VOLUMEN 17NMERO 2

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VOLUMEN 17NMERO 2

JULIO A DICIEMBRE 2013ISSN: 1870-6525

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

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

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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 1822, 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, Jerey 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 conferenciaOperads 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, Jerey 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

Jerey Giansiracusa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Morfismos, Vol. 17, No. 2, 2013, pp. 156

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 Dierential 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. Thurstons result in Section 3.4, but the idea is asfollows:

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

: 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 and Conf[p, q;K3,Rn] is a pullback spaceconstructed from an evaluation map and a projection map. The fiber of 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 of 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 (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 Thurstons, 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...