Bousfield-Kan spectral sequence that can beassociated to any cosimplicial space, and in particular to the cosimplicialspace defined by Sinha [47] that models the Taylor tower for Kn. Thespaces in this cosimplicial model are slight modifications of the Fulton-MacPherson compactification of Conf(p,Rn) and the maps “double”and “forget” points. In particular, the doubling maps are motivated by

Configuration space integrals and knots 51

the situation from Figure 19. In addition, this turns out to be the samespectral sequence (up to regrading) as the one defined by Vassiliev [53]which motivated the original definition of finite type knot invariants.

Theorem 5.3.3. The homology spectral sequence described above col-lapses rationally at the E2 page for n ≥ 3.

This theorem was proved for n ≥ 4 in [34], for n = 3 on the diagonalin [27], and for n = 3 everywhere in [37] and [48]. Theorem 5.2.6 can alsobe interpreted as the collapse of this spectral sequence on the diagonalfor n = 3.

Idea of proof. The vertical differential in the spectral sequence is theordinary internal differential on the cochain complexes of configurationspaces (the vertical one has to do with doubling configuration points,and this has to do with Figure 19). By Theorem 5.3.1, this differentialcan be replaced by the zero differential, and hence the spectral sequencecollapses. Some more sophisticated model category theory techniquesare required for the case n = 3.

Remark 5.3.4. Collapse is also true for the homotopy spectral sequencefor n ≥ 4 [1].

So for n ≥ 4, the homology of the E2 page is the homology of Kn.A more precise way to say this is

Theorem 5.3.5. For n ≥ 4, H∗(Kn;Q) is the Hochschild homology ofthe Poisson operad of degree n−1, which is the operad obtained by takingthe homology of the little n-cubes operad.

For more details on the Poisson operad of degree n− 1, see [52, Sec-tion 1]. Briefly, this is the operad encoding Poisson algebras, i.e. gradedcommutative algebras with Lie bracket of degree n−1 which is a gradedderivation with respect to the multiplication. One can define a differ-ential (the Gerstenhaber bracket), and the homology of the resultingcomplex is the Hochschild homology of the Poisson operad.

In summary, H∗(Kn;Q) is built out of H∗(Conf(p,Rn);Q), p ≥ 0,which is understood. In fact, this homology can be represented combi-natorially with graph complexes of chord diagrams. This therefore givesa nice combinatorial description of H∗(Kn;Q), n ≥ 4. The case n = 3is not yet well understood, and the implications of the collapse are yetto be studied. The main impediment is that, even though the spectral

52 Ismar Volic

sequence collapses, it is not clear what the spectral sequence convergesto.

It would be nice to rework the results described in this section forspaces of links. For ordinary embedded links and braids, things shouldwork the same, but homotopy links are more challenging since we donot yet have any sort of a convergence result for the Taylor (multi)towerfor this space.

Acknowledgements

The author would like to thank Sadok Kallel for the invitation towrite this article and the referee for a careful reading.

Ismar VolicDepartment of Mathematics,Wellesley College,106 Central Street,Wellesley, MA 02481,ivolic@wellesley.edu

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Morfismos, Vol. 17, No. 2, 2013, pp. 57–69

Topological chiral homologyand configuration spaces of spheres

Oscar Randal-Williams 1

Abstract

We compute the rational homology of all spaces of finite configu-rations on spheres. Our tool is a bar spectral sequence which canbe viewed as coming from the notion of “topological chiral homol-ogy”, though we give a self-contained construction of the spectralsequence.

2010 Mathematics Subject Classification: 55R80, 55T05, 57T30.Keywords and phrases: configuration space, spectral sequence.

1 Result

For a topological manifold M , let Cn(M) denote the space of n un-ordered distinct points in M . The purpose of this short note is to givea proof of the following.

Theorem 1.1. Suppose d ≥ 2 is even. Then

H∗(Cn(Sd);Q) =

⎧⎪⎨

⎪⎩

Q in degree 2d− 1 n ≥ 3

Q in degree d n = 1

0 n = 0, 2.

Suppose d ≥ 3 is odd. Then

H∗(Cn(Sd);Q) =

Q in degree d n ≥ 1

0 n = 0.

1Supported by the Herchel Smith Fund, ERC Advanced Grant No. 228082, andthe Danish National Research Foundation (DNRF) through the Centre for Symmetryand Deformation.

57

Morfismos, Vol. 17, No. 2, 2013, pp. 57–69

58 O. Randal-Williams

This theorem is not new (for example, it may be quickly deducedfrom [7, Theorem 18]), but the method of proof we offer is different and,we feel, somewhat interesting.

2 A “multiplicative” decomposition of configu-ration spaces

Let Cn(M,X) denote the space of n unordered points in a topologicalmanifold M , labeled by the space X, and let

C(M,X) :=!

n≥0

Cn(M,X).

If the manifold M has dimension d, fix an embedding e : Rd → Mand let S(t) ⊂ M denote the image under e of the sphere of radius tcentred at the origin of Rd. Define Bn to be"(t0, . . . , tn; c) ∈ Rn+1

>0 × C(M,X) | t0 < · · · < tn, c ∩ S(ti) = ∅ ∀i#.

Let di : Bn → Bn−1 be the map that forgets ti, and ϵ : B0 → C(M,X)be the map that forgets t0; with this structure B• is a semi-simplicialtopological space, augmented over C(M,X).

Proposition 2.1. The map |ϵ| : |B•| → C(M,X) is a weak homotopyequivalence.

Proof. The augmented semi-simplicial space ϵ : B• → C(M,X) is a“topological flag complex” in the sense of [3, Definition 6.1]. Further-more, it satisfies the conditions of [3, Theorem 6.2]: conditions i) andii) are clear, and for condition iii) we observe that for any configurationc, and any finite collection of elements of the set

t ∈ R>0 | c ∩ S(t) = ∅,

which is the set of vertices over the configuration c, there exists anotherelement of that set which is larger than them all, as the set is infinite(because the configuration c is finite). Theorem 6.2 of [3] then impliesthat the augmentation map induces a weak homotopy equivalence ongeometric realisation.

The space C((0, 1) × Sd−1, X) is an H-space via stacking cylindersend-to-end. If we write M for M \ e(D1) then the spaces C(M,X)and C(Rd, X) are right and left H-modules over C((0, 1) × Sd−1, X)respectively. Thus, fixing a field F,

Configuration spaces of spheres 59

(i) A := H∗(C((0, 1)× Sd−1, X);F) is a ring,

(ii) H∗(C(M,X);F) is a right A-module,

(iii) D := H∗(C(Rd, X);F) is a left A-module.

Proposition 2.2. There is a spectral sequence

(2.1) E2s,∗ := TorsA(H∗(C(M,X);F), D) =⇒ H∗(C(M,X);F).

Proof. Let us write Dt ⊂ Rd for the open ball of radius t, and Dt forits closure. There is a map

ϕ : Bn −→ C(M,X)× (C((0, 1)× Sd−1, X))n × C(Rd, X)

given by the canonical identifications of

(i) M \ e(Dtn) with M ,

(ii) Dti+1 \Dti with (0, 1)× Sd−1,

(iii) Dt0 with Rd.

The product of ϕ with the map to Rn+1>0 given by (t0, . . . , tn; c) &→

(t0, t1 − t0, t2 − t1, . . . , tn − tn−1) is a homeomorphism, and so ϕ is ahomotopy equivalence. Furthermore, it is clear that ϕ gives an identi-fication of semi-simplicial objects in the homotopy category of spaces,where

C(M,X)× (C((0, 1)× Sd−1, X))• × C(Rd, X)

is such a semi-simplicial object via the H-space and H-module structuremaps (and the simplicial identities hold by the homotopy associativityof these maps).

If we filter |B•| by skeleta |B•|(k), then we have identifications

|B•|(k)/|B•|(k−1) ∼= ∆k ×Bk/∂∆k ×Bk

∼= Sk ∧ (Bk)+.

The spectral sequence for this filtration has

E1s,t = Hs+t(|B•|(s), |B•|(s−1);F) ∼= Ht(Bs;F)

and, following [9, §5], we see that under this identification the differentiald1 : E1

s,t → E1s−1,t is given by

!si=0(−1)i(di)∗, the alternating sum of

60 O. Randal-Williams

the maps induced on homology by the face maps. By the identificationϕ and the Kunneth theorem we have

H∗(Bs;F) ∼= H∗(C(M,X);F)⊗H∗(C((0, 1)× Sd−1, X);F)⊗s

⊗H∗(C(Rd, X);F)∼= H∗(C(M,X);F)⊗A⊗s ⊗D

and by inspection of the differential d1, the chain complex (E1∗,∗, d

1)

agrees with the bar complex B(H∗(C(M,X);F), A,D). Thus the E2

page has the description claimed.

Everything in sight has an extra grading: for any manifold N , thereis a canonical decomposition H∗(C(N,X)) =

!n≥0H∗(Cn(N,X)). We

call this the multiplicity grading, and write the grading of an elementas (h,m) where h is the homological grading and m is the multiplicitygrading.

Remark 2.3. The notion of topological chiral homology, c.f. [4, §5.3.2],[1], [8], roughly speaking associates to a framed En-algebra A (in topo-logical spaces) and an n-manifold N a space

"N A. The association

N $→"N A is covariant, and sends disjoint union to cartesian product.

In particular, for an n-manifold N with boundary the space"[0,1]×∂N A

is an A∞-algebra, as for each configuration of m little 1-cubes thereis an embedding

#m[0, 1] × ∂N → [0, 1] × ∂N to which"−A can be

applied.It can be shown that

"N A is a

"[0,1]×∂N A-module (right or left, as"

[0,1]×∂N A has a canonical antiinvolution). Furthermore, if ∂N = ∂N ′

then there is a natural equivalence

$%

NA

&⊗L!

[0,1]×∂N A

$%

N ′A

&−→

%

N∪∂NN ′A,

from the (derived) tensor product of these two A∞-modules. This givesa bar spectral sequence

TorsH∗(![0,1]×∂N A)

$H∗

$%

NA

&, H∗

$%

N ′A

&&=⇒ H∗

$%

N∪∂NN ′A

&.

If we take A = C(Rd, X) to be the free Ed-algebra on a spaceX, thenthe topological chiral homology

"N A may be shown to be homotopy

Configuration spaces of spheres 61

equivalent to C(N,X), so taking N = M \ int(Dd) and N ′ = Dd weobtain a spectral sequence

TorsH∗(C([0,1]×∂Sd−1,X))

!H∗(C(M,X)), H∗(C(Rd, X))

"⇒ H∗(C(M,X))

which agrees with ours.

3 The structure of A and D in characteristiczero

Let X = ∗ and F = Q. In this section we wish to give a generatorsand relations description of the ring A and the left A-module D, andconstruct an explicit resolution (which will have length 1) of D as anA-module.

For a smooth manifold with boundary M , a choice of boundarycomponent E gives a stabilisation map

sE : Cn(M) −→ Cn+1(M).

Let τM+ denote the fibrewise one-point compactification of the tangentbundle of M , and Γn(M) denote the space of sections of this bundlewhich are compactly supported in the interior of M , and which havedegree n. There is an “electric charge”, or “scanning”, map

S : Cn(M) −→ Γn(M),

cf. [5]. We shall need the following result. We state it for integralhomology, though we only need it for rational homology.

Proposition 3.1. The map S induces an injection on integral homol-ogy, and an isomorphism on integral homology in degrees 2∗ ≤ n.

Proof. This is obtained by combining the main results of [5] and [6].

Remark 3.2. In the following, for a set S we write Q[S] for the freecommutative Q-algebra on the set S, Q⟨S⟩ for the free noncommutativeQ-algebra on the set S, and QS for the free Q-vector space on theset S.

62 O. Randal-Williams

3.1 The disc: C(Rd)

We write [n] ∈ H0(Cn(Rd);Q) for the class of any configuration of npoints; these satisfy sE∗([n]) = [n + 1], and [n] has bidegree (0, n). Wealso write τ ∈ Hd−1(C2(Rd);Q) for the image of the fundamental classunder the map

(3.1) Sd−1 −→ C2(Rd)

which sends x to the configuration 0, x, which has bidegree (d− 1, 2).

Proposition 3.3. The class τ2 ∈ H2(d−1)(C4(Rd);Q) is zero, τ and [1]commute, and the induced map

φ :

!Q[[1]] d odd

Q[[1], τ ]/(τ2) d even−→ H∗(C(Rd)) = D

is an isomorphism.

Proof. The scanning map in this case is

S : Cn(Rd) −→ ΩdnS

d.

By a theorem of Serre, ΩdnS

d has trivial rational homotopy groups if dis odd, so also has trivial rational homology, and has a single nontrivialrational homotopy group πd−1(Ωd

nSd) ⊗ Q ∼= Q if d is even. It is a

simple calculation that in this case it also has a single nontrivial rationalhomology group in degree (d − 1), and we claim that as long as n ≥ 2the class S∗(τ · [n− 2]) is a generator. By the homotopy commutativityof the diagram

C2(Rd) S !!

sn−2E

""

Ωd2S

d

−·S([n−2])≃""

Cn(Rd) S !! ΩdnS

d,

and the injectivity of S∗, it suffices to prove that τ ∈ Hd−1(C2(Rd);Q)is nontrivial. But C2(Rd) is homeomorphic to RPd−1, an orientablemanifold, and the map (3.1) has degree ±2, so τ is nothing but (±)twice the fundamental class of RPd−1, hence nontrivial. (For d odd,Hd−1(RPd−1;Q) = 0 so the class τ is zero.)

It is clear that τ and [1] commute, by geometric considerations (themultiplication on C(Rd) extends to an Ed-algebra structure). The class

Configuration spaces of spheres 63

S∗(τ2) lies in a group which is zero (as Ωd4S

d has trivial rational ho-mology in degree 2(d− 1), by the above discussion), and S∗ is injectiveso τ2 = 0 and we obtain an induced map φ as in the statement of theproposition.

This map is clearly an isomorphism in multiplicity grading 0 or 1,and it remains to show that φn : Q[n], τ · [n− 2] → H∗(Cn(Rd);Q) isan isomorphism for n ≥ 2. But S∗ φn is an isomorphism, and S∗ isinjective, so φn is an isomorphism too, as required.

It is convenient to note, as we did in the proof, that the class τ isdefined for all d, but is zero if d is odd.

3.2 The cylinder: C((0, 1)× Sd−1)

We perform an analysis similar to the above. There is a map

(3.2) Sd−1 −→ C1((0, 1)× Sd−1)

sending x to the one-point configuration (12 , x), and we let

∆ ∈ Hd−1(C1((0, 1)× Sd−1);Q)

be the image of the fundamental class, so deg(∆) = (d − 1, 1). Byidentifying Rd with (0, 1) × Dd−1

− , the lower half of the cylinder, weobtain an inclusion

C(Rd) −→ C((0, 1)× Sd−1)

and we write [n] and τ for the images of the elements defined in theprevious section.

Proposition 3.4. In the ring A = H∗(C((0, 1)×Sd−1);Q) the relations

τ is central [1] ·∆ = ∆ · [1]− τ

hold. The induced map

φ :

!Q[[1],∆] d odd

Q⟨[1], τ,∆⟩/(τ2, [[1],∆] = −τ, τcentral) d even−→ A

is an isomorphism.

64 O. Randal-Williams

Proof. First consider the proposed relation [1] · ∆ = ∆ · [1] − τ inHd−1(C2((0, 1) × Sd−1);Q). This follows from the homology betweenthe cycles ∆ · [1] − [1] · ∆ and τ , which may be seen in the diagramabove of cycles in the configuration space of two points on the cylinder.The fact that τ is central follows from similar geometric considerations(clearly τ and [1] commute, then one sees from a similar figure that[∆, τ ] is homologous to a cycle which is supported in a disc, but bythe previous section H2(d−1)(C3(Rd);Q) = 0 so there are no nontrivialhomology classes of this dimension supported in a disc).

The target of the scanning map in this case is map∂n([0, 1]×Sd−1, Sd),the space of continuous maps f : [0, 1]×Sd−1 → Sd which send 0, 1×Sd−1 to the basepoint ∗ ∈ Sd, and which have degree n in the sensethat the induced map

f∗ : Hd([0, 1]× Sd−1, 0, 1× Sd−1;Z) −→ Hd(Sd, ∗;Z)

sends the relative fundamental class to n times the fundamental class.The collection of all these mapping spaces fit into a fibration sequence

(3.3) ΩdSd −→ map∂([0, 1]× Sd−1, Sd)p−→ ΩSd,

where p restricts a map to the interval [0, 1]×∗. By taking the adjointin the middle mapping space, we can express it as Ωmap(Sd−1, Sd).This exhibits the fibration sequence (3.3) as obtained from looping theevaluation fibration

Ωd−1Sd −→ map(Sd−1, Sd) −→ Sd

so it is a principal fibration. Moreover, the evaluation fibration has asection, given by the inclusion of the constant maps, so after looping itsplits as a product: thus (3.3) is a trivial fibration.

Let us compute the rational homology of the H-space map∂([0, 1]×Sd−1, Sd) with its Pontrjagin ring structure. Suppose first that d isodd. Then Ωd

nSd has trivial rational homology, and ΩSd = ΩΣ(Sd−1)

Configuration spaces of spheres 65

so by the Bott–Samelson theorem has rational Pontrjagin ring the free(non-commutative) algebra on !H∗(Sd−1;Q), i.e. Q[u] where the class uis obtained from the map Sd−1 → ΩSd adjoint to the identity map. Asp∗(S∗(∆)) = u and p and S are H-space maps, it follows that

Q[[1]±1,S∗(∆)] −→ H∗(map∂([0, 1]× Sd−1, Sd);Q)

is an isomorphism of rings.Suppose now that d is even. We again have that H∗(ΩSd;Q) ∼=

Q[u] as a ring. We have the path components [n] ∈ H0(map∂([0, 1] ×Sd−1, Sd);Q) for n ∈ Z as well as the classes S∗(∆) and S∗(τ), which wewill simply call ∆ and τ again to save space, and we obtain an inducedmap

Q⟨[1]±1,∆, τ⟩(τ2, [[1],∆] = −τ, τ central)

−→ H∗(map∂([0, 1]× Sd−1, Sd);Q).

It follows from the fact that p∗(S∗(∆)) = u and that (3.3) is a fibrationsequence of H-spaces which is trivial as a fibration of spaces (so thehomology Serre spectral sequence is one of rings, in fact even of Hopfalgebras c.f. [2, §5]) that this map is surjective, but by the splitting of(3.3) and counting dimensions it follows that in fact it is an isomorphism.

From these two calculations it follows that the induced map φ inthe statement of the proposition is injective. To see that it is an iso-morphism, note that the cokernel vanishes after stabilisation by [n] (asφ induces an isomorphism after inverting [1]), so it is enough to showthat if x, of bidegree (k(d− 1),m), is such that

(3.4) x · [n] = A ·∆k · [n+m− k] +B ·∆k−1 · τ · [n+m− k − 1]

for n ≫ 0, then if m− k < 0 then A and B are zero, and if m− k = 0then B is zero. We prove this by induction on the multiplicity grading ofx: if m = 0 then the class x is in the homology of C0((0, 1)×Sd−1) = ∗,and the claim follows.

For the induction step, we use a map

t∗ : H∗(Cn((0, 1)× Sd−1);Q) −→ H∗(Cn−1((0, 1)× Sd−1);Q)

constructed as follows. Let π : Cn,1((0, 1)× Sd−1) → Cn((0, 1)× Sd−1)denote the n-fold covering space whose total space consists of a config-uration of n points in (0, 1) × Sd−1 with one distinguished point, andπ forgets which point is distinguished. There is a map f : Cn,1((0, 1)×

66 O. Randal-Williams

Sd−1) → Cn−1((0, 1)×Sd−1) which removes the distinguished point, andwe let t∗ be the composition of the transfer map for the finite coveringπ followed by f∗. The construction of t∗ shows that it is a derivationfor the H-space multiplication, and it is easy to compute that

t∗(∆) = 0 t∗(τ) = 0 t∗([1]) = [0].

If m− k < 0 then applying t∗ to (3.4) n times annihilates ∆k · [n+m− k] and ∆k−1 · τ · [n+m− k − 1], and we obtain

n! · x+ y · [1] = 0

for y some expression in iterated applications of t∗ to x. The class y isof bidegree (k(d − 1),m − 1) and satisfies the analogue of (3.4), so byinduction both A and B are zero, which finishes the proof in this case.

If m− k = 0 then applying t∗ to (3.4) n times gives

n! · x+ y · [1] = A · n! ·∆k

and substituting back into (3.4) gives y ·[n+1] = −n!·B ·∆k−1 ·τ ·[n−1].But y has bidegree (k(d−1),m−1) so by induction B = 0, which finishesthe proof in this case.

3.3 D as an A-module (d even)

The left A-module structure on D is given by

[k] • (τ ϵ · [n]) = τ ϵ · [n+ k] τ • (τ ϵ · [n]) = τ ϵ+1 · [n]

and

∆ • (τ ϵ · [n]) = n · τ ϵ+1 · [n− 1].

The first two are clear and the last follows from ∆ • [0] = 0 (as C1(Rd)is contractible, so has trivial homology in degree (d− 1)), ∆ • τ = 0 (asC3(Rd) has trivial homology in dimension 2(d− 1)), and the commuta-tion relation [∆, [1]] = τ . One verifies that

0 !! Σd−1,1A ·∆ !! A[0] #→[0]

!! D !! 0

is an exact sequence of left A-modules.

Configuration spaces of spheres 67

3.4 D as an A-module (d odd)

The left A-module structure on D is given by [k] • [n] = [n + k] and∆ • [n] = 0. One verifies that

0 !! Σd−1,1A ·∆ !! A[0] "→[0]

!! D !! 0

is an exact sequence of left A-modules.

4 Configuration spaces of spheres

We apply the spectral sequence (2.1) to M = Sd, for d even. In this caseM = Dd so H∗(C(M);Q) = D, but it has a right A-module structure.This is induced from the left A-module structure by the antiinvolutionof A, which in turn is induced by reflecting the first coordinate of thecylinder around 1

2 . It is easy to see that this antiinvolution : A → Ais given on generators by

[n] = [n] τ = −τ ∆ = ∆.

Concretely, the right module structure is given by

(−) • (∆i · τ ϵ · [n]) := ([n] · (−τ)ϵ ·∆i) • (−).

We have shown that D has a length 1 resolution by free A-modules,

so the complex Σd−1,1D−•∆−→ D computes the E2 page of the spectral

sequence. This map is given explicitly by

(τ ϵ · [n]) #→ (τ ϵ · [n]) •∆ = ∆ • (τ ϵ · [n]) = n · τ ϵ+1 · [n− 1],

so is as shown in Figure 1. By inspection there can be no furtherdifferentials in the spectral sequence, and we immediately see the correctrational homology for C0(Sd) = ∗, C1(Sd) = Sd, and C2(Sd) ≃ RP d,and also deduce that for all n ≥ 3, Cn(Sd) has the rational homologyof S2d−1. The analysis for d odd is the same, but a little easier as theelement τ does not appear.

5 Final remarks

The reader will realise that the multiplicative decomposition techniquedescribed in Section 2 admits many variations. Let us describe one,

68 O. Randal-Williams

Figure 1: E1 page of the spectral sequence for d even, with multiplicitygrading along the horizontal axis, and Tor grading along the verticalaxis. The homological grading is not shown.

which leads to what appears to be a difficult calculation in homologicalalgebra. For the manifold M = S1 × Sd−1 we may let Bn be!(p0, . . . , pn; c) ∈ (S1)n+1 × C(M)

""""pi distinct, cyclically ordered,

c ∩ pi× Sd−1 = ∅

#

and as in Section 2 we may show that the augmentation |B•| → C(M)is a homotopy equivalence. Following the proof of Proposition 2.2, wefind that the E1 page of the resulting spectral sequence is now the cyclicbar complex for the algebra A = H∗(C((0, 1)× Sd−1);F), so we have aspectral sequence

E2s,∗ = HHs(A,A) =⇒ H∗(C(S1 × Sd−1);F).

starting with the Hochschild homology of the algebra A with coefficientsin itself. For F = Q, one ought to be able to use the calculation inSection 3.2 of the algebra A to study this spectral sequence, but thehomological algebra seems to be a lot harder.

Oscar Randal-WilliamsCentre for Mathematical Sciences,Wilberforce Road,Cambridge CB3 0WB, UK,o.randal-williams@dpmms.cam.ac.uk

References

[1] Andrade R., From manifolds to invariants of En-algebras, arXiv:-1210.7909 [math.AT],(30 Oct 2012).

Configuration spaces of spheres 69

[2] Browder W., On differential Hopf algebras, Trans. Amer. Math.Soc. Transactions of the American Mathematical Society 1071963, 153–176.

[3] Galatius S.; Randal-Williams O., Homological stability for modulispaces of high dimensional manifolds, arXiv:1203.6830 [math.AT],(30 Mar 2012).

[4] Lurie J., Higher Algebra, 2012.

[5] McDuff D., Configuration spaces of positive and negative particles,Topology 14 (1975), 91–107.

[6] Randal-Williams O., Homological stability for unordered congura-tion spaces, Q. J. Math 64:1 (2013), 303–326.

[7] Salvatore P., Configuration spaces on the sphere and higher loopspaces, Cohomological Methods in Homotopy Theory, BarcelonaConference on Algebraic Topology, Bellaterra, Spain (1998),Progress in Mathematics, 196 2001, 375–395.

[8] Salvatore P., Conguration spaces with summable labels, Q. J. Math64:1 (2013), 303–326.

[9] Segal G., Classifying spaces and spectral sequences, Inst. HautesEtudes Sci. Publ. Math. 34:1 (1968), 105–112.

[10] Weiss M., What does the classifying space of a category classify?,Homology Homotopy Appl. 7:1 (2005), 185–195.

Morfismos, Vol. 17, No. 2, 2013, pp. 71–100

Cooperads as symmetric sequences

Benjamin Walter

Abstract

We give a brief overview of the basics of cooperad theory usinga new definition which lends itself to easy example creation andverification and avoids common pitfalls and complications causedby nonassociativity of the composition operation for cooperads.We also apply our definition to build the parenthesization andcosimplicial structures exhibited by cooperads and give examples.

2010 Mathematics Subject Classification: 18D50; 16T15, 17B62.Keywords and phrases: Cooperads, operads, coalgebras, Kan extensions.

1 Introduction

In the current work we discuss cooperads in generic symmetric monoidalcategories from the point of view of symmetric sequences. Fix a sym-metric monoidal category (C, ⊗). Let us roughly recall the standardframework.

Operads encode algebra structures. The tautological example is theendomorphism operad of an object end (A) = n Hom(A⊗n, A). Oper-ads have a natural grading by levels expressing the “arity” of different“operations” (for example, end (A)(n) = Hom(A⊗n, A)). The symmet-ric group Σn acts on the n-ary operations of an operad (for end (A)(n)this action is by permutation of the A⊗n). A graded object with Σn-actions is called a “symmetric sequence.” Operads are further equippedwith a composition product identifying the result of plugging opera-tions into each other (for example, end (A) end (A) → end (A)). Veryroughly, an operad is “a bunch of objects with a rule for plugging theminto each other”.

Operads encode algebra structures via maps of operads (preservingsymmetric group actions and composition structure). So, for example,

71

72 B. Walter

there is an operad lie of formal Lie bracket expressions modulo Lie re-lations, along with a composition rule identifying the result of pluggingbracket expressions into each other. A map of operads lie → end(A)identifies a specific endomorphism of A for each formal Lie bracket ex-pression, in such a way that composition of Lie bracket expressionsis compatible with the composition of corresponding endomorphisms.This gives A the structure of a Lie algebra.

Coalgebra structures can also be defined via operads. The coendo-morphisms of an object coend(A) =

!nHom(A,A⊗n) also form an

operad: It is graded, with symmetric group action, and has a naturalmap coend(A)coend(A) → coend(A) also given by plugging thingsinto each other. Replacing end by coend changes algebra structures tocoalgebra structures. For example a map of operads lie → coend(A)identifies a coendomorphism of A for each Lie bracket expression, thusgiving A a Lie coalgebra structure.

This is a common point of view (see e.g. [11]), but there is an alterna-tive. For clarity, we will continue with the example of Lie algebras. A Liealgebra structure is maps lie(n) → Hom(A⊗n, A) which is equivalent tomaps lie(n)⊗A⊗n → A (ignore Σn-actions for the moment). Dually, aLie coalgebra structure is maps lie(n) → Hom(A,A⊗n) which is equiva-lent to maps lie(n)⊗A → A⊗n which is equivalent to A →

"lie(n)

#∗⊗

A⊗n. (Dualizing lie(n) should not introduce trouble, because it is finitedimensional.) The level-wise dual object lie =

!n

"lie(n)

#∗has struc-

ture dual to that of lie. This is a cooperad. (The precise definition isthe subject of the current paper.)

Experience [9] [10] has shown that it is sometimes more useful todirectly work with cooperads and cooperad structures when describingcoalgebras rather than continually referring all the way back to operadsand operad structures. Also sometimes coalgebras can have a morenatural expression as coalgebras over cooperads, rather than coalgebrasover operads. Just as operads can be thought of as “a bunch of objectswhich are plugged into each other”, cooperads can be thought of as “abunch of objects where subobjects are removed or quotiented”.

Unfortunately category theory causes a slight hitch when attemptingto blindly dualize operad structure to define cooperads. The dual ofoperad composition is cooperad composition, which is similar exceptfor some colimits being replaced by limits. The problem comes whenlooking at associativity. In a symmetric monoidal category ⊗ is leftadjoint (to Hom) so it will commute with colimits. This allows operadcomposition products to be associative (e.g. (lielie)lie = lie(lie

Cooperads as Symmetric Sequences 73

lie)). However, this will generally not happen for cooperad composition(e.g. (lieˇ• lie ) • lie = lie • (lie • lie )). This issue crops up forexample, in the cooperadic cobar constructions of Ching in his thesis [4]and arXiv note [5].

We work by defining a new composition product – a compositionproduct of tree-functors. The motivating intuition is that the composi-tion product of two symmetric sequences should not itself be a symmetricsequence – in particular its group of symmetries is much too large. Mapsto and from the tree-functor composition product can be expressed asmaps to and from universal extensions, which yields the classical operadand cooperad composition products. Using the tree-functor compositionproduct (rather than its Kan extension) when describing or defining co-operads greatly simplifies bookkeeping; though it turns out that, foroperads, it doesn’t really make a difference.

We begin by introducing the notation of wreath product categories.These are inspired by the wreath product categories of Berger [2], andat the most basic level are merely Groethendieck constructions. Wreathproduct categories are defined so that they will be the natural sourcecategory for iterated composition products of symmetric sequences. Weuse this to give a simple definition of cooperads and prove all of thestandard structure holds. Then we describe comodules and coalgebras.We finish with simple examples related to work in [9], [10], and [13].

In the sequel [12] we use the structure presented here to build cofreecoalgebras, connecting to the constructions of Fox [6] and Smith [11].

We assume that the reader is comfortable with the category theorynotions of adjoint functors and Kan extensions, as well as basic simplicialand cosimplicial structures. A familiarity with the classical definitionsof operads and their modules/algebras is not required, but would behelpful.

2 Wreath product categories

This section is divided into two parts. In the first subsection, we de-fine wreath product categories using functors to the category of finitesets. Our definition is related to, but more general than, the dual of re-fined partitions of sets as used in literature by e.g. Arone-Mahowald [1].The salient difference between wreath categories and refined partitionsis that wreath categories incorporate the empty-set (see Remark 2.8).In the second subsection, an equivalent definition is given in terms of

74 B. Walter

labeled level trees – a more familiar category for the discussion of oper-ads.

The neophyte reader (or the reader looking for an immediate con-nection to classical constructions) may find it useful to read §2.2 before

§2.1.

2.1 Wreath Products

Write Σn for the category of n-element sets and set isomorphisms andΣ∗ =

!n≥0Σn for the category of all finite sets and set isomorphisms

(Σ0 = ∅). Our notation reflects the fact that a functor Σn → C is merelyan object of C with a Σn-action.

There is an alternative way to express Σn. Write FinSet for thecategory of finite sets and all set maps, and write [n] for the category

1f1−→ 2

f2−→ · · ·fn−1−−−→ n. Then Σ∗ is equivalent to the category of functors

[1] → FinSet and natural isomorphisms. We generalize this to definewreath product categories.

Definition 2.1. The wreath product category Σ≀n∗ is the category of

contravariant functors [n]→ FinSet and natural isomorphisms.

Remark 2.2. We will write objects of Σ≀n∗ as sequences of maps of finite

sets, indexed in the following manner.

S1f1←− S2

f2←− · · ·fn−1←−−− Sn

Since [n] ∼= [n]op, the use of contravariant functors in Definition 2.1 ispurely cosmetic. Using covariant functors would change nothing, exceptthat indices would not line up as perfectly later on.

Note that we are clearly defining the levels of a simplicial category.Before continuing in that direction, however, we will explain our choiceof notation via an equivalent, hands-on definition of wreath productswith a generic category A.

Definition 2.3. The wreath product category Σn ≀ A is the categorywith

• objects Obj(Σn ≀ A) ="Ass∈S

## S ∈ Obj(Σn), As ∈ Obj(A)$

given by n-element sets of decorated objects of A;

• and morphisms%σ; φtt∈T

&: Att∈T −→ Bss∈S given by a set

isomorphism σ : T → S and a set of A-morphisms φt : At → Bσ(t).

Cooperads as Symmetric Sequences 75

The wreath product category Σ∗ ≀A is given by Σ∗ ≀A :=!

n≥0Σn ≀A.

Remark 2.4. Σ0 ≀A is the empty category, since Ass∈∅ = ∅. Further-more Σ∗ ≀ Σ0

∼= Σ∗∼= Σ1 ≀ Σ∗. These equivalences are given by writing

objects of Σ∗ ≀ Σ0 as"∅s#s∈S

and objects of Σ1 ≀ Σ∗ as"S⋆

#and using

the facts that ∅ is initial and a one point set ⋆ is final in FinSet. Wemake further use of these equivalences later. Note that Σ∗ ≀ Σ1 ! Σ∗

because one point sets are not initial in FinSet.

The following proposition is easy to check.

Proposition 2.5. Definitions 2.1 and 2.3 agree:

• Σ≀2∗∼= Σ∗ ≀ Σ∗, and more generally

• Σ≀n∗∼= Σ∗ ≀

$Σ≀n−1∗

% ∼=

n& '( )Σ∗ ≀ (· · · ≀ (Σ∗ ≀ Σ∗)).

Proof Sketch: The object$S

f←− T

%of Σ≀2

∗ corresponds to the object"f−1(s)s

#s∈S

of Σ∗ ≀ Σ∗.

The object"As

#s∈S

corresponds to$S

π←−

*

s∈S

As

%where π is the

map on the coproduct induced by πs : As → s ⊂ S.

Using notation from Definition 2.3, the endomorphisms of the wreathproduct category Σn ≀ Σm correspond to the automorphisms of an n-element set of m-element sets S = A1, . . . , An with |Ai| = m. Ele-ments within each Ai can be permuted by Σm and the Ai “blocks” arepermuted by Σn – this is the wreath product group Σn ≀ Σm. Thus, afunctor Σn ≀ Σm → C is an object of C equipped with an action of thewreath product group Σn ≀ Σm. We view Σ∗ ≀ Σ∗ as a generalization ofthis basic example – the “blocks” Ai no longer need to be same size,and there can be an arbitrary number of them.

We return to the simplicial structure of the collection of wreathproducts

!nΣ

≀n∗ . Recall that there are standard “face” functors

∂ni : [n]→ [n− 1]

for 1 ≤ i ≤ (n− 1), given by composing morphisms or forgetting 1 (forreasons to be explained shortly, we do not use the “forget n” face map,∂nn).

∂n1 (1

f1−→ · · ·fn−1−−−→ n

%=

$2→ · · ·→ n

%

∂ni

$1

f1−→ · · ·fn−1−−−→ n

%=

$1→ · · ·→ (i− 1)

fifi−1−−−−→ (i+ 1)→ · · ·→ n

%

76 B. Walter

Furthermore, (because we do not allow the use of ∂nn functors) any

chain of (n− 1) compositions ∂2i2 · · · ∂n

in: [n]→ [1] equals the functor

γn : [n]→ [1] which forgets all but the top object.

γn!1

f1−→ · · ·fn−1−−−→ n

"=

!n"

We will write ∂ni and γn also for the induced functors ∂n

i : Σ≀n∗ → Σ≀(n−1)

∗ ,

for 1 ≤ i ≤ (n − 1), and γn : Σ≀n∗ → Σ∗. When n is clear from context

we may write merely ∂i and γ.

Remark 2.6. In the notation of Definition 2.3, the map γ2 = ∂21 :

Σ∗ ≀ Σ∗ → Σ∗ is given by Stt∈T &→#

T St. All other ∂ni and γn are

induced by this (see Proposition 2.10).

Before describing the degeneracy maps, we explain the missing ∂nn .

Recall that Σ∗ is equivalent to the full subcategory $Σ∗ = Σ1 ≀ Σ∗ ⊂Σ∗ ≀ Σ∗ of functors sending 1 to a one-element set. More generally, Σ≀n

∗

is equivalent to the full subcategory $Σ≀n∗ = Σ1 ≀ Σ

≀n∗ ⊂ Σ≀n+1

∗ of functorssending 1 to a one-element set. Objects of $Σ≀n

∗ are sequences of set maps

⋆f0←− S1

f1←− · · ·fn−1←−−− Sn

Under this correspondence the face functors ∂ni : $Σ≀n

∗ → $Σ≀n−1∗ , for

1 ≤ i ≤ (n− 1), are all given by composition; however the functor ∂nn is

not.

∂ni

!⋆

f0←− S1f1←− · · ·

fn−1←−−− Sn

"=

!⋆← · · ·← Si−1

fi−1fi←−−−− Si+1 ← · · ·← Sn

"

(Our indexing convention is for the one point set to be ⋆ = S0 in $Σ≀n∗ ).

Operad and cooperad structure is induced by structure of $Σ≀n∗ and

∂ni . Instead of working with this directly, we use the equivalent cat-

egories and functors Σ≀n∗ and ∂n

i ; because in practice keeping track ofthe final, one point set at the bottom of each sequence is unnecessarilytedious.

We continue with the degeneracies of the simplicial structure, whichare most conveniently written via the equivalent categories $Σ≀n

∗ . In thisnotation, the degeneracy functors sni : $Σ≀n

∗ → $Σ≀n+1∗ for 0 ≤ i ≤ n are

the doubling maps.

sni!⋆

f0←− S1f1←− · · ·

fn−1←−−− Sn

"=

!⋆← · · ·← Si

Id←− Si ← · · ·← Sn

"

Cooperads as Symmetric Sequences 77

Note that defining the degeneracy sn0 on the level of Σ≀n∗ requires picking

a distinguished one point set. A reader averse to making choices shouldreplace all Σ∗, ∂i, etc. by !Σ∗, ∂i, etc. from now on.

It is classical that the degeneracies sni−1 and sni are each sections of

the face map ∂n+1i on the level of [n]. Thus face and degeneracy maps

combine to give a collection of categories and functors:

· · · Σ∗ ≀ Σ∗ ≀ Σ∗ ≀ Σ∗ Σ∗ ≀ Σ∗ ≀ Σ∗ Σ∗ ≀ Σ∗ Σ∗

where the dashed, left-pointing arrows are sections of their neighboringright-pointing arrows and all pairs of neighboring right-pointing arrowsare coequalized by an arrow out of their target. Under the correspon-dence Σ≀n

∗∼= !Σ≀n

∗ ⊂ Σ≀n+1∗ , this is very explicitly a simplicial category

with the bottom level as well as the first and last face maps removed;equivalently, an augmented simplicial category with two extra degen-eracies.

Remark 2.7. We could express all of the standard face maps ∂ni , 1 ≤

i ≤ n, as compositions by writing Σ≀n∗∼= Σ

≀n∗ = Σ1 ≀Σ

≀n∗ ≀Σ0 ⊂ Σ≀n+2

∗ , thefull subcategory of functors sending (n+2) to the empty-set and 1 to aone-element set. Then ∂n

n becomes:

∂nn

"⋆

f0←− S1f1←− · · ·

fn−1←−−− Sn

fn←− ∅#=

"⋆← S1 ← · · ·← Sn−1

fn−1fn←−−−−− ∅

#

The Σ≀n∗ fit together to make an (unaugmented) simplicial category with

two extra degeneracies. In the next section, the levels of this will begiven an alternate definition and called ∅n. This structure is useful forconstructing algebras and coalgebras instead of operads and cooperads.

Remark 2.8. Another construction which has been useful in the pastfor describing and working with operads uses the category of sets equi-pped with iterated refinements of partitions where morphisms are givenby set isomorphisms respecting all partition equivalences (see Arone-Mahowald [1] and Ching [4]) . A partition of a set S is equivalent to asurjective set map S → T where T is the partition set. An iteratedpartition of a set S is equivalent to a functor from [n] to the categoryof finite sets and surjections (instead of the category of finite sets andall set maps). This is sufficient for describing operads and cooperads

78 B. Walter

which are trivial in “0-arity”. So partitions cannot be used to describe,for example, an operad of algebras over an algebra. Also missing 0-aritymeans that partitions cannot work with algebras (or coalgebras) as justa special case of modules (or comodules).

Before continuing with the next subsection, we will combine Defi-nitions 2.1 and 2.3 to get a more general definition of wreath productswith generic categories, necessary to discuss associativity.

Definition 2.9. The wreath product category Σ≀n∗ ≀ A is the category

with

• Obj(Σ≀n∗ ≀A) =

!"F, Ass∈F (n)

#| F ∈ Obj(Σ≀n

∗ ), As ∈ Obj(A)$

• morphisms"Φ; φss∈F (n)

#:"F, As

#→

"G, Bt

#given by

a natural isomorphism Φ : F → G and a set of A-morphismsφs : As → B(Φn)(s)

Objects of Σ≀n∗ ≀A can be written as sequences of set maps

S1f1←− S2

f2←− · · ·fn−1←−−− Sn−1

fn←−%As

&s∈Sn

.

Morphisms are level-wise set isomorphisms accompanied by (at the top

level) A-maps As → Bt (where φ : Sn

∼=−→ Tn with φ(s) = t). Inthe following subsection we will give an alternate way to describe theseobjects via labeled trees.

Proposition 2.10. Wreath product is associative:

(Σ∗ ≀ Σ∗) ≀ Σ∗∼= Σ∗ ≀ (Σ∗ ≀ Σ∗) ∼= Σ≀3

∗ .

More generally, Σ≀n∗ ≀ Σ

≀m∗∼= Σ≀n+m

∗ . Furthermore, the face maps ∂ni are

all induced by γ2 = ∂21 as

∂ni = Id ≀ γ2 ≀ Id : Σ≀i−1

∗ ≀"Σ∗ ≀ Σ∗

#≀ Σ≀n−i−1

∗ −→ Σ≀i−1∗ ≀

"Σ∗

#≀ Σ≀n−i−1

∗ .

For example ∂31 = γ2 ≀ Id and ∂3

2 = Id ≀ γ2.

2.2 Level trees

In this subsection we connect the wreath product constructions of theprevious subsection with the classical, visual, method of describing op-erads via trees.

Cooperads as Symmetric Sequences 79

For our purposes a tree is a (nonempty) non-cyclic, connected, finitegraph whose vertices are distinguished as: a “root vertex” of valency1, a (possibly empty) set of “leaf vertices” of valency 1, and all othervertices called “interior vertices”. We require each tree to have a rootand at least one interior vertex; however, we do not require that interiorvertices have valency > 1 – despite the oxymoron (in particular, we allowthe tree with a root, an “interior vertex” but no leaves as in Figure 1). Atree isomorphism is an isomorphism of vertex and edge sets, preservingroot and leaf distinctions.

For convenience of notation we will orient all edges of our trees sothat they point towards the root vertex; when drawing trees, we will notexplicitly indicate this orientation, but rather always position the rootat the bottom and the leaves at the top, with the understanding that alledges point downwards. We will denote interior vertices with a darkeneddot •, but we will not draw the root or leaf vertices – instead we willindicate only the edges connecting to them. Also for convenience, wewill draw trees on the plane, however we consider them as non-planarobjects. In particular, we will not assert any planar orderings on verticesor edges.

There is a natural height function on the vertices of trees – assigningto each vertex the number of vertices on the path between it and theroot (the vertex adjacent to the root has height 0; the root has height-1). A “level n tree” is a tree whose leaves all have height n and whoseinterior vertices have height < n. A “level tree” is a tree which is leveln for some n. Note that a level n tree may have branches without leaveswhich contain no interior vertices of height (n − 1), as in Figure 1. Inparticular, a tree with no leaves may be level n as well as level (n+ 1),etc.

•• ••

••

• •• •

• • •

••• •

• • •• • • •

Figure 1: Some examples of level 2 trees and a level 4 tree

If v is the target of the directed edge e then we say e is an “incomingedge” of v and we write In(v) for the set of incoming edges of v. In ourdrawings, incoming edges are edges abutting a vertex from above. Each

80 B. Walter

non-root vertex also has one “outgoing edge” (the abutting edge on thepath from the vertex to the root), which will be drawn connecting tothe vertex from below.

Definition 2.11. A labeled level tree is a level tree equipped with

labeling isomorphisms lv : Sv∼=−−→ In(v)v from finite sets to the sets

of incoming edges at each vertex. Let Ψ be the category of all labeledlevel trees with morphisms given by tree isomorphisms. Let Ψn be thefull subcategory of Ψ consisting of only level n trees.

Since there is always only one incoming edge at the root, and neverany incoming edges at leaves, we may equivalently label only the incom-ing edges at interior vertices.

Definition 2.12. Given a category A define the wreath product cate-gory Ψ ≀A to be the category of all labeled level trees whose leaves aredecorated by elements of A; morphisms are given by tree isomorphismsequipped with A-morphisms between the leaf decorations compatiblewith the induced isomorphism of leaf sets. Let Ψn ≀ A be the full sub-category of this consisting of only level n trees.

•i

•i j

•ji k

••

Ai

i

•

Ai Aj

i j

•

Ai AjAk

ji k

•

Figure 2: Some objects of Ψ1 and of Ψ1 ≀A

It is standard to note that the category Σ∗ may be identified withthe category Ψ1 of labeled level 1 trees. In this vein, the wreath productcategory Σ∗ ≀A may be identified with Ψ1 ≀A. More generally, the wreathproduct category Σ∗ ≀ Σ∗ is equivalent to the category Ψ2 of all labeledlevel 2 trees; and the iterated wreath product category Σ≀n

∗ is equivalentto Ψn the category of all labeled level n trees.

Proposition 2.13. There are equivalences of categories:

Ψ1∼= Σ∗, Ψ1 ≀A ∼= Σ∗ ≀A, Ψn

∼= Σ≀n∗ , and Ψn ≀A ∼= Σ≀n

∗ ≀A.

Example 2.14. The elements of !Σ≀2∗ corresponding to the Ψ2 elements

in Figure 3 are given by the following chains of maps in FinSet:

•"⋆← ∅ ← ∅

#.

Cooperads as Symmetric Sequences 81

•i1 i2

• •

•i

•j

•i1 i2

• •j1 j2

• •i1 i2 i3

• • •j1 j2 j3 k1 k2

Figure 3: Some objects of Ψ2

•!⋆← i2, i2← ∅

".

•!⋆← i

f←− j

"where f(j) = i.

•!⋆← i1, i2

f←− j1, j2

"where f(js) = i1.

•!⋆ ← i1, i2, i3

f←− j1, j2, j3, k1, k2

"where f(js) = i1 while

f(kt) = i3.

Under this identification, the functor γ2 = ∂21 : Ψ2 → Ψ1 operates

by forgetting the height 1 vertices on a level 2 tree. Paths from theheight 0 interior vertex to leaves (on level 2) are replaced by edges; thelabeling of each such edge is given by the path labeling of the path whichit replaces, as in Figure 4.

γ2 = ∂21 :

•i1 i2 i3

• • •j1 j2 j3 k1 k2

%−→•

i1j1 i1j2 i1j3 i3k1 i3k2

Figure 4: An example of γ2 : Ψ2 → Ψ1

Similarly, the functors γn : Ψn → Ψ1 operate by forgetting all inte-rior vertices except for those of height 0; replacing paths by edges carry-ing the paths’ labels. The face functors ∂n

i : Ψn → Ψn−1 for 1 ≤ i ≤ n−1are given by forgetting only the vertices of level i of a level n tree. (Thedisallowed face functor ∂n

n would forget the leaves.) The degeneracyfunctors sni : Ψn → Ψn+1 for 0 ≤ i ≤ n are given by “doubling” –replace each vertex v at level i by two vertices connected by a directededge ev, attached to the tree such that all incoming edges connect tosource vertex of ev and the outgoing edge connects to the target vertex(for the labeling, allow each edge to label itself lt(ev) : ev → ev).Note that the degeneracy snn doubles the leaf vertices – the leaves of theresulting tree are the sources of the edges ev .Remark 2.15. We very purposefully do not use the notation Υ for ourcategory of level trees, since that notation is already commonly used

82 B. Walter

s10 :ji k

• !−→

ji k

•e

•

s11 :ji k

• !−→

ji k

•• •e2e1 e3

•

Figure 5: An example of s10, s11 : Ψ1 → Ψ2

to denote the category consisting of all trees. The category Ψ differsfrom this both on the level of objects (only level trees) and on thelevel of morphisms (only isomorphisms of trees – in particular, no “edgecontraction” maps).

Remark 2.16. Note that Ψ is not isomorphic to the category Ψ∗ =!nΨn. Write ∅n for the full subcategory of Ψn consisting of trees with

no leaves. Then ∅n is a full subcategory of ∅n+1. In terms of the Ψn,the category Ψ itself is given by

Ψ ∼= Ψ1

"

∅1

Ψ2

"

∅2

Ψ3

"

∅3

Ψ4 · · ·

In the notation of the previous subsection, an element of ∅n is equivalentto a contravariant functor [n] → FinSet sending n to the empty-set asin Remark 2.7.

3 Symmetric sequences, composition products,

and cooperads

3.1 Symmetric Sequences

Let (C,⊗, 1⊗) be a symmetric monoidal category with monoidal unit 1⊗.In order to have all desired Kan extensions exist, we will further requirethat C is cocomplete. Write ⋆C for the final object of C. [In order todualize to operads, we would require C be complete with initial object∅C .]

Definition 3.1. A symmetric sequence is a functor A : Σ∗ → C.

Recall that a functor Σ∗ → C is equivalent to a sequence of objectsA(n)n≥0 of C along with a symmetric group action on each A(n).We will make use of this viewpoint when convenient without further

Cooperads as Symmetric Sequences 83

comment. If A is a symmetric sequence, then we will refer to A(n) asthe “n-ary part of A” since for operads it will encode n-ary algebraoperations. (The “0-ary operations” require no input. For example, inthe category of algebras over a field, elements of the base field are all0-ary operations.)

3.2 Composition of Symmetric Sequences

We define a “product” operation on symmetric sequences. It is impor-tant to note that our product will not itself be a symmetric sequence.Instead it is a larger diagram, reflecting a larger group of symmetries.The traditional composition product of operads as well as our cooperadcomposition product are Kan extensions of this symmetric sequenceproduct.

Definition 3.2. Given A1, . . . , An : Σ∗ → C define (A1 ! · · · ! An) :

Σ≀n∗ → C by

(⋆

f0←− S1f1←− · · ·

fn−1←−−− Sn

)$−→

⊗

0≤i≤n−1

⎛

⎝⊗

s∈Si

Ai+1(f−1i (s)

)⎞

⎠

with the convention that ⋆ = S0.Define A1 • · · · •An to be the right Kan extension of A1 ! · · ·!An

over the map γ : Σ≀n∗ −→ Σ∗.

Σ∗ A1•···•An := RγA1!···!An

Σ∗ ≀ · · · ≀ Σ∗A1!···!An

γ

Cι

Write ι : (A1 • · · · • An) γ → A1 ! · · · ! An for the universal naturaltransformation.

[Dually, to construct operads , we would define A1 · · · An to bethe left Kan extension over γ.]

Remark 3.3 (For young readers). The symmetric sequence A • B iscompletely determined by the property that every natural transforma-

tion Cγζ−→ A ! B factors uniquely as Cγ

ξ−→ (A • B)γ

ι−→ A ! B.

Dually, A B is determined by the the unique factorization of every

transformation A!Bρ−→ Dγ as A!B

π−→ (A B)γ

β−→ Dγ.

84 B. Walter

Using the notation of Definition 2.9, we can generalize the abovedefinition slightly in order to discuss associativity.

Definition 3.4. Given A : Σ≀n∗ → C and B : A → C, define (A ! B) :

Σ≀n∗ ≀A −→ C by

(A!B

)(F, Ass∈F (n)

)= A(F ) ⊗

⎛

⎝⊗

s∈F (n)

B(As)

⎞

⎠ .

Remark 3.5 (Completed tensor product). The discussion of coalgebrasvia cooperads in the introduction glossed over an important subtlety.The dual of the endomorphism operad is

∐

n

Hom(A∗, (A⊗n)∗) =∐

n

Hom(A∗, (A∗)⊗n)

where ⊗ is the “completed” tensor product (if C∗ is a category wherethis exists). The product ⊗ is a right adjoint (rather than left adjoint)to Hom. In cases where it exists, ⊗ is usually something like “formal,infinite linear combinations of elements a⊗ b”. For coalgebras to satisfya maximal number of duality properties with algebras, the completedtensor product (if it exists) should make an appearance once we begindiscussing coalgebras; but it is not used in the construction of cooperads.Note that in categories satisfying good finiteness conditions (for examplefinitely generated projective bimodules over a commutative ring), A∗ ⊗B∗ = (A⊗B)∗ = A∗ ⊗B∗.

Short calculations yield the following propositions.

Proposition 3.6. The operation ! is associative:

(A1 !A2)!A3∼= A1 !A2 !A3

∼= A1 ! (A2 !A3).

Proposition 3.7. Given A, B symmetric sequences, A •B is given by

(A •B)(n) =∏

k≥0

⎛

⎝∏

!ri =n

A(k) ⊗B(r1)⊗ · · · ⊗B(rk)

⎞

⎠Σk

.

Note that • is probably not associative. This will be discussed ingreater detail in the next section (see Proposition 4.1). The operation! is clearly functorial. If F : A1 → A2 and G : B1 → B2 are natural

Cooperads as Symmetric Sequences 85

transformations of functors A1, A2 : Σ≀n∗ → C and B1, B2 : Σ≀m

∗ → C,then we write (F !G) : (A1 !B1) → (A2 !B2) for the induced natural

transformation of functors Σ≀n∗ ≀ Σ≀m

∗ → C.

In the following subsections, we define cooperad structure and, inparallel, build the cosimplicial structure induced on

!nA

!n by the

simplicial structure of wreath product categories!

nΣ≀n∗ .

3.3 Cocomposition and Coface Maps

Definition 3.8. A symmetric sequence with cocomposition is (A, ∆)where ∆ is a cocomposition natural transformation ∆ : A γ2 −→ A!Aof functors Σ∗ ≀ Σ∗ → C compatible with the face maps ∂3

1 = (γ2 ≀ Id)and ∂3

2 = (Id ≀ γ2).Write ∆ for the associated universal natural transformation of sym-

metric sequences ∆ : A −→ A •A.

In other words, the following diagram of functors Σ∗ ≀ Σ∗ ≀ Σ∗ → Cshould commute.

(1)

(A!A)(γ2 ≀ Id) ∆≀Id

Aγ3∆

∆

A!A!A

(A!A)(Id ≀ γ2) Id≀∆

The upper path uses the factorization γ3 = ∂21 ∂3

1 = γ2 (γ2 ≀ Id) andthe lower path uses the factorization γ3 = ∂2

1 ∂32 = γ2 (Id ≀ γ2).

Applying Proposition 2.10, we may generalize ∆ to the followingmaps.

Definition 3.9. For a given a symmetric sequence with cocomposition(A, ∆) define associated natural transformations ∆n

i : A!(n−1)∂ni →

A!n, for 1 ≤ i ≤ (n− 1), which apply ∆ at position i. (Thus ∆ = ∆21.)

These natural transformations induce coface maps on!

nA!n in the

following manner. Since γn−1 ∂ni = γn and ∂n

i is epi, transformationsB γn → A!(n−1) ∂n

i are equivalent to transformations Bγn−1 → A!(n−1)

(where B : Σ∗ → C is some symmetric sequence). Therefore there is anequality of right Kan extensions Rγn

"A!(n−1) ∂n

i

#= Rγn−1

"A!(n−1)

#=

A•(n−1). (We will make extensive use of this equality in later sectionswithout further comment.)

86 B. Walter

Define ∆ni : A•(n−1) → A•n to be the following map.

(2)A•(n−1) A•n

Rγn

!A!(n−1) ∂n

i

" Rγn (∆ni )

Rγn

!A!n

"

Under right Kan extension, Diagram (1) translates to the followingdiagram of symmetric sequences.

(3)

A • A∆3

1

A

∆

∆

A •A •A

A • A∆3

2

Combined with Proposition 2.10, this generalizes to the following.

Proposition 3.10. Let (A, ∆) be a symmetric sequence with cocompo-sition. Then the transformation ∆n

i : A•(n−1) → A•n equalizes the twotransformations

∆n+1i , ∆n+1

i+1 : A•n⇒ A•(n+1).

More generally, ∆n+1j ∆n

i = ∆n+1i ∆n

j−1 for j > i.

Corollary 3.11. Let (A, ∆) be a symmetric sequence with cocomposi-tion. There are canonical, unique maps ∆[n] : A → A•n. (Given bytaking any chain of compositions ∆n

in· · ·∆1

i1.)

3.4 Counit and Codegeneracies

Write 1 for the functor 1 : Σ∗ → C given by

1(T ) =

#1⊗ if |T | = 1,

⋆C otherwise.

We will call 1 the “counit” symmetric sequence. [The dual definition ofthe “unit” symmetric sequence would use ∅C .]

Definition 3.12. A counital symmetric sequence is (A, ϵ) where A isa symmetric sequence and ϵ is a natural transformation to the counitϵ : A → 1.

Note that being counital is equivalent to the existence of a mapA(1) → 1⊗. We will not require the map A(1) → 1⊗ to be equippedwith a section. In the next subsection, we will use the following basicequality whose proof can be read off of Figure 5.

Cooperads as Symmetric Sequences 87

Lemma 3.13. The following functors Σ∗ → C are equal.

!1!A

"s10 = A =

!A! 1

"s11.

More generally, the following functors Σ≀n∗ → C are equal.

#!A!i

"! 1!

!A!(n−i)

"$sni = A!n

In the footsteps of Lemma 3.13 we define the following generaliza-tion.

Definition 3.14. Given a counital symmetric sequence (A, ϵ) defineassociated natural transformations ϵni : A!(n+1)sni → A!n, for 0 ≤ i ≤n, to be the following compositions.

A!(n+1)sni A!n

#!A!i

"!A!

!A!(n−i)

"$sni

(Id!ϵ!Id) sni#!

A!i"! 1!

!A!(n−i)

"$sni

Define ϵ00 = ϵ : A → 1.

These natural transformations induce codegeneracies in the follow-ing manner. Since γn+1 sni = γn, the universal transformation

A•(n+1) γn+1 → A!(n+1)

induces a transformation A•(n+1) → Rγn

!A!(n+1) sni

". Define ϵni :

A•(n+1) → A•n to be the following composition.

(4)A•n

A•(n+1) Rγn

!A!(n+1) sni

" Rγn (ϵni )

Rγn

!A!n

"

Similar to Proposition 3.10, the corresponding properties of sni implythe following.

Proposition 3.15. Let (A, ϵ) be a counital symmetric sequence. Thenthe transformation ϵn−1

i : A•n → A•(n−1) coequalizes the two transfor-mations ϵni , ϵ

ni+1 : A•(n+1) ⇒ A•n. More generally ϵn−1

i ϵnj = ϵn−1j−1 ϵ

ni for

j > i.

88 B. Walter

3.5 Cooperads and Cosimplicial Structure

Definition 3.16. A cocomposition operation on a counital symmetricsequence respects the counit if the following diagram of natural trans-formations Σ∗ → C commutes.

(5)

Aγ2s10∆s10 (A!A)s10

(ϵ!Id)s10 (1!A)s10 =

A Id

=

=

A

Aγ2s11∆s11

(A!A)s11 (Id!ϵ)s11(A! 1)s11 =

A counital cooperad is a counital symmetric sequence with cocom-position which respects the counit.

By applying Proposition 2.10 and using the simplicial structure ofwreath product categories, the requirement in Definition 3.16 implies amore general statement.

Proposition 3.17. If (O, ∆, ϵ) is a cooperad, then the following com-position is equal to the identity IdO!n , for j = (i− 1), i.

O!n = O!n ∂n+1i snj

∆n+1i snj

−−−−−−→ O!(n+1) snjϵnj

−−→ O!n

Furthermore, the following compositions are equal if j < i− 1.

O!n (∂n+1i snj )

∆n+1i snj

O!(n+1) snjϵnj

O!n

O!n (sn−1j ∂n

i+1)ϵn−1j ∂n

i+1

O!(n−1)∂ni+1

∆ni+1

O!n

as well as the similar statement for j > i.

We have now almost completed the proof of the following.

Theorem 3.18. If (O, ∆, ϵ) is a cooperad, then the collection O•nnalong with coface maps ∆n

i and codegeneracy maps ϵni defines a coaug-mented cosimplicial symmetric sequence with two extra codegeneracies.

(6) O O • O O•3 O•4 · · ·

Cooperads as Symmetric Sequences 89

Proof. In Propositions 3.10 and 3.15, we have already shown the cosim-plicial identities ∆n+1

j ∆ni = ∆n+1

i ∆nj−1 and ϵn−1

i ϵnj = ϵn−1j−1 ϵ

ni .

It remains only to consider the compositions ∆n+1i ϵnj . These come

from the right Kan extension over γn of the statements of Proposi-tion 3.17. Note that the right Kan extension

Rγn

!O!n ∂n+1

i snj∆n+1

i snj−−−−−−→ O!(n+1) snj

"

is equal to the composition

Rγn+1

#O!n ∂n+1

i

$ ∆n+1i−−−−−→ Rγn+1

#O!(n+1)

$−→ Rγn

#O!(n+1) snj

$.

Corollary 3.19. There are unique transformations ∆[n] : O → O•n.These are equal to any combination of parenthesization maps and co-composition maps from their source to their target.

4 Cooperads via • versus !

We will now connect the cooperad structures defined in the previoussections with the classical methods which would attempt to use onlythe induced product • on symmetric sequences. When using •, the lackof associativity introduces extra “parenthesization” maps, which mustbe dealt with carefully.

4.1 Parenthesization Maps.

From now on, let A,B,C be generic symmetric sequences and (O, ∆, ϵ)be a generic counital cooperad.

Proposition 4.1. There are canonical “parenthesization” natural trans-formations:

(A •B) • C

A •B • C

A • (B • C)

More generally there are parenthesization maps to A1 •· · · •An from anyparenthesization of this expression.

90 B. Walter

Proof. We show the existence of the map (A •B) •C → A •B •C. Theother maps are similar.

The universal natural transformation (A•B) γ2 −→ (A!B) inducesa natural transformation of functors (Σ∗ ≀ Σ∗) ≀ Σ∗ −→ C:

!(A •B)! C

"∂31 −→ (A!B)! C = A!B ! C.

The desired map is induced by taking the right Kan extension Rγ3 ofthe diagram above.

(A •B) • C A •B • C

Rγ3

#!(A •B)! C

"∂31

$Rγ3(A!B ! C)

Remark 4.2 (On the associativity of •). Without making further as-sumptions, it is not true that (A • B) • C ∼= A • B • C ∼= A • (B • C).This would follow from the existence of natural equivalences

!Rγ2(A!

B)"! C ∼= R∂3

1(A ! B ! C) as well as the corresponding equivalence

using ∂32 . However, this will generally only occur in the unlikely event of

the symmetric monoidal product commuting with categorical products.The situation contrasts starkly with that of the operad composition

product, defined dual to • using left rather than right Kan extensions. IfC is a closed monoidal category, then ⊗ is a left adjoint, so it will in par-ticular commute with categorical coproducts and left Kan extensions. Inthis case the parenthesization maps for the operad composition productare isomorphisms and the operad composition product is associative.

In practice, authors have generally dealt with this in the past by ei-ther not using cooperads at all, or by (implicitly or explicitly) restrictingtheir categories so that (· · · )Σn = (· · · )Σn and

%=

&. In this (very

special case) A •B = A B and there is no problem. Alternately, heavyrestrictions can be placed on C and/or on the category of symmetric se-quences to force ⊗ to commute with

%. For example, in the category of

finitely generated, injective bimodules over a commutative ring, ⊗ = ⊗which is a right adjoint.

Proposition 4.3. Parenthesization maps are associative.

For example the following diagrams commute.

(7)

(A •B • C) •D!(A •B) • C

"•D A •B • C •D

(A •B) • C •D

Cooperads as Symmetric Sequences 91

(8)

(A •B) • C •D

(A •B) • (C •D) A •B • C •D

A •B • (C •D)

Proof of 4.3. It is enough to consider Diagrams (7) and (8). Commu-tativity is shown by writing the diagrams as right Kan extensions. Thediagrams above are Rγ4 of the following diagrams of functors Σ≀4

∗ → C.(7’)

!(A •B • C)!D

"(γ3 ≀ Id)

#!(A •B) • C

"!D

$(γ3 ≀ Id) A!B ! C !D

!(A •B)! C !D

"∂41

(8’)!(A •B)! C !D

"∂41

!(A •B)! (C •D)

"(γ2 ≀ Id ≀ γ2) A!B ! C !D

!A!B ! (C •D)

"∂43

Diagram (7’) is just − !D applied to the following universal diagram(in which the upper-left map is Rγ3 of the lower-right).

(7”)

(A •B • C) γ3

!(A •B) • C

"γ3 A!B !C

!(A •B)! C

"∂31

Diagram (8’) commutes because the upper and lower composition areboth equal to

!(A •B)! (C •D)

"(γ2 ≀ Id ≀ γ2)

ι1!ι2−−−−−−→ (A!B)! (C !D)

Where ι1 : (A • B) γ2 → A ! B and ι2 : (C •D) γ2 → C ! D are theuniversal natural transformations from their respective Kan extensions.

4.2 Cooperad Structures

We relate parenthesization maps with cooperad structure. By the func-toriality of !, there are natural transformations

Id!∆ : A!O → A! (O •O) and ∆! Id : O !A → (O • O)!A,

92 B. Walter

where A is any symmetric sequence. Define the maps Id •∆ and ∆ • Idto be the natural transformations induced on right Kan extensions viafunctoriality of Kan extension. For example

∆ • Id = Rγ2

!∆! Id

": O • A −→ (O • O) • A.

By alternately letting A be a parenthesization of O•k and using functo-riality of • this defines maps originating in any parenthesization of O•n.For example

!(Id • Id) •∆

"• Id :

!(O •O) • O

"• O −→

!(O •O) • (O • O)

"• O.

Theorem 4.4. The following diagrams commute (unlabeled maps areparenthesization):

(O •O) • O

O • O

∆•Id

∆31

O • O • O

O • (O • O)

O •O

Id•∆

∆32

O • O • O

More generally, parenthesization maps convert Id•∆•Id (and its paren-thesizations) to ∆4

2, etc.

Proof. We show the first diagram commutes. The second diagram andmore general statement are proven the same.

Consider the diagram below, where maps marked ι are all universaltransformations of right Kan extensions (RFX)F

ι−→ X.

(9)!(O •O) • O

"γ3

ι ∂31

1⃝2⃝

!(O • O)!O

"∂31

ι!Id!O •O

"γ3

(∆•Id) γ3

ι ∂31

∆31 γ

3

(O !O) ∂31

∆!Id

(∆!Id) ∂31

3⃝

(O • O • O) γ3 ι O !O !O

Parallelograms 1⃝ and 3⃝ commute by functoriality of right Kan ex-tension. The left side of parallelogram 1⃝ is Rγ2 of the right side, andthe left side of parallelogram 3⃝ is Rγ3 of the right side. Triangle 2⃝

commutes by functoriality of ! (recall that ι∆ = ∆).

Applying Rγ3 along the outside of Diagram (9) yields the following

Cooperads as Symmetric Sequences 93

(where the map labeled ∗ is the parenthesization map):

(10)

(O • O) • O = (O •O) • O

∗O •O

∆•Id

∆31 O • O • O = O •O • O

Example 4.5. The following diagram is commutative (the unlabeledmaps are parenthesizations):

(O • O) • O(∆•Id)•Id !

(O • O) • O"• O

O • O • O ∆•Id•Id

∆41

(O • O) • O • O O • O • O • O

Theorem 4.4 has the following corollary:

Corollary 4.6. Commutativity of the following diagrams are equivalent.

(11)

(A!A)(γ2 ≀ Id) ∆≀Id

Aγ3∆

∆

A!A!A

(A!A)(Id ≀ γ2) Id≀∆

(12)

A • A ∆•Id (A • A) • A

A

∆

∆

A •A •A

A • AId•∆

A • (A • A)

Thus a cooperad could equivalently be defined as (O, ∆, ϵ) whereO is a symmetric sequence, ∆ : O → O • O so that the analog ofDiagram (12) commutes, and∆ is compatible with the counit ϵ : O(1) →1⊗.

5 Comodules and Coalgebras

Throughout this section, let (O, ∆O, ϵ) be a counital cooperad andM bea symmetric sequence. The definition of cooperad comodules mirrorsthat of coalgebra comodules (dual to algebra modules). The benefitof viewing cooperads as symmetric sequences is that coalgebras over acooperad can be viewed, essentially, as a special type of comodule.

94 B. Walter

5.1 Comodules

Definition 5.1. A left O-comodule is (M, ∆M ) whereM is a symmetricsequence and ∆M : M γ2 → O !M is compatible with ∂3

1 and ∂32 and

s10.

That is, the following diagrams (analogous to Diagrams (1) and (5))should commute.

(13)

(O !M) (γ2 ≀ Id) ∆!Id

M γ3

∆M

∆M

O !O !M

(O !M) (Id ≀ γ2) Id!∆M

(14) M γ2s10∆M s10 (O !M) s10

(ϵ!Id) s10 (1!M) s10 =

M Id

=

M

As with cooperads, we write ∆M for the induced universal trans-formation to the right Kan extension ∆M : M → O • M . There areinduced transformations ∆n+1

i :!O!(n−1) !M

"∂n+1i → O!n !M and

∆n+1i : O•(n−1) •M → O•n •M .

Theorem 5.2. Analogous to Theorem 3.18 there is a canonical coaug-mented cosimplicial complex as below.

M O •M O•2 •M O•3 •M · · ·

Corollary 5.3. There are unique transformations ∆[n]M : M → O•(n−1)•

M . These are equal to any combination of parenthesization maps andcocomposition maps from their source to their target.

Remark 5.4. Right O-comodules could be defined similarly. Howeverrecent experience suggests that right comodules are most interestingin the non-counital case; in which situation we should use partial co-composition products rather than cocomposition products. This movesbeyond the scope of the current work.

5.2 Coalgebras

Let a be an object of C and A be a symmetric sequence. Note that acan be viewed as a functor a : Σ0 → C. Recall the descriptions of the

Cooperads as Symmetric Sequences 95

category ∅n in Remarks 2.16 and 2.7. We may view ∅n either as the

category of level n trees with no leaves; or as Σ≀(n−1)∗ ⊂ Σ≀(n+1)

∗ , the fullsubcategory consisting of chains of set maps of the following form.

⋆f0←− S1

f1←− · · ·fn−2←−−− Sn−1

fn←− ∅

Note that the category Σ≀0∗ consists of only the trivial chain (⋆ ← ∅).

This is equivalent to Σ0.

The face and degeneracy maps of Σ≀(n+1)∗ induce the following face

and degeneracy maps on Σ≀(n−1)∗ . (We introduce an index shift below

so that ∂ni and snj map from ∅n = Σ

≀(n−1)∗ .)

!∂ni : Σ

≀(n−1)∗ → Σ

≀(n−2)∗ , for 1 ≤ i ≤ (n− 1), and n > 1

sni : Σ≀(n−1)∗ → Σ

≀n∗ , for 0 ≤ i ≤ n and n ≥ 1

The degeneracy map snn doubles ∅, recognizing that a tree without leaves

of level n is also of level (n+1). Note that ∂21 : Σ

≀1∗ → Σ0 coequalizes all

chains of face maps from Σ≀(n−1)∗ to Σ

≀1∗ . We write γn for the composition

γn = (∂2i2· · · ∂n

in).

Under the identification Σ≀n∗ ⊂ Σ≀(n+2)

∗ , Definition 3.2 of symmet-ric sequence composition restricts to a functor (A1 ! · · ·An−1 ! a) :

Σ≀(n−1)∗ → C. For example, A! a is given by the following.

"⋆

f0←− Sf1←− ∅

#(−→ A(S)⊗

$%

s∈S

a(∅)

&

= A(S)⊗ a⊗|S|

The right Kan extension of Definition 3.2 restricts to a right Kan ex-

tension over γn : Σ≀(n−1)∗ → Σ0, yielding the following functor.

A1 • · · · • An−1 • a = Rγn(A1 ! · · ·!An−1 ! a) : Σ0 −→ C

For example, (A • a) ='

k≥0

(A(k) ⊗ a⊗k

)Σk

.

Remark 5.5 (Completed tensor product). In categories with a com-pleted tensor product ⊗, the completed coendomorphisms !coend(A) =*

nHom(A, A⊗n) form a cooperad. Cocomposition is dual to the com-position operation on the operad end(A∗). Dualizing the classical alge-bra definition, coalgebras should be equivalent to objects equipped with

96 B. Walter

cooperad maps !coend(A) → O. The completed tensor should be a right

adjoint to Hom so that this is equivalent to maps A → O(n) ⊗A⊗n. Inthis case, the definition of coalgebras above should use ⊗ rather than⊗ (though ⊗ should still be used for the cooperad). A more detailedsurvey of this issue is beyond the scope of the current work, which isintended to focus on cooperads.

Definition 5.6. A coalgebra over the cooperad (O, ∆, ϵ) is (c, ∆c) wherec is an object of C and ∆c : c γ2 → O ! c is compatible with face maps∂21 = (γ2 ≀ Id), ∂2

2 = (Id ≀ γ2) and degeneracy s10.

That is, the following diagrams (analogous to Diagrams (13) and(14)) should commute.

(15)

(O ! c)(γ2 ≀ Id) ∆!Id

c γ3

∆c

∆c

O !O ! c

(O ! c)(Id ≀ γ2) Id!∆c

(16) c γ2s10∆c s10 (O ! c) s10

(ϵ!Id) s10 (1! c) s10 =

c Id

=

c

Statements and proofs about left comodules translate into state-ments and proofs about coalgebras by converting ∂n

i , sni into ∂n

i , sni .

Essentially, coalgebras are left comodules which are concentrated in 0-arity. Write ∆c for the induced map (in C) ∆c : c → O • c. As withcomodules we have ∆n+1

i :!O!(n−1) ! c

"∂n+1i → O!n ! c inducing

∆n+1i : O•(n−1) • c → O•n • c.

Theorem 5.7. The comultiplication ∆c defines a canonical coaugment-ed cosimplicial complex (in C)

c O • c O•2 • c O•3 • c · · ·

Corollary 5.8. There are unique C-maps ∆[n] : c −→ O•(n−1)•c. Theseare equal to any combination of parenthesization maps and cocomposi-tion maps from their source to their target.

Cooperads as Symmetric Sequences 97

6 Examples

We end with a two simple examples of cooperads which are not duals ofstandard operads. Both of these are constructed via quotient/contrac-tion operations. The (directed) graph cooperad is used in [9] and thecontractible ∆ complex operad is a generalization.

6.1 The Graph Cooperad

Given a finite set S, a contractible S-graph is a connected, acyclic graphwhose vertex set is S. The unoriented graph cooperad has gr(S) equalto the free Z module generated by all contractible S-graphs. The co-composition natural transformation ∆ : gr γ2 → gr ! gr is defined asfollows.

Given two graphs G and K, a quotient map of graphs q : G " Kis a surjective map from vertices of G onto vertices of K such thatq(v1, v2) =

!q(v1), q(v2)

"defines a map sending edges of G to edges and

vertices (if q(v1) = q(v2)) of K, surjecting onto the edges. Note thatif q : G " K is a quotient map and v is a vertex of K, then q−1(v)is a subgraph of G. A graph contraction is a quotient map where eachq−1(v) is a connected subgraph. Note that there is a bijection betweenthe edges of G and the edges of K union those of the q−1(v).

Suppose G is an S-graph and f : S " T is a surjection of sets.Given t ∈ T , let f−1(t) be the maximal subgraph of G supported bythe vertices of f−1(t). We say that f induces a graph contraction on Gif f−1(t) is contractible for each t. In this case, we define the inducedcontracted graph (G/f) to have vertices T with an edge from vertex t1to t2 if there is an edge in G from the subgraph f−1(t1) to the subgraphf−1(t2).

Cocomposition ∆ takes the element!T

f←− S

"of Σ∗ ≀Σ∗ to the map

gr(S) −→ gr(T )⊗#$

t∈T

gr(f−1(t))%

which takes a S-graph G to (G/f)⊗!&

t∈T f−1(t)"if f defines a graph

contraction on G, and sends G to 0 otherwise. Since the quotient opera-tion described previously is clearly associative, this defines a symmetricsequence with cocomposition. The counit map sends S-graphs with onlyone vertex to 1 ∈ Z and kills all others.

The (directed) graph cooperad is similar to the unoriented graphcooperad. In the category of directed, contractible S-graphs define

98 B. Walter

−gr(S) = gr(S)/ ∼, where ∼ identifies reversing the orientation of anedge with multiplication of a graph by −1. Cocomposition on gr givesa well-defined map on −gr since reversing an arrow in G will reverseexactly one arrow either in the quotient graph G/f or in one of thef−1(t).

Free nilpotent coalgebras over the graph cooperad can be writtenas free Z modules generated by all (finite) graphs with vertices labeledby the primitive elements. The cooperad structure operates by ripping

out subgraphs. Explicitly ∆ takes!T

f←− ∅

"to the map which takes a

graph G to the sum of terms (G/f)⊗!#

t∈T f−1(t)"taken over all set

maps f : Vert(G) → T . [With the convention that G/f = 0 if f doesnot define a graph contraction on G.] Note that

∆(Tf←− ∅)

will kill a graph G for all sets T with |T | > |Vert(G)| (because graphcontraction maps cannot increase the number of vertices). Thus thiscoalgebra is nilpotent. We will leave the full proof that this is a freenilpotent coalgebra for the sequel.

The graph cooperad generalizes to the following.

6.2 The CDC Cooperad

By a ∆-complex, we mean what Hatcher [8, Appendix] calls a “singu-lar ∆-complex” or “s∆-complex”. Essentially this is a CW complexwhose cells are all (oriented) simplices and whose attaching maps factorthrough face maps of the simplex. Given a set S, an S∆-complex is a∆-complex whose 0-cells are labeled by elements of S. The CDC coop-erad has cdc(S) equal to the free Z module generated by contractibleS∆-complexes. Cocomposition is defined similar to that for gr.

If T is a subset of the 0-cells of a ∆-complex X, write T for themaximal CW subcomplex of X supported by T . Quotient maps for∆-complexes are CW quotient maps. We say a quotient map X ! Yis a contraction if the inverse image of each 0-cell of Y is a contractiblesubcomplex of X. If X is a S∆-complex then a set surjection f : S ! Tinduces a CW contraction on X if f−1(t) is contractible for each t ∈ T .In this case, we define (X/f) to be the quotient of X by the sub CW-

complexes f−1(t). The cocomposition map of cdc takes (Tf←− S) to

the map which sends the S∆-complex X to (X/f) ⊗!#

t∈T f−1(t)"if

f induces a CW contraction on X and 0 otherwise.

Cooperads as Symmetric Sequences 99

Acknowledgments

I would like to thank Dev Sinha, whose questions led to the incep-tion of this work; as well as Michael Ching who resolved many of myearly confusions. Also Clemens Berger, Bruno Vallette, and Jim Mc-Clure listened to early versions of these ideas and provided invaluablefeedback. Most of all, I must thank Kallel Sadok and the MediterraneanInstitute for Mathematical Sciences (MIMS) for an invitation to speakat the conference on “Operads and Configuration Spaces” in June 2012,which led me to finally revising and clarifying these ideas which havebeen on paper and bouncing around in my head for almost six years.This work is based on the notes from my series of talks at MIMS.

Benjamin Walter

Mathematics Research and Teaching Group,

Middle East Technical University,

Northern Cyprus Campus,

Kalkanli, Guzelyurt, KKTC

via Mersin 10, Turkey

benjamin@metu.edu.tr

References

[1] Mahowald Arone; Greg; Mark, The Goodwillie tower of the iden-tity functor and the unstable periodic homotopy of spheres, Invent.Math. 135 (1999), 743–788.

[2] Clemens B., Iterated Wreath Product of the Simplex Categoryand Iterated Loop Spaces, arXiv (2005).

[3] Block R., Recognizable formal series on trees and cofree coalgebraicsystems, Journ. Alg. 215 (1999), 543–573.

[4] Ching M.,Bar constructions for topological operads and the Good-willie derivatives of the identity, Geom. Topol. 9 (2005), 833–933.

[5] Ching M., A note on the composition product of symmetric se-quences, arXiv:math/0510490v2 (2012).

[6] Fox T., The construction of cofree coalgebras, JPAA 84 (1993),191–198.

[7] Hazewinkel M., Cofree coalgebras and multivariable recursiveness,JPAA 183 (2003), 61–103.

100 B. Walter

[8] Hatcher A., Algebraic Topology, Cambridge Univ. Press, (2002).

[9] Sinha D.; Walter B., Lie coalgebras and rational homotopy theory,I: Graph coalgebras, Homology, Homotopy and Applications 13:2, (2011), 1–30.

[10] Sinha D.; Walter B., Lie coalgebras and rational homotopy theory,II: Hopf invariants, Trans. Amer. Math. Soc 365: 2 (Feb. 2013),861–883.

[11] Smith J., Cofree coalgebras over operads, Top. and Appl., 133(2003), 105–138.

[12] Walter B.,Cofree coalgebras over cooperads, in preparation

[13] Walter B., Lie algebra configuration pairing, arXiv:1010.4732,(2010).

Morfismos, Vol. 17, No. 2, 2013, pp. 101–125

Moduli spaces and modular operads

Jeffrey Giansiracusa 1

Abstract

We describe a generalised ribbon graph decomposition for a broadclass of moduli spaces of geometric structures on surfaces (withnonempty boundary), including moduli of spin surfaces, r-spinsurfaces, surfaces with a principle G-bundle, surfaces with mapsto a background space, surfaces with Higgs bundle, etc.

2010 Mathematics Subject Classification: 57M50, 57M15, 18D50, 18D05,58D27.Keywords and phrases: Ribbon graphs, moduli spaces, mapping classgroup, arc complex, 2-categories, cyclic operads.

1 Introduction

This paper is an expansion of some ideas that I first talked about in2012 in the MIMS conference on Operads and Configuration Spaces.Here I shall give a more detailed account, though still not a completeone, of a certain theorem about modular envelopes. The full details willappear in a future paper; in this note I will try to be expository andfocus on illuminating the central ideas without being overly concernedby technical details that might otherwise obscure some of the conceptualclarity of the arguments.

Fix a class ψ of geometric structures on surfaces. For example, onecould take orientations, principal G-bundles, or spin structures, etc.Associated to any surface Σ is the space ψ(Σ) of all such structures onthat surface. Taking the homotopy quotient by the diffeomorphism groupyields a homotopy theoretic moduli space of surfaces with ψ-structure.If we consider surfaces with some marked intervals along the boundary,and ψ-structures that have a fixed value on each marked interval, thenwe can glue the intervals together and the result is a modular operad.

101

102 Jeffrey Giansiracusa

denoted Mψ. (If, instead of a single fixed value on the intervals, weallow one of several fixed valued then the result is instead a colouredmodular operad). These moduli spaces are the objects we wish to study.The idea of this work is to decompose them, in a sense, into modulispaces of discs with ψ-structure. The modular operad Mψ contains asub-cyclic operad Dψ of moduli spaces of discs with ψ-structure. Ourmain result is that Mψ is freely generated (in a homotopical sense) as amodular operad over this sub-cyclic operad. I.e., the derived modularenvelope of Dψ is weakly equivalent to Mψ.

This result was inspired by the work of Costello. As part of hisgroundbreaking work in the homotopy theory of open-closed topologicalfield theories [9], he gave a new perspective on the very important ideaof describing the moduli space of Riemann surfaces with ribbon graphsin [8, 10]. He proved that the derived (i.e., homotopy invariant) modularenvelope of the associative operad gives a model for the modular operadof moduli spaces of Riemann surfaces with open-string type gluing forthe compositions. A point in this modular envelope can be described asa graph equipped with lengths on all of its edges and a cyclic order ofthe edges incident at each vertex — i.e., a metric ribbon graph. Thusthe moduli space of ribbon graphs is equivalent to the moduli space ofRiemann surfaces.

Costello’s proof used geometry and analysis on a certain partialcompactification of the moduli space of Riemann surfaces. Thus itappears his argument is not suited to more homotopy theoretic contextssuch as the one considered in this paper. In [13], I gave a different proofof Costello’s modular envelope theorem. This proof instead rested onthe well-known contractibility of the arc complex of a surface. This newargument lead to an adaptation to dimension 3: the derived modularenvelope of the framed little 2-discs is equivalent to the modular operadof moduli spaces of 3-dimensional handlebodies.

Here we instead focus of refining and generalising the argument of[13] in dimension 2. When the structures being considered are principalG-bundles then we expect this result will lead to a G-equivariant versionof Costello’s open-closed TFT theorem.

2 Operads

A cyclic operad in C is a functor P from the category of finite sets andbijections to C together with composition maps

Moduli spaces and modular operads 103

P(I)⊗ P (J)ij→ P(I # J ! i, j),

for i ∈ I and j ∈ J , satisfying an associativity condition and natural in(I, i) and (J, j). One can think of P as a collection of abstract “electricalcircuit components,” where P(I) as a set/space of components withterminals given by the set I. The composition maps correspond to wiringterminals together to produce new components; terminals can only beglued in pairs (no trivalent connections) and in a cyclic operad twocomponents can only be glued together in at most one place. Allowingmultiple gluings leads to the following definition.

A modular operad in C is a cyclic operad Q together with natural

self-composition maps Q(I)i,j→ Q(I ! i, j) that commute with the

cyclic operad composition maps and with each other.

Example 2.1. 1. The commutative modular operad is the constantfunctor sending each finite set to a point.

2. The associative cyclic operad Assoc sends I to the set of cyclicorders on I.

We will need a slight generalisation in which there are different typesof terminals and two terminals can only be connected if they are of thesame type. The types are called colours. Fix a set Λ, which we willcall the set of colours. A Λ-coloured set I consists of a finite set witha map to Λ. A morphism I → I ′ of coloured sets is a bijection thatrespects the colours. A coloured cyclic operad P is a functor from thecategory of coloured sets to C together with a collection of compositionmaps ij as before, but now only defined when i and j have the samecolour. A coloured modular operad is defined analogously, where theself-composition maps ij also only defined when i and j have the samecolour.

2.1 Homotopy theory of cyclic and modular operads

Berger and Batanin [7] have recently constructed fully satisfactoryQuillen model category structures on cyclic and modular operads. Whentalking about derived constructions such as the derived modular envelope,one could work with the model category structures. However, we take amore pragmatic approach, since the modular envelope is the only functorwe ever have to derive, and our construction of the derived functor willbe manifestly homotopy invariant due to the homotopy invariance of

104 Jeffrey Giansiracusa

homotopy colimits. We need only the following definition. A morphismP → P ′ of cyclic or modular operads in spaces is a weak equivalence ifeach map of spaces P(I) → P ′(I) is a weak equivalence.

3 Category theory and homotopy theory

3.1 The nerve of a category

Let C be a category, which we assume is small, meaning that the objectsform a set rather than a class. (E.g., The category of all sets is notsmall, but the category of all subsets of a fixed big set is small.) Wecan associate a simplicial set (and hence a space) with C via the nerveconstruction.

The nerve of C , denoted N(C ) is the simplicial set whose 0-simplicesare the objects of C , 1-simplices are the morphisms of C , 2-simplicesare the 2-simplex shaped diagrams in C

X0

X1 X2,!!

f""

gf

##g

and so on. In general, the n-simplices N(C )n are the set of composablen-tuples of morphisms,

X0f1→ X1

f2→ · · · fn→ Xn.

We will write BC for the geometric realisation of the nerve.It is easy to see that a functor F : C → D induces a map of nerves

N•C → N•D . A natural transformation F → F ′ induces a homotopybetween the corresponding maps. From this it follows that an adjointpair (F,G) induces a homotopy equivalence of nerves.

If a category C has an initial object u then there is a naturaltransformation form the constant functor with value u to the identity,and so the nerve of C is contractible. Likewise, existence of a final objectimplies contractibility of the nerve.

3.2 The fundamental group of the nerve of a category

While computing the higher homotopy groups of a space is usually verydifficult, there is a convenient recipe for computing the fundamentalgroup of the nerve of a category.

Moduli spaces and modular operads 105

Given C , let C [C−1] denote the category formed by adjoining inversesto all of the arrows in C (see [6, §1.1]) This localisation is clearly agroupoid (all morphisms are invertible). If two objects of a groupoidlie in the same connected component then their automorphism groupsare isomorphic (by an isomorphism that is unique up to conjugation).Given an object x of C , the fundamental group of NC based at x iscanonically isomorphic to the automorphism group of x in the groupoidC [C−1].

3.3 Strict 2-categories

A strict 2-category is a category enriched in Cat . I.e., it consists of aclass of objects ObjC , a category HomC (a, b) for each pair of objects,and composition functors

HomC (a, b)×HomC (b, c) → HomC (a, c)

that are strictly associative and for which a unit exists in HomC (a, a).The objects of the hom categories are called 1-morphisms and themorphisms of the hom categories are called 2-morphisms.

Example 3.1. Let T op2 denote the strict 2-category whose objects arespaces, and for which Hom(X,Y ) is the groupoid of maps and homotopyclasses of homotopies. For example, the groupoid of morphisms froma point to a circle is equivalent to the group Z (i.e., one object andautomorphism group Z).

Example 3.2. Let Cat2 denote the strict 2-category whose objects aresmall categories and whose hom categories are the categories of functorsand natural transformations.

Example 3.3. Since a set can be considered as a category with nonon-identity morphism, an ordinary category can be considered as a2-category in which there are no non-identity 2-morphisms.

Remark 3.4. In this paper, all strict 2-categories that arise will havethe property that all 2-morphisms are in fact isomorphisms. Such a2-category is sometimes called a (2,1)-category. Strict 2-categories are arestricted class of 2-categories. More generally, one often wants to workwith weak or lax 2-categories, where the associativity and unit conditionsonly hold up to natural transformations (which must then satisfy someconditions). We will have no need of these more sophisticated notions inthis paper.

106 Jeffrey Giansiracusa

A strict 2-functor between strict 2-categories F : C → D is a mapF : ObjC → D together with a functor Hom(a, b) → Hom(F (a), F (b))for each pair of objects such that these functors are strictly compatiblewith the composition functors. We will usually abbreviate and call thisa functor.

We will leave it as an exercise to spell out precisely what a naturaltransformation between strict 2-functors is.

A strict 2-category C has a nerve N(C ) that is a bisimplicial set.It is constructed as follows. First one replaces all the hom categorieswith their nerves to obtain a simplicial category. Then the nerve of thissimplicial category yields a bisimplicial set that is the nerve of C . Asin the case of 1-categories, a strict 2-functor induces a map of nerves,and a natural transformation induces a homotopy. In particular, observethat if F : C → Cat2 is a strict 2-functor then taking the realisation ofthe nerve pointwise yields a strict 2-functor C → T op2.

3.4 Over categories and Quillen’s Theorems A and B

Let F : A → B be a functor. Given an object b ∈ B, one can define acategory of objects in A over b. This is denoted F ↓ b (or B ↓ b whenF is the identity functor) and is called the over category of F basedat b (some people instead call it the comma category). Its objects arepairs consisting of an object a ∈ A and a morphism g : F (a) → b in

B. A morphism from F (a)g→ b to F (a′)

g′→ b consists of a morphismh : a → a′ in A such that the diagram

F (a) F (a′)

b!!g

""F (h)

## g′

in B commutes. Observe that a morphism f : b → b′ in B induces afunctor f∗ : (F ↓ b) → (F ↓ b′). There is also a canonical projectionfunctor (F ↓ b) → A given by forgetting the morphism to b.

Over categories can be thought of as a category-theoretic analogueof homotopy fibres. In fact, Quillen’s Theorems A and B are instancesof this analogy. If the homotopy fibre of a map is contractible then themap is a weak equivalence.

Theorem 3.1 (Quillen’s Theorem A). Let F : A → B be a functorand suppose that for each object b of B the nerve of the over categoryF ↓ b is contractible. Then F induces a weak equivalence of nerves.

Moduli spaces and modular operads 107

This is a special case of a more general theorem that allows one toidentify the homotopy fibre of a map of nerves induced by a functor.

Theorem 3.2 (Quillen’s Theorem B). Let F : A → B be a functor andsuppose for each morphism f : b → b′ in B the corresponding functorf∗ : (F ↓ b) → (F ↓ b′) induces a weak equivalence of nerves. Then(F ↓ b) → A → B is a homotopy fibre sequence.

The construction of over categories and Quillen’s Theorems A and Bhave extensions to strict 2-categories. See [1] and [2] for details. Givena strict 2-functor F : A → B and an object x ∈ ObjB, there is an over2-category (F ↓ x). It objects are pairs (a ∈ ObjA, f : F (a) → x). A

morphism from F (a1)f1→ x to F (a2)

f2→ x consists of a morphism g :a1 → a2 in A and a 2-morphism in B from f1 to f2 F (g). A 1-morphismx → x′ in B induces a strict translation 2-functor (F ↓ x) → (F ↓ x′).

Theorem 3.3 (Theorem B for 2-categories,[2]). If all of the translationfunctors induce homotopy equivalences then

N(F ↓ x) → N(A) → N(B)

is a homotopy fibre sequence for any object x.

3.5 Left Kan extension

Let A ,B,C be categories. Given a functor f : A → B, precompositionwith f sends a functor B → C to a functor A → C . This defines afunctor

f∗ : Fun(B,C ) → Fun(A ,C ).

It turns out that this f∗ admits a left adjoint f!, which is called the leftKan extension.

3.6 Left Kan extensions and homotopy left Kan exten-sions

Let A , B and C be categories with C cocomplete. Consider functors

A C

B.

!!F

""G

108 Jeffrey Giansiracusa

Recall that the left Kan extension of F along G is a functor

G!F : B → C

defined on objects by the colimit

G!F (b) = colim(G↓b)

F jb,

where (G ↓ b) is the comma category of objects in C over b and jb : (G ↓b) → C forgets the morphism to b (to simplify the notation we will oftenomit writing jb). Left Kan extensions possess a universal property: thefunctor G!F comes with a natural transformation

F ⇒ G!F P

that is initial among natural transformations from F to functors factoringthrough P .

If C is a Quillen model category (such as topological spaces or chaincomplexes) then there is a homotopy invariant (or, derived) versionknown as the homotopy left Kan extension LG!F ; it is given by theformula

LG!F (b) = hocolim(G↓b)

F jb.

This construction is homotopy invariant in the following sense: a naturaltransformation F → F ′ that is a pointwise homotopy equivalence in-duces a natural transformation LG!F → LG!F ′ that is also a pointwisehomotopy equivalence. In fact, this is the left derived functor of left Kanextension with respect to the projective model structure on the functorcategories.

There is a homotopy coherent version of the universal property forhomotopy left Kan extensions. See [5, Proposition 6.1] for the details.

Note that there is a “Fubini theorem” for both ordinary and homotopycolimits,

colimA

F ∼= colimB

G!F and hocolimA

F≃→ hocolim

BLG!F.

3.7 Homotopy colimits, the Grothendieck constructionand Thomason’s Theorem

At several points we shall be taking homotopy colimits of diagrams inT op obtained from diagrams in Cat by applying the classifying space

Moduli spaces and modular operads 109

functor B (i.e. geometric realisation of the nerve) pointwise. Here webriefly recall a couple of useful tools for this situation.

Given a functor F : C → Cat , the Grothendieck constructionon F , denoted

!C F is the category in which objects are pairs (x ∈

C , y ∈ F (x)), and a morphism (x, y) → (x′, y′) consists of an arrowf ∈ homC (x, x′) and an arrow g ∈ homF (x′)(f∗y, y

′). This constructionsatisfies an associativity condition: if F : C → Cat and G :

!C F → Cat

are functors then sending c ∈ ObjC to!F (c)G defines a functor

!F G :

C → Cat and there is a natural equivalence of categories

"!

C FG ≃

"

C

#"

FG

$.

Thomason’s Theorem [11, Theorem 1.2] asserts that there is a naturalhomotopy equivalence,

hocolimC

BF≃−→ B

#"

CF

$.

As a special case, if C is actually a group G (a category with a singleobject ∗ and all arrows invertible), then BF (∗) is a space with a Gaction, and B(

!G F ) is homotopy equivalent to the homotopy quotient

(BF (∗))hG.If C = ∆op

semi then F is a semi-simplicial category, BF is a semi-simplicial space, and B(

!∆op

semiF ) ≃ hocolimBF is equivalent to the

geometric realisation of this semi-simplicial space.There is a 2-categorical version of the above. First of all, given a

strict 2-category C and a strict 2-functor F : C → Cat2 there is aGrothendieck construction that produces a strict 2-category

!C F over

C . An object of!C F is a pair (x ∈ ObjC , y ∈ ObjF (x)). A 1-morphism

(x, y) → (x′, y′) is a pair (f1, f2). where f1 : x → x′ is a 1-morphismin C and f2 : F (f1)(y) → y′ is a morphism in F (x′). A 2-morphism(f1, f2) → (g1, g2) consists of a 2-morphism α : f1 ⇒ g1 in C (whichgives a natural transformation α∗ from F (f1) to F (g1)) such that thediagram (in F (x′))

F (f1)(y) y′

F (g1)(y)!!

α∗

""f2

##

g2

110 Jeffrey Giansiracusa

commutes. This 2-categorical Grothendieck construction satisfies the ob-vious analogue of the associativity condition satisfied by the 1-categoricalconstruction. Also, a 2-functor C → T op2 has a homotopy colimit, anda 2-categorical version of Thomason’s theorem holds [2]: if F : C → Cat2

is a 2-functor then

hocolimC

BF ≃ B

!

CF.

We again refer the reader to [2] and the references there for furtherdetails.

3.8 Graphs and Costello’s graph category

For us, a graph Γ will consist of a set V of vertices, a set H of half-edges,an incidence map in : H → V , and an involution ι : H → H, calledthe edge flip, that specifies how the half-edges are glued together. Thefree orbits of ι are the edges of the graph, denoted E(Γ), so each edgeconsists of a pair of half-edges. The fixed points are called the legsand are denoted L(Γ). The incidence map sends each half-edge to thevertex that it meets. A graph Γ has a topological realisation |Γ| as a1-dimensional CW complex with a 0-cell for each vertex and a 1-cellfor each edge and leg. A graph is a tree if its topological realisation iscontractible, and a forest if it is a union of trees.

A corolla is a graph that consists of a single vertex and a number oflegs incident at it. If a graph is a disjoint union of corollas then the edgeflip map is the identity and so giving a union of corollas is equivalent togiving a triple (V,H, in : V → H). Associated with a graph Γ are twodisjoint unions of corollas. The first is given by forgetting the edge flipand is denoted ν(Γ) (cutting each edge into a pair of legs). The secondis denoted π0Γ; it has one vertex for each connected component of thegraph and one leg for each leg of the original graph Γ.

Costello [8] introduced a category Graphs in which the objects aredisjoint unions of corollas and morphisms are given by graphs. In intuitiveterms, we think of a morphism as assembling a bunch of corollas intoa graph Γ followed by contracting all edges so that what remains isagain a union of corollas (the result is π0Γ). Composition of morphismsis defined by iterating this process. More precisely, the objects aretriples, (V,H, in : V → H); a morphism from (V1, H1, in1 : V1 → H1)to (V2, H2, in2 : V2 → H2) is represented by a graph Γ together with anisomorphism from the source to ν(Γ) and an isomorphism from the targetto π0Γ. There is an obvious notion of equivalence on these data and the

Moduli spaces and modular operads 111

set of morphisms is defined as the set of equivalence classes. Alternatively,one can describe the set of morphisms as follows. A morphism consistsof involution a on H1 so that (V1, H1, in1 : V1 → H1, a) defines a graphΓ together with an isomorphism of π0Γ with the union of corollascorresponding to the target. To define the composition of morphisms weuse this second description. A composable pair of morphisms is given bya union of corollas, an involution a1 on the half edges, and then a secondinvolution a2 on the set of fixed points of the first. The composition isgiven by the involution that is equal to a1 on the free orbits of a1 and isequal to a2 on the fixed points of a1.

Graphs also form a category in a different way, where the objectsare graphs and the morphisms are given by contracting a set of treesubgraphs to points.

Disjoint union makes Graphs into a symmetric monoidal category.We will be interested in the symmetric monoidal subcategory Forests ⊂Graphs which has only those objects containing no 0-valent components(i.e., no isolated vertices) and only those morphisms that are forests (i.e.,disjoint unions of trees); the inclusion functor will be denoted ℓ. We willalso be interested in the over category of this inclusion. Fix a union ofcorollas x and consider the over category ℓ ↓ x.

Proposition 3.5. The category ℓ ↓ x is canonically equivalent to thecategory whose objects are graphs with legs identified with the legs of xand whose morphism are given by contracting a collection of disjointtrees down to points.

3.9 Coloured graphs

Fix a set Λ of colours. A Λ-coloured graph is a graph together withan element of Λ assigned to each edge and each leg. One can form acategory of Λ-coloured graphs, generalised Costello’s category Graphs,in which the objects are disjoint unions of corollas and the morphismsare coloured graphs.

3.10 Cyclic and modular operads as functors

Costello introduced his categories of graphs in order to reformulate thedefinition of cyclic and modular operads in terms more amenable todoing homotopy theoretic constructions.

Proposition 3.6. The category of cyclic operads in C is equivalentto the category of symmetric monoidal functors Forests → C , and

112 Jeffrey Giansiracusa

the category of modular operads in C is equivalent to the category ofsymmetric monoidal functors Graphs → C .

The idea is that a cyclic (or modular) operad P determines a functorby sending the n-corolla ∗n to the space P(n), and it sends a disjointunion of several corollas ∗n1# · · ·#∗nk to the product P(n1)⊗ · · ·⊗P(nk).Gluing legs together is sent to the map induced by the correspondingcomposition map.

Coloured cyclic and modular operads can of course also be describedas symmetric monoidal functors, using the categories of coloured graphs.

3.11 Some examples: the commutative and associativeoperads

We will mainly be concerned with the case when the ambient categoryin which our operads live is the category T op of topological spaces.

The commutative operad Comm is the cyclic operad that is theconstant functor Gr → T op sending each graph to a single point ∗.Clearly the commutative operad can also be considered as a modularoperad.

A ribbon structure on a graph is a choice of cyclic ordering of thehalf-edges incident at each vertex. A graph with ribbon structure iscalled a ribbon graph. The associative operad Assoc is the cyclic operadthat sends each graph γ to the discrete space consisting of one pointfor each ribbon structure on γ. It is not hard to see that if γ → γ′

is a contraction of a tree subgraph then there is a canonical bijectionbetween ribbon structures on γ and on γ′. There is also a canonicalbijection between ribbon structures on γ and on its atomisation, andthis provides the natural isomorphism required in the definition of acyclic operad.

3.12 Modular envelopes

Restriction from Graphs to Forests defines a forgetful functor frommodular operads to cyclic operads. This functor admits a left adjoint,Mod, called the modular envelope. We think of the modular envelope ofa cyclic operad as the modular operad it freely generates. The modularenvelope can be constructed via left Kan extension along the inclusionForests → Graphs .

By replacing the Kan extension with the derived Kan extension, wehave the derived modular envelope functor LMod.

Moduli spaces and modular operads 113

4 Moduli of geometric structures on surfaces

4.1 Surfaces with collars

Let Σ be a surface with boundary and corners (by corners, we meanthat it is locally modelled on the positive quadrant [0,∞)2 ⊂ R2). Theboundary of Σ is canonically partitioned into smooth strata, each ofwhich is either a circle of an interval. A boundary interval is a boundarystratum that is an interval. Let J ⊂ Σ be a boundary interval. Acollar of J is a smooth embedding φ of (−1, 0]× [0, 1] into Σ that sendsboundary to boundary and is a diffeomorphism of 0× [0, 1] onto J . Asurface equipped with a finite set of disjoint boundary intervals equippedwith disjoint collars is called a collared surface. If the collars are labelledby a set I then we call the surface I-collared.

Suppose Σ1 and Σ2 are I-collared surfaces. A diffeomorphism of I-collared surfaces Σ1 → Σ2 is a diffeomorphism of the underlying surfacesthat respects the labelling and parametrization of the collars.

Suppose Σ is a surface with disjoint boundary intervals J1 and J2equipped with disjoint collars φ2 and φ2 respectively. One can glue thesetwo boundary intervals together and obtain a new smooth surface. Thisis done as follows: let Σ′ = Σ ! (J1 ∪ J2)/ ∼, where we identify φ1(x)with φ2(x) for each x ∈ (−1, 0)× [0, 1].

4.2 Sheaves of geometric structures

Let Surf be the category enriched in T op of finite type surfaces (possiblywith boundary and corners) and open embeddings.

Definition 4.1. A smooth sheaf ψ on "Surf is an enriched functor"Surf → T op that sends pushout squares to homotopy pullback squares.

Remark 4.2. Smooth sheaves of this type have been studied in [3], wherethey are called homotopy sheaves and their relation with Goodwillie-Weiss embedding calculus of functors is explored. This notion also couldgo under the name of ∞-stacks.

Here we will think of the space ψ(Σ) as the space of geometricstructures of a given type on Σ. Below is a list of interesting examplesof some of the kinds of structures that one can consider within thisdefinition.

Example 4.3. 1. Orientations: ψ(Σ) is the set of orientations on Σ.

114 Jeffrey Giansiracusa

2. Almost complex structures: Since the space of almost complexstructures (for a fixed orientation) is contractible, this is equivalentto simply taking orientations.

3. Principal G-bundles: The space associated with a surface Σ is thespace of maps Σ → BG.

4. Maps to a background space X: The space ψ(Σ) is the space ofmaps Σ → X.

5. Spin, Spinc and r-spin can all be described in terms of the spaceof lifts of the classifying map Σ → BSO(2) of the tangent bundle.

6. Foliations: ψ(Σ) is the geometric realization of the simplicial spacewhose space of p-simplices is the space of codimension 2 foliationsof Σ×∆p that are transverse to the boundary of the simplex.1

One way to produce examples of smooth sheaves is to take sectionsof a bundle that is functorially associated with the tangent bundle. LetX be a space with an action of GL2(R). Given a surface Σ, let P → Σbe the GL2(R)-principal bundle associated with the tangent bundle andconsider the bundle P ×GL2(R) X → Σ.

Proposition 4.4. Sending Σ to the space of sections of P ×GL2(R) X

(with the compact-open topology) defines a smooth sheaf ψX on !Surf .Similarly, if X is a smooth manifold on which GL2(R) acts smoothly,then sending Σ to the space of smooth sections (with the smooth topology)defines a smooth sheaf.

Those smooth sheaves arising in this way will be called tangential.

Remark 4.5. A priori, the definition of a smooth sheaf appears moregeneral than the definition of tangential smooth sheaf. Not every smoothsheaf is tangential, such as the example of foliations in the list above.However, every smooth sheaf admits a tangnetial approximation andsometimes the approximation is actually equivalent to the original sheaf.In more detail, as described in [4, p. 16–17], given an smooth sheaf ψ,there is associated a tangential sheaf τψ and a canonical comparisonmorphism ψ → τψ. Moreover, (a version of) Gromov’s h-principle givesconditions under which this comparison morphism is a weak equivalenceof sheaves.

1The author thanks the referee for suggesting the inclusion of this example.

Moduli spaces and modular operads 115

A smooth sheaf ψ will be called connected if ψ((−1, 0)× I) is con-nected. Assuming ψ is connected, we can choose a basepoint ∗ ∈ψ((−1, 0)× I). Suppose J ⊂ ∂Σ is a boundary interval equipped with acollar φ. We say that a section s ∈ ψ(Σ) is trivial at J if φ∗(s) restrictsto the chosen basepoint ∗ on (−1, 0)× I.

Remark 4.6. In proving the main theorem of this paper, the assumptionthat ψ is connected can be discarded if one is willing to work with colouredcyclic and modular operads instead of ordinary (single colour) cyclic andmodular operads.

If Σ is a surface equipped with a collection of disjoint collaredboundary intervals J1, . . . , Jn, we write

!ψ(Σ) ⊂ ψ(Σ)

for the subspace consisting of sections that are trivial at the boundaryintervals Ji.

4.3 The monoid of geometric structures on a strip

Consider the unit square I × I equipped with a collar at each of theintervals 0× I and 1× I oriented in the same direction. We writeAψ for the space ψ′(I × I) of sections that are trivial at each side of thesquare because this space will play a particularly important role in theresults ahead.

Proposition 4.7. Gluing squares side to side endows the space Aψwith an A∞ composition making it into a group-like A∞ monoid; thehomotopy inverse map is induced by rotating the square 180 degrees.Fixing a collared boundary interval J on a surface Σ, there is a rightA∞ action of Aψ on !ψ(Σ) by gluing the right side of a square to J , anda left A∞ action given gluing the left edge of a square to J .

We will not spell out the proof of this here; it is straightforward buttechnical because of the necessity of using some machinery to handleA∞ monoids and their actions.

Proposition 4.8. Let J1 and J2 be two disjointly collared boundaryintervals on a surface Σ, and let Σ′ be the result of gluing J1 to J2.There is a homotopy equivalence

ψ(Σ′) ∼ !ψ(Σ)hAψ

116 Jeffrey Giansiracusa

where the action of Aψ is as follows. Given a square with ψ structureK ∈ Aψ, we glue the left edge of one copy of K to J1 and glue the leftedge of a second copy of K to J2.

This proposition is the key topological input in our generalised ribbongraph. The proof is rather technical and so it will be postponed forthe future paper. The idea is straightforward and we explain it now.Gluing a square at J1 has the effect of simply changing the trivializationof the ψ structure at J1, and this action is transitive in a homotopicalsense, so the homotopy quotient of this action is equivalent to the spaceof ψ-structures on Σ that are not necessarily trivial at J1. Thus thehomotopy quotient appearing in the proposition builds a model for thespace of ψ-structures on Σ such that are not necessarily trivial at J1 andJ2 but are required to agree at these collars. Giving such a structure isequivalent to giving a structure on the glued surface Σ′.

4.4 A 2-categorical model for the category of surfaces

In defining the modular operad of moduli spaces of ψ-structures, rather

than the category !Surf of surfaces and open embeddings, we will needa slightly different category Let Surf denote the topological categorywhose objects are collared surfaces. In rough language, a morphismΣ1 → Σ2 is a gluing of some collared boundary intervals together followedby a diffeomorphism. More precisely, the space of morphism is the disjointunion over all surfaces Σ′ obtained from Σ by gluing a number of pairsof boundary intervals together of the space of diffeomorphisms Σ′ → Σ2.We let Discs ⊂ Surf denote the full subcategory whose objects aredisjoint unions of discs each having at least 1 collared boundary interval

These topological categories are difficult to work with, so it is conve-nient to replace them with more combinatorial models that will workjust as well for our purposes. Let Surf 2 and Discs2 denote the strict2-categories with the same objects as Surf and Discs respectively,but with each space of diffeomorphisms replaced by the groupoid ofdiffeomorphisms and isotopy classes of isotopies.

Proposition 4.9. Given a collared surface Σ, the nerve of the categoryHomSurf 2(Σ,Σ) is weakly equivalent to the space Hom!Surf

(Σ,Σ) ∼=Diff(Σ).

Proof. When X is a disc or annulus with no collared boundary compo-nents then the diffeomorphism group is homotopy equivalent to a circle.

Moduli spaces and modular operads 117

For any fixed diffeomorphism, there is a Z worth of isotopy classes ofisotopies from it to itself, and the nerve of this Z gives the desired circle.In all other cases there is at most one isotopy class of isotopies betweenany two diffeomorphisms and the components of the diffeomorphismgroup are weakly contractible.

Given a smooth sheaf ψ and a collared surface Σ, we have introducedthe space !ψ(Σ) of sections of ψ that are trivial at the collared boundaryintervals. One sees that !ψ determines a continuous functor Surf → T op,which in turn determines a strict 2-functor Surf → T op2 that we shalldenote by the same symbol.

In order to talk about cyclic and modular operads, we will needto versions of the above 2-categories in which the collared boundaryintervals are labelled by a fixed finite set.

We define a strict 2-functor

S : Graphs → (Cat2 ↓ Surf 2)

by sending a union of corollas τ to the strict 2-category of collaredsurfaces with components identified with the components of τ andcollared boundary intervals compatibly identified with the legs of τ . Thefunctor to Surf 2 is given by forgetting the extra identification data. Wealso define

D : Forests → (Cat2 ↓ Discs2)

analogously. One can check that these functors S and D are actuallysymmetric monoidal and hence they define cyclic and modular operadsrespectively (albeit in somewhat odd looking ambient categories). Avalue S(τ) of S consists of a 2-category C and a functor P : C → Surf .Composing the !ψ with P gives a functor which we will write (with amoderate abuse of notation) as S(τ) → T op2.

4.5 The modular operad of moduli spaces

Associated with a smooth sheaf ψ and a diffeomorphism type of surfaces[Σ], there is a (homotopy theoretic) moduli space of surfaces diffeomor-phic to Σ and equipped with a ψ-structure. This moduli space is simplythe homotopy quotient

ψ(Σ)hDiff(Σ).

If we consider collared surfaces equipped with ψ-structures that aretrivial at the collared intervals then these moduli spaces collectively form

118 Jeffrey Giansiracusa

a modular operad. However, rigorously defining this modular operad sothat it is strictly associative is somewhat subtle. As a first approximation,given a finite set I, the corresponding space of the modular operad isthe disjoint union

(1)!

Σ

"ψ(Σ)hDiff(Σ)

where Σ runs over a set of representatives of diffeomorphism classesof surfaces equipped with a set of disjoint collared boundary intervalslabelled by I. The composition maps i,j and ij should be inducedby gluing collared boundaries. However, with this construction, thecomposition maps would only be associative up to A∞ homotopy.

One way to resolve this issue and strictify the composition mapsis to use all surfaces (within some set-theoretic universe) rather thanselecting one representative from each diffeomorphism class. To this endwe make the following definition.

Definition 4.10. The moduli space modular operad Mψ associatedwith a connected smooth sheaf ψ is defined by sending an object τ ∈Obj Graphs (i.e., a union of corollas) to the homotopy colimit

Mψ(τ) = hocolimS(τ)

"ψ.

Similarly, we define a cyclic operad Dψ of moduli spaces of discs by

Dψ(τ) = hocolimD(τ)

"ψ.

(To make sense of these symbols, please recall the abuse of notationmentioned above at the end of the previous subsection.)

In light of Proposition 4.9 above, Mψ evaluated on a corolla with Ilegs yields a space homotopy equivalent to the homotopy quotient thethe homotopy-theoretic moduli space of (1).

We can now state our main theorem.

Theorem 4.1. The derived modular envelope of the cyclic operad Dψ

is weakly equivalent as a modular operad to Mψ.

5 Arc systems in a surface

In this section we shall prove that the space of decompositions of asurface into discs is contractible.

Moduli spaces and modular operads 119

5.1 Arcs and cutting

Let Σ be a collared surface with nonempty boundary. An arc in Σ isan embedding [0, 1] → Σ that sends the interior of the interval to theinterior of the surface, sends the endpoints to the boundary, and meetsthe boundary transversally. We consider two arcs to be equivalent ifthey differ by reversing the direction of the interval via t "→ 1 − t. IfΣ is equipped with any collars then we require that the arc is disjointfrom the collars. An arc system is a finite (possibly empty) collection ofdisjoint arcs in Σ that divide the surface in regions homeomorphic todiscs (the regions will be diffeomorphic to polygons rather than discssince they will have corners), each of which touches at least one arc orcollared boundary component.

Observe that a surface Σ equipped with an arc system A can bycut along the arcs of an arc system to yield a disjoint union of discs.Moreover, each of the boundary intervals created by the cutting can becollared uniquely up to diffeomorphism so that the resulting union ofdiscs can be considered as a collared surface with one collared boundaryinterval for each collared boundary interval on the original surface plusone for each arc in the arc system. We shall denote this union of collareddiscs by ΣA.

An arc system has a dual graph with one vertex for each region inthe complement of the arcs, an edge for each arc, and a leg for eachcollar on the surface. We say that an arc system is reduced if its dualgraph has the minimum possible number of bivalent vertices (1 in thecase of a disc or annulus and zero in all other cases). An orientationon the surface induces a ribbon structure on the graph (i.e., a cyclicordering of the half edges incident at each vertex).

5.2 The category of arc systems

A diffeomorphism of Σ sends arc systems to arc systems. An isotopyfrom an arc system A to an arc system B is a 1-parameter family of arcsystems At such that A0 = A and A1 = B. A bijection from the arcs ofA to the arcs of B is said to be admissible if it can be induced by anisotopy.

Arc systems form a category A (Σ): the objects are arc systems anda morphism A → B consists an isotopy class of isotopies from A to asubsystem of B. We let A r (Σ) denote the full subcategory of reducedarc systems. There is a reduction functor R : A (Σ) → A r(Σ) that is

120 Jeffrey Giansiracusa

defined by replacing each collection of parallel arcs with a single arc anddeleting any arc that is parallel to a collared boundary interval (this isnon canonical the case of a disc or annulus, but a choice can be made inorder to define the functor).

Theorem 5.1. The nerve of A (Σ) is contractible.

Proof. The proof is divided into three cases. (1) Σ is a disc withoutcollars. (2) Σ is an annulus without collars. (3) Σ is any other collaredsurface.

(1) The dual graph of an arc system in the disc is a planar tree, andthe set of univalent vertices (corresponding to arcs that bound discscontaining no other arcs) inherits a cyclic order from the disc. Let Λbe the category of finite nonempty cyclically ordered sets and degree 1maps (Connes’s cyclic category). There is a functor q : A → Λ given bysending an arc system to the set of univalent vertices of its dual graph.The nerve of Λ is known to be equivalent to BS1, and we will identifythe map induced by the above functor with the map ES1 → BS1.

First, we show that the homotopy fibre of the map is S1. For anyobject [n] ∈ Λ, consider the over category q ↓ [n]. Let Z denote thecategory with a single object and a Z worth of endomorphisms. Thereis a functor r : Z → (q ↓ [n]) given by sending the single object tothe arc system consisting of a single arc and sending the generatingautomorphism to the automorphism given by rotating the disc through360 degrees. Over any object of q ↓ [n], the over category of thefunctor r has a nonempty set of objects and by unwinding the definitionscarefully one can see that there is a unique isomorphism between anytwo objects. Thus the nerve of any over category of r is contractibleand Quillen’s Theorem A implies that r induces a homotopy equivalenceof nerves. By considering a generator of the fundamental group onecan check that any morphism [n] → [m] in Λ induces a translationfunctor (q ↓ [n]) → (q ↓ [m]) that is a homotopy equivalence on nerves.Quillen’s Theorem B thus says that BZ → A → Λ gives a homotopyfibre sequence upon passing to nerves. The nerve of the fibre is S1 andthe nerve of the base is BS1.

To conclude that the total space is ES1, we need only check thatthe inclusion of the fibre S1 into the total space is trivial on π1. Thegenerator of π1 of the fibre is represented by a rotation of the discthrough 360 degrees. Given a symmetric configuration of 3 arcs in thedisc, there is an automorphism given by rotation by 120 degrees, and thecube of this automorphism is the 360 degree rotation. A calculation in

Moduli spaces and modular operads 121

Figure 1: Calculation in the localised category representing the totalspace, showing the the generator of the fundamental group of the fibreis trivial in the total space. The unlabelled arrows in this diagram areall the evident inclusions of arc systems.

the localised category using §3.2 will show that this 120 degree rotation istrivial in the localisation. The calculation is represented by the diagramin figure 5.2.

(2) The dual graph of an arc system in the annulus is a chain ofbivalent vertices (corresponding to those arcs which go from the innerboundary to the outer one) with some trees attached (corresponding tonested sets of arcs that have both ends on the same boundary circle).This chain on bivalent vertices inherits a cyclic order from the annulus,and sending an arc system to this set defines a functor from A to Λ.Using arguments similar to case (1) above, one can conclude that thehomotopy fibre of the map of nerves is S1 and the fibre sequence is infact S1 → ES1 → BS1.

(3) In this case we use a category-theoretic reformulation of anargument from Hatcher [12]. Let A denote the category of isotopyclasses of arc systems and admissible bijections. Since, in this case, everyadmissible bijection is induced by a unique isotopy class of isotopies, thecanonical functor A → A is an equivalence of categories.

Fix an arc x. We will show that the identity functor on A and

122 Jeffrey Giansiracusa

A

x

Sx(A)

Figure 2: The effect of the arc surgery operator Sx.

the constant functor sending any arc system to x are homotopic afterpassing to nerves by constructing a zigzag of natural transformationsbetween functors A → A , starting with the identity and finishing withthe constant functor.

We define an operator Sx from arcs to arcs as follows. If x and yare disjoint (up to isotopy) then Sx(y) = y. If x and y intersect then,moving them by isotopies, we may put them in a position so that theycross transversally and the number of intersection points is minimal (e.g.,choose a metric and use the geodesic representatives of their isotopyclasses). We may now cut y at each point where it meets x and slidesthe resulting endpoints along x until they reach the boundary of thesurface, as shown in figure 5.2. We can extend Sx to a map from arcsystems to arc systems by applying it to each of the arcs in a system.

Let S1 : A → A be the functor that sends an arc system A toA ∪ Sx(A), let S2 be the functor that sends A to Sx(A), let S3 be thefunctor that sends A to x ∪ Sx(A), and let S4 be the constant functorsending any arc system to x. It is straightforward to see that there arenatural transformations

id→ S1 ← S2 → S3 ← S4

induced by the evident inclusions of arc systems. Upon passing to nerves,this zigzag of natural transformation yields the desired homotopy fromthe identity map to the constant map.

Corollary 5.1. The nerve of A r(Σ) is contractible.

Proof. Let i denote the inclusion A r !→ A and observe that for anyarc system A the over category i ↓ A has an initial object given by

Moduli spaces and modular operads 123

any choice reduction R(A) and morphism R(A) → A. The result thenfollows from Quillen’s Theorem A and Theorem 5.1.

6 Sketch of the proof Theorem 4.1

Let ψ be a smooth sheaf and recall that we have defined the cyclicoperad Dψ of moduli spaces of discs, and the modular operad Mψ ofmoduli spaces of surfaces. We will construct a chain of weak equivalencesbetween the derived modular envelope LMod(Dψ) and Mψ.

By the construction of the derived Kan extension along ℓ : Forests →Graphs (evaluated at τ as the homotopy colimit over the over categoryℓ ↓ τ , we see that the derived modular envelope of Dψ, evaluated on aunion of corollas γ, is given by

hocolimγ∈ℓ↓τ

Dψ = hocolimγ∈ℓ↓τ

hocolimD(γ)

!ψ.

By the Fubini theorem for homotopy colimits, this is weakly equivalentto hocolim!

ℓ↓τ D!ψ.

An object of"ℓ↓τ D is a graph γ (equipped with an identification

π0γ ∼= τ) and a decoration of each vertex by a disc. Gluing the discstogether as prescribed by the graph results in a surface Σ ∈ S(τ).Moreover, the collection of arcs in Σ along which the gluing was performedyield an arc system. This defines a strict 2-functor.

Lemma 6.1. The above construction defines an equivalence of 2-cate-gories

"ℓ↓τ D ≃

"S(τ) A .

The inverse of the equivalence is constructed by cutting along thearcs, and it is denoted κ. Note that it is not a strict 2-functor but onlya lax 2-functor. Hence

hocolim!ℓ↓τ D

!ψ ≃ hocolim!S(τ) A

!ψ κ,

which is weakly equivalent to

hocolim!S(τ) A r

LR!( !ψ κ),

where R :"S(τ) A →

"S(τ) A r is the arc system reduction functor.

We now come to the key step in the argument. As explained in[13], Kan extension along R can be thought of as integrating out the

124 Jeffrey Giansiracusa

bivalent vertices in the dual graphs to the arc systems, and this has thefollowing effect. For each edge in the dual graph, one builds a 2-sidedbar construction for the monoid Aψ of geometric structures on a stripacting on the the spaces associated to the vertices at either end of theedge. By Proposition 4.8 we then have the following result.

Lemma 6.2. LR!( !ψ κ) ≃ !ψ π, where π :"S(τ) A r → S(τ) forgets the

arc system.

In other words, starting with the space of all pairs of an unreducedarc system and a ψ-structure trivial on the arcs, and then integrating outthe bivalent vertices in the dual graph yields a space equivalent to thespace of pairs of a reduced arc system and a ψ-structure not-necessarilytrivial on the arcs.

Finally, by Corollary 5.1, the homotopy colimit of !ψ π over"S(τ) A r

is equivalent to the homotopy colimit of !ψ overS(τ), which is preciselythe definition of Mψ(τ).

AcknowledgementI thank the MIMS for hosting the conference at which I first spoke

about this work, and Sadok Kallel for organizing the meeting. I wouldalso like to thank the anonymous referee for some useful comments andsuggestions.

Jeffrey GiansiracusaDepartment of Mathematics,Swansea University,Singleton Park, Swansea, Wales,SA2 8PP, United Kingdom,j.h.giansiracusa@swansea.ac.uk

References

[1] Carrasco P.; Cegarra A. M.; Garzon A. R., Nerves and classifyingspaces for bicategories, Algebr. Geom. Topol. 10:1 (2010), 219–274.

[2] Cegarra, A. M., Homotopy fiber sequences induced by 2-functors,J. Pure Appl. Algebra 215:4 (2011), 310–334.

[3] Boavida de Brito P.; Weiss M., Manifold calculus and homotopysheaves, Homology Homotopy Appl. 15:2 (2013), 361–383.

[4] Ayala D., Geometric cobordism categories, arxiv:0811.2280 (2008).

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[5] Cordier J.-M.; Porter T., Homotopy coherent category theory, Trans.Amer. Math. Soc. 349:1 (1997), 1–54.

[6] Gabriel P.; Zisman M., Calculus of Fractions and Homotopy The-ory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag New York Inc., New York, 1967.

[7] Berger C.; Batanin M., Homotopy theory for algebras over polyno-mial monads, arXiv:1305.0086 (2013).

[8] Costello K., The A–infinity operad and the moduli space of curves,arXiv:math/0402015, (2004).

[9] Costello K., Topological conformal field theories and Calabi-Yaucategories, Adv. Math. 210:1 (2007), 165–214.

[10] Costello K., A dual version of the ribbon graph decomposition ofmoduli space, Geom. Topol. 11 (2007), 1637–1652.

[11] Thomason R. W., Homotopy colimits in the category of smallcategories, Math. Proc. Cambridge Philos. Soc. 85:1 (1979), 91–109.

[12] Hatcher A., On triangulations of surfaces, Topology Appl. 40:2(1991), 189–194.

[13] Giansiracusa J., The framed little 2-discs operad and diffeomor-phisms of handlebodies, J. Topol. 4:4 (2011), 919–941.

Morfismos se imprime en el taller de reproduccion del Departamento de Matema-ticas del Cinvestav, localizado en Avenida Instituto Politecnico Nacional 2508, Colo-nia San Pedro Zacatenco, C.P. 07360, Mexico, D.F. Este numero se termino de im-primir en el mes de marzo de 2014. El tiraje en papel opalina importada de 36kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

Contents - Contenido

Configuration space integrals and the topology of knot and link spaces

Ismar Volic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Topological chiral homology and configuration spaces of spheres

Oscar Randal-Williams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Cooperads as symmetric sequences

Benjamin Walter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Moduli spaces and modular operads

Jeffrey Giansiracusa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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