Morfismos, Vol 18, No 2, 2014

VOLUMEN 18NÚMERO 2

JULIO A DICIEMBRE 2014ISSN: 1870-6525

Chief Editors - Editores Generales

• Isidoro Gitler • zelaznoGsuseJ

Guest Editors - Editores Invitados

• Ernesto Lupercio • Miguel A. Xicotencatl

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-

-ametaMedotnematrapeD,aicnelaVaruaL.arSalaadigiridriebedaicnednopticas del Cinvestav, Apartado Postal 14-740, Mexico, D.F. 07000, o por correo

:noicceridalaocinortcele [email protected].

VOLUMEN 18NÚMERO 2

JULIO A DICIEMBRE DE 2014ISSN: 1870-6525

noicacilbupanuse,4102erbmeicidaoiluj,2oremuN,81nemuloV,somsfiroMsodaznavAsoidutsEedynoicagitsevnIedortneCleropadatidelartsemes

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, [email protected], 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-

aledsadartsulIsatsiveRysenoicacilbuPedarodacfiilaCnoisimoCalropsodledsacitametaMedotnematrapeDleroposerpmI.noicanreboGedaıraterceS

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

marzo de 2015 con un tiraje de 50 ejemplares.

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

-nocsoledlaicrapolatotnoiccudorperaladibihorpetnematcirtseadeuQlednoicazirotuaaiverpnis,noicacilbupaledsenegamiesodinet Cinvestav.

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

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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.

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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.

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

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• Papers in category (a) might be written in Spanish; all other papers proposed forpublication in Morfismos shall be written in English, except those for which theEditoral Board decides to publish in another language.

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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.

• Los artıculos del tipo (a) podran estar escritos en espanol. Los demas trabajos deberanestar redactados en ingles, salvo aquellos que el Comite Editorial decida publicar enotro idioma.

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In memory of Samuel Gitler, a brilliant mathemati-cian and an extraordinary human being. Samuel guid-ance will be much missed.

A la memoria de Samuel Gitler, un matematico genialy, ademas, un ser humano excepcional. Extranaremosprofundamente las ensenanzas de Samuel.

Samuel Gitler (July 14, 1933 – September 9, 2014) was one of the foremostthinkers in the history of Mexican mathematics. He graduated under NormanSteenrod with a thesis in algebraic topology from Princeton University in 1960and, throughout the XX century, he published famous works on the applica-tions of homotopy theory to various geometric problems, most famously theimmersion of manifolds into euclidian spaces problem.

As a monument to his relentless creativity he continued to produce beautifulmathematics into the XXI century with his work on toric topology, a field inwhich he became a leader during the last stretch of his career.

This volume represents the proceedings of the conference SAM80 that, tocelebrate his 80th birthday, took place on September of 2013 at El Colegio

Nacional in Mexico City. A large group of some of the best topologists in theworld (on the next page you can find the list of speakers) converged in MexicoCity to discuss the state of the art of algebraic topology on various fields ofinterest to Gitler. Professor Gitler didn’t miss a beat and was active askingquestions and at all the mathematical discussions of the conference.

The volume contains three remarkable contributions.

On the first one, Don Davis uses ku-cohomology to obtain new resultsregarding the topological complexity of 2-torsion lens spaces using a very deli-cate linear algebra argument. The second contribution, by Kee Lam and DuaneRandall, avoids using spectral sequences in a truly elegant method for comput-ing the geometric dimension of stable vector bundles over spheres using onlyK-theory. Finally, the third contribution is an announcement of an importantmethod for producing an explicit presentation by generators and relations ofthe equivariant and ordinary cohomology rings (with rational coefficients) ofany regular nilpotent Hessenberg variety Hess(h) of type A by Hiraku Abe,Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda.

The very high quality of the results presented here and the truly inter-national character of the contributors are a testament to Sam’s influence inmodern mathematics.

Ernesto LupercioMiguel XicotencatlMexico City, April 2015.

Sam’s 80A Conference to Celebrate Sam Gitler’s 80th Birthday

Schedule:

September 25 (Wednesday)

10:00 – 10:50 Opening ceremony

10:50 – 11:40 Don Davis (Lehigh University)Stable geometric dimension: Old work with Markand Sam (and Martin)

11:40 – 12:00 COFFEE BREAK

12:00 – 12:50 Fred Cohen (University of Rochester)An excursion into moment-angle complexes, polyhedralproducts, and their applications

12:50 – 13:40 Mikiya Masuda (Osaka City University)Toric origami manifolds in toric topology

13:40 – 15:30 LUNCH

15:30 – 16:20 Bill Browder (Princeton University)How big a finite group can act freely on a product of spheres?

16:20 – 17:10 Ralph L. Cohen (Stanford University)Gauge theory, loop groups, and string topology


17:30 – 18:20 Soren Galatius (Stanford University)Manifolds and moduli spaces

1

September 26 (Thursday)

10:00 – 10:50 Jesus Gonzalez (Cinvestav-IPN)Topological robotics

10:50 – 11:40 Martin Bendersky (Hunter College, CUNY)Structure of The Polyhedral Product and Related Spaces


12:00 – 12:50 Dennis Sullivan (SUNY, Stony Brook)From Topology to Analysis

12:50 – 13:40 Douglas C. Ravenel (University of Rochester)A Solution to the Arf-Kervaire Invariant Problem

13:40 – 15:30 LUNCH

15:30 – 16:20 Kee Yuen Lam (University of British Columbia, Canada)Solution of the Yuzvinsky conjecture for certain types of matrices

16:20 – 17:10 Victor M. Buchstaber (Steklov Mathematical Institute, Russia)Toric topology and Grassmann manifolds


17:30 – 18:20 Soren Galatius (Stanford University)The method of scanning

2

September 27 (Friday)

10:00 – 10:50 Ernesto Lupercio (Cinvestav-IPN)Non-commutative Toric Varieties

10:50 – 11:40 Tony Bahri (Rider University)A generalization of the standard topological constructionof toric manifolds and applications involving variousoperations on simplicial complexes


12:00 – 12:50 Taras Panov (Moscow State University)Complex geometry of moment-angle manifolds

12:50 – 13:40 Alberto Verjovsky (UNAM)Poincare theory for compact abelian one-dimensionalsolenoidal groups

13:40 – 15:30 LUNCH

15:30 – 16:20 Santiago Lopez de Medrano (UNAM)Projective moment-angle complexes (first steps)

16:20 – 17:10 Alejandro Adem (University of British Columbia)A classifying space for commutativity in Lie groups


17:30 – 18:20 Soren Galatius (Stanford University)Homological stability for moduli spaces

3

Contents - Contenido

Topological complexity of 2-torsion lens spaces and ku-(co)homology

Donald M. Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Geometric dimension of stable vector bundles over spheres

Kee Yuen Lam and Duane Randall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

The equivariant cohomology rings of regular nilpotent Hessenberg varieties inLie type A: Research Announcement

Hiraku Abe, Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda . . . . 51

Morfismos, Vol. 18, No. 2, 2014, pp. 1–40

Topological complexity of 2-torsion

lens spaces and ku-(co)homology

Donald M. Davis

Abstract

We use ku-cohomology to determine lower bounds for the topolog-ical complexity of mod-2e lens spaces. In the process, we give analmost-complete determination of ku∗(L

∞(2e))⊗ku∗ku∗(L∞(2e)),

proving a conjecture of Gonzalez about the annihilator ideal of thebottom class. Our proof involves an elaborate row reduction ofpresentation matrices of arbitrary size.

2000 Mathematics Subject Classification: 55M30, 55N15.Keywords and phrases: Topological complexity, lens space, K-theory.

1 Main Theorems

The determination of the topological complexity of topological spaceshas been much studied since its introduction by Farber in [2]. The(normalized) topological complexity, TC(X), of a space X is 1 less thanthe smallest number of open subsets of X ×X over which the fibrationPX → X ×X, which sends a path σ to (σ(0), σ(1)), has a section. See[4] and [5] for an expanded discussion of this concept, especially as itrelates to lens spaces.

Let L2n+1(t) denote the standard (2n+1)-dimensional t-torsion lensspace, and let b(n, e), as defined in [5], denote the smallest integer k suchthat there exists a map

(1) L2n+1(2e)× L2n+1(2e) → L2k+1(2e)

which when followed into L∞(2e) is homotopic to a restriction of theH-space multiplication of L∞(2e) = BZ/2e. In [4], it is proved that

2b(n, e) ≤ TC(L2n+1(2e)) ≤ 2b(n, e) + 1.

1


Topological complexity of 2-torsion

lens spaces and ku-(co)homology

Donald M. Davis

Abstract

We use ku-cohomology to determine lower bounds for the topolog-ical complexity of mod-2e lens spaces. In the process, we give analmost-complete determination of ku∗(L

∞(2e))⊗ku∗ku∗(L∞(2e)),

proving a conjecture of Gonzalez about the annihilator ideal of thebottom class. Our proof involves an elaborate row reduction ofpresentation matrices of arbitrary size.

2000 Mathematics Subject Classification: 55M30, 55N15.Keywords and phrases: Topological complexity, lens space, K-theory.

1 Main Theorems

The determination of the topological complexity of topological spaceshas been much studied since its introduction by Farber in [2]. The(normalized) topological complexity, TC(X), of a space X is 1 less thanthe smallest number of open subsets of X ×X over which the fibrationPX → X ×X, which sends a path σ to (σ(0), σ(1)), has a section. See[4] and [5] for an expanded discussion of this concept, especially as itrelates to lens spaces.

Let L2n+1(t) denote the standard (2n+1)-dimensional t-torsion lensspace, and let b(n, e), as defined in [5], denote the smallest integer k suchthat there exists a map

(1) L2n+1(2e)× L2n+1(2e) → L2k+1(2e)

which when followed into L∞(2e) is homotopic to a restriction of theH-space multiplication of L∞(2e) = BZ/2e. In [4], it is proved that

2b(n, e) ≤ TC(L2n+1(2e)) ≤ 2b(n, e) + 1.

1

2 Donald M. Davis

Thus the following theorem yields a lower bound for TC(L2n+1(2e)).Here and throughout α(n) denotes the number of 1’s in the binaryexpansion of n.

Theorem 1.1. If e ≥ 2 and e ≤ α(m) < 2e, then

b(m+ 2α(m)−e − 1, e) ≥ 2m− 2α(m)−e.

This immediately implies the following result for topological com-plexity, which might be considered our main result.

Corollary 1.2. If e ≥ 2 and e ≤ α(m) < 2e, then

TC(L2m+2α(m)−e+1−1(2e)) ≥ 4m− 2α(m)−e+1.

Other results follow from this and the obvious relation b(n+1, e) ≥b(n, e). The author believes that this result contains all lower boundsfor b(n, e) implied by 2-primary connective complex K-theory ku. In [6],a much stronger conjectured lower bound for b(n, e) is given, with thesame flavor as our theorem. Their conjecture depends on conjecturesabout BP ∗(L2n+1(2e) × L2n+1(2e)), while our theorem depends on atheorem about ku∗(L2n+1(2e)× L2n+1(2e)).

Our first new result for topological complexity is

TC(L2m+7(2α(m)−2)) ≥ 4m− 8 if α(m) ≥ 4.

Our theorem is proved by applying ku∗(−) to the map (1), obtain-ing a contradiction under appropriate choice of parameters. Our mainingredient is the almost-complete determination of ku4n−2d(L2n(2e) ×L2n(2e)). It is well-known that ku∗ = Z(2)[u] with |u| = −2 and thatits 2e-series satisfies

[2e](x) =

2e∑i=1

(2e

i

)ui−1xi.

It is proved in [3, Proposition 3.1] that

(2) kuev(L2n(2e)× L2n(2e)) = ku∗[x, y]/(xn+1, yn+1, [2e](x), [2e](y)),

where |x| = |y| = 2. One of our main accomplishments is to give a moreuseful description of ku4n−2d(L2n(2e)× L2n(2e)).

Topological Complexity 3

On the other hand, ku-homology, ku∗(L2e), of the infinite-dimensionallens space L2e = L∞(2e) is the ku∗-module generated by classes zi,i ≥ 0, of grading 2i+ 1 with relations

i∑�=0

(2e

�+1

)u�zi−�, i ≥ 0.

Here |u| = 2 in ku∗. Also, ku∗(L2e × L2e) contains ku∗(L2e) ⊗ku∗

ku∗(L2e) as a direct ku∗-summand. We define

Me := ku∗(Σ−1L2e)⊗ku∗ ku∗(Σ

−1L2e).

It is a ku∗-module on classes [i, j] := zi ⊗ zj of grading 2i+2j, i, j ≥ 0,with relations

(3)i∑

�=0

(2e

�+1

)u�[i− �, j], i, j ≥ 0, and

j∑�=0

(2e

�+1

)u�[i, j − �], i, j ≥ 0.

The desuspending was just for notational convenience. Note that thecomponent of Me in grading 2d, which we denote by Gd, is isomorphicto ku4n−2d(L2n(2e)× L2n(2e)) under the correspondence

uk[i, j] ↔ ukxn−iyn−j .

Much of our work goes into an almost-complete description of Me. Theresult is described in Section 2.

In [5, Theorem 2.1], it is proved that the ideal

Ie := (2e, 2e−1u, 2e−2u3·2−2, 2e−3u3·22−2, . . . , 21u3·2

e−2−2, u3·2e−1−2)

annihilates the bottom class [0, 0] of Me, and in [5, Conjecture 2.1] it isconjectured that Ie is precisely the annihilator ideal of [0, 0] in Me. Oneof our main theorems is that this conjecture is true.

Theorem 1.3. For e ≥ 1, the annihilator ideal of [0, 0] in Me is pre-cisely Ie.

This is immediate from our description of Me in Section 2. See theremark preceding Theorem 2.2.

After describing Me in Section 2, we use this description in Section3 to prove Theorem 1.1. In Section 4, we prove our result for M6, andin Section 5, we explain how this proof generalizes to arbitrary Me.

4 Donald M. Davis

Finally, in Section 6, we give a different proof of Theorem 1.3 for e ≤ 5,one which is easily checked by a simple computer verification.

The author wishes to thank Jesus Gonzalez for suggesting this prob-lem, some guidance as to method, and for providing some computerresults which were very helpful for finding a general proof.

2 Description of Me

Our approach to describing Me is via an associated matrix Pe of poly-nomials, which we row reduce. The row-reduced form of Pe is quitecomplicated, and involves some polynomials which are not completelydetermined. That is why we call our description “almost complete.” Inthis section, we approach the description of Me in three steps.

First we give an introduction to our method, define the polynomialmatrices Pe, and give in Table 1, without proof, the reduced form ofP4, obtained without a computer. The result for P4 is not used inour general proof, but provides a useful example for comparison. JesusGonzalez obtained an equivalent result using Mathematica.

Next we give in Theorem 2.1 an almost-complete description of thereduced form of P6. This incorporates all aspects of the general reducedPe, but is still describable in a moderately tractable way. Finally wegive in Theorem 2.2 the general result for Pe, which involves a plethoraof indices.

Let Gd denote the component of Me in grading 2d. Our ordered setof generators for Gd is

(4) [0, d], . . . , [d, 0], u[0, d− 1], . . . , u[d− 1, 0], . . . , ud[0, 0].

Our final presentation matrix of Gd will be a partitioned matrix

M0,0 M0,1 . . . M0,d...

......

...Md,0 Md,1 . . . Md,d

,

where Mi,j is a (d+1− i)-by-(d+1− j) Toeplitz matrix. The columnsin a block Mi,j correspond to monomials uj [−,−].

We will use polynomials to represent the submatrices Mi,j . A poly-nomial or power series p(x) = α0 + α1x + α2x

2 + · · · corresponds to aToeplitz matrix (of any size) with (j+k, j) entry equal to αk. Thus thematrix is


a0 0 0

α1 α0 0. . .

α2 α1 α0

α3 α2 α1. . .

α4. . .

. . .. . .

......

......

We define Pe to be the polynomial matrix associated to the partitionedpresentation matrix of Me corresponding to the generators (4) and re-lations (3). In (11) we depict P6.

We let

pn(x) =xn − 1

x− 1= 1 + x+ · · ·+ xn−1.

We will display a single upper-triangular matrix of polynomials, whoserestriction to the first d+1 columns yields a presentation of Gd for all d.For example, we will see that the first 8 columns for the reduced formof P4 are

16 0 0 4xp2(x) 0 0 0 2xp6(x)8 0 4p3(x) 0 0 0 2p7(x)

8 0 0 0 0 08 0 0 0 0

4 0 0 04 0 0

4 04

.

This implies that a presentation matrix of G7 is as below.

16I8 0 0 M0,3 0 0 0 M0,7

8I7 0 M1,3 0 0 0 M1,7

8I6 0 0 0 0 08I5 0 0 0 0

4I4 0 0 04I3 0 0

4I2 04I1

,

where It is a t-by-t identity matrix, and

6 Donald M. Davis

M0,3 =

0 0 0 0 04 0 0 0 04 4 0 0 00 4 4 0 00 0 4 4 00 0 0 4 40 0 0 0 40 0 0 0 0

, M0,7 =

02222220

,

M1,3 =

4 0 0 0 04 4 0 0 04 4 4 0 00 4 4 4 00 0 4 4 40 0 0 4 40 0 0 0 4

, M1,7 =

2222222

.

The precise reduced form of P4 is as in Table 1 below. We do notoffer a proof here, but can prove it by the methods of Section 4. Weoften write pk instead of pk(x).

The abelian group that the associated matrix of numbers presentshas 276 generators and 276 relations. This associated matrix of numbersis almost, but not quite, in Hermite form. For example, the polynomialin position (2, 17) contains terms such as 2x5, and so the associatedmatrix of numbers will have some 2’s sitting far above 2’s at the bottomof the column. For the matrix to be Hermite, all nonzero entries abovea 2 at the bottom should be 1’s. We could obtain such a polynomial inposition (2, 17) by subtracting (x5 + x6 + x9 + x10) times row 17 fromrow 2. We have chosen not to do this here because it will be importantto our reduction that the first three nonzero entries in column 17 are12p3(x

4) times the corresponding entries of column 9.

By restricting to G1, the 8 in position (1, 1) shows that 8u[0, 0] = 0in M4. Similarly, by restriction to G4, the 4 in position (4, 4) impliesthat 4u4[0, 0] = 0. We also obtain 2u10[0, 0] = 0 and u22[0, 0] = 0 fromthe matrix. The Hermite form of the associated matrix of numbersimplies that 8[0, 0], 4u3[0, 0], 2u9[0, 0], and u21[0, 0] are all nonzero, andthis implies Theorem 1.3 when e = 4.

Next we describe the reduced form of P6. We let Pi,j denote theentry in row i and column j, where the numbering of each starts with 0.


Throughout the paper, the same notation Pi,j will be used for entriesin the matrix at any stage of the reduction.

Table 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 16 0 0 4xp2 0 0 0 2xp6 0 0 0 0 0 0 0 xp14 0

1 8 0 4p3 0 0 0 2p7 0 0 0 0 0 0 0 p15 0

2 8 0 0 0 0 0 2x2p2(x2) 2xp6 0 0 0 0 0 0 x2p6(x

2)

3 8 0 0 0 0 0 2x2p2(x2) 0 0 0 0 0 0 0

4 4 0 0 0 2p3(x2) 2xp2(x

3) 0 0 0 0 0 0 p7(x2)

5 4 0 0 0 2p3(x2) 0 0 0 0 0 0 0

6 4 0 0 0 0 0 0 0 0 0 0

7 4 0 0 0 0 0 0 0 0 0

8 4 0 0 0 0 0 0 0 0

9 4 0 0 0 0 0 0 0

10 2 0 0 0 0 0 0

11 2 0 0 0 0 0

12 2 0 0 0 0

13 2 0 0 0

14 2 0 0

15 2 0

16 2

17 18 19 20 21 22

0 0 0 0 x7p4(x2) x5p2(x

2)p4(x3) 0

1 0 0 0 x6p8 x4p2(x12) 0

2 xp6p3(x4) 0 x3p3p2(x

2)p2(x7) x4p4p2(x

7) xp2p2(x16) + x8p2(x

3) 0

3 x2p6(x2) 0 0 x5p2(x

3)p2(x4) x4p2(x

7)p4 0

4 xp2(x3)p3(x

4) 0 x3p2(x2)p2(x

7) x4p2(x2)p2(x

6) xp2(x9)(1 + x2p3 + x6) 0

5 p7(x2) 0 0 x5p2p2(x

4) x4p2(x2)p2(x

6) 0

6 0 x4p2(x4) 0 x2p6(x

2) x5p2p2(x4) 0

7 0 0 x4p2(x4) 0 x2p6(x

2) 0

8 0 0 0 x4p2(x4) 0 0

9 0 0 0 0 x4p2(x4) 0

10 0 p3(x4) 0 x2p2(x

6) 0 0

11 0 0 p3(x4) 0 x2p2(x

6) 0

12 0 0 0 p3(x4) 0 0

13 0 0 0 0 p3(x4) 0

14 0 0 0 0 0 0

15 0 0 0 0 0 0

16 0 0 0 0 0 0

17 2 0 0 0 0 0

18 2 0 0 0 0

19 2 0 0 0

20 2 0 0

21 2 0

22 1

Theorem 2.1. The reduced form of the matrix P6 is upper-triangularwith diagonal entries

Pi,i =

64 i = 0

32 1 ≤ i ≤ 3

16 4 ≤ i ≤ 9

8 10 ≤ i ≤ 21

4 22 ≤ i ≤ 45

2 46 ≤ i ≤ 93

1 i = 94.

8 Donald M. Davis

Other than these, the nonzero entries are as described below.

a. There are none in columns 0–2, 4–6, 10–14, 22–30, 46–62, and94.

b. The nonzero entries in columns 3, 7–9, 15–17, 31–33, and 63–65are as below.

3 7 8 9 15 16 17

0 16xp2 8xp6 4xp14

1 16p3 8p7 4p15

2 8x2p2(x2) 8xp6 4x2p6(x

2) 4xp6p3(x4)

3 8x2p2(x2) 4x2p6(x

2)

4 8p3(x2) 8xp2(x

3) 4p7(x2) 4xp2(x

3)p3(x4)

5 8p3(x2) 4p7(x

2)

31 32 33 63 64 65

0 2xp30 xp62

1 2p31 p63

2 2x2p14(x2) 2xp6p7(x

4) x2p30(x2) xp6p15(x

4)

3 2x2p14(x2) x2p30(x

2)

4 2p15(x2) 2xp2(x

3)p7(x4) p31(x

2) xp2(x3)p15(x

4)

5 2p15(x2) p31(x

2)

c. The nonzero entries in columns 18–21, 34–37, and 66–69 are as inTable 2. Here B refers to everything in the 18–21 block except the4p3(x

4)-diagonal near the bottom. The •s along a diagonal refer tothe entry at the beginning of the diagonal. Each letter q refers toa polynomial. These polynomials are, for the most part, distinct.The meaning of the diagram is that, except for the diagonal nearthe bottom, each entry in the middle portion equals 1

2p3(x8) times

the corresponding entry in the left portion, and similarly for theright portion, as indicated. More formally, for 18 ≤ j ≤ 21 andi < j − 8,

Pi,j+16 =12p3(x

8) · Pi,j and Pi,j+48 =14p7(x

8) · Pi,j .

d. Similarly, the nonzero elements in columns 38 to 45 (other thanPi,i) are as in Table 3. If C denotes all the entries except the2p3(x

8)-diagonal near the bottom, then columns 70 to 77 are filled


exactly with C · 12p3(x

16) together with a p7(x8)-diagonal going

down from (22, 70).

Table 218

19

20

21

34

35

36

37

66

67

68

69

00

04q

4q

10

04q

4q

20

4q

4q

4q

30

04q

4q

B·1 2

p3(x

8)

B·1 4

p7(x

8)

40

4q

4q

4q

50

04q

4q

64x4p2(x

4)

04q

4q

70

��

04q

80

0��

0

90

00

��

10

4p3(x

4)

04q

4q

2p7(x

4)

p15(x

4)

11

0��

04q

��

��

12

00

��

0��

��

13

00

0��

��

��

e. Finally, columns 78 to 93 have a form very similar to Table 3with q instead of 2q and rows going from 0 to 61. The lower twodiagonals are x16p2(x

16) coming down from (30, 78) and p3(x16)

coming down from (46, 78), and these are the only non-leadingnonzero entries in column 78.

10 Donald M. Davis

Table 3

38 39 40 41 42 43 44 45

0 0 0 2q 2q 2q 2q 2q 2q

1 0 0 2q 2q 2q 2q 2q 2q

2 0 2q 2q 2q 2q 2q 2q 2q

3 0 0 2q 2q 2q 2q 2q 2q

4 0 2q 2q 2q 2q 2q 2q 2q

5 0 0 2q 2q 2q 2q 2q 2q

6 0 0 2q 2q 2q 2q 2q 2q

7 0 0 0 2q 2q 2q 2q 2q

8 0 0 0 0 2q 2q 2q 2q

9 0 0 0 0 0 2q 2q 2q

10 0 0 2q 2q 2q 2q 2q 2q

11 0 0 0 2q 2q 2q 2q 2q

12 0 0 0 0 2q 2q 2q 2q

13 0 0 0 0 0 2q 2q 2q

14 2x8p2(x8) 0 0 0 2q 0 2q 2q

15 0��

0 0 0 2q 0 2q

16 0 0��

0 0 0 2q 0

17 0 0 0��

0 0 0 2q

18 0 0 0 0��

0 0 0

19 0 0 0 0 0��

0 0

20 0 0 0 0 0 0��

0

21 0 0 0 0 0 0 0��

22 2p3(x8) 0 0 0 2q 0 2q 2q

23 0��

0 0 0 2q 0 2q

24 0 0��

0 0 0 2q 0

25 0 0 0��

0 0 0 2q

26 0 0 0 0��

0 0 0

27 0 0 0 0 0��

0 0

28 0 0 0 0 0 0��

0

29 0 0 0 0 0 0 0��

Now we state the general theorem, of which Theorem 1.3 is an imme-diate consequence, since the first occurrence of 2e−� along the diagonaloccurs in (3 · 2�−1 − 2, 3 · 2�−1 − 2).


Theorem 2.2. Let Pi,j denote the entries in the reduced polynomialmatrix for Me. The nonzero entries are

i. For 0 ≤ s ≤ e−1 and 3 ·2s−2 ≤ i < 4 ·2s−2 and 0 ≤ t ≤ e−1−s,

Pi,i+2s+1(2t−1) = 2e−1−s−tp2t+1−1(x2s).

ii. For 0 ≤ s ≤ e− 1 and 2 · 2s − 2 ≤ i < 3 · 2s − 2, Pi,i = 2e−s and,for 2 ≤ t ≤ e− s,

Pi,i+2s(2t−1) = 2e−s−tx2sp2t−2(x

2s).

iii. For 3 ≤ t ≤ e and 2t + 2t−2 − 2 ≤ j ≤ 2t + 2t−1 − 3, there arepossibly nonzero entries Pi,j = 2e−tqi,j for 0 ≤ i < j − 2t−1, andalso, for 1 ≤ v ≤ e− t,

Pi,j+2t(2v−1) = 2e−t−vp2v+1−1(x2t−1

)qi,j .

This generalizes Table 1 and Theorem 2.1. Note that some of theentries of type ii are among the entries of type iii. Note also that p1(x) =1, and that in part i for s = e− 1, we usually just consider the smallestvalue of i.

3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1 by proving the equivalent state-ment

(5) if 0 ≤ t < e and α(m) = t+ e, then b(m+ 2t − 1, e) ≥ 2m− 2t.

The case t = 0 is elementary ([5, (1.3)]) and is omitted. We will firstprove the following cases of (5) and then will show that all other casesfollow by naturality.

Theorem 3.1. For 1 ≤ t < e,

(6) b(3 · 2t−1 − 1 + 2t+1B, e) ≥ 2t+2B if α(B) = e+ t− 1,

and

(7) b(2t − 1 + 2tB, e) ≥ (2B − 1)2t if α(B) = e+ t.

12 Donald M. Davis

These are the cases m = 2α(B)−e(4B + 1) and m = 2α(B)−eB ofTheorem 1.1 or (5).

Proof. We focus on (6), and then discuss the minor changes requiredfor (7). Let n = 3 · 2t−1 − 1 + 2t+1B and suppose there is a map

L2n(2e)× L2n(2e) → L2t+3B−1(2e)

as in (1). Precompose with the self-map (1,−1) of L2n(2e) × L2n(2e),where −1 is homotopic to the Hopf inverse of the identity. Then, as in[1], we obtain

(x− y)2t+2B = 0 ∈ ku∗(L2n(2e)× L2n(2e)).

The result (6) will follow from showing that

(x− y)2t+2B �= 0 ∈ ku2(2n−d)(L2n(2e)× L2n(2e))

with n = 3 · 2t−1 − 1 + 2t+1B and d = 3 · 2t − 2. This group is iso-morphic to the component group Gd for Me whose presentation matrixwas described in Section 2. The ordered set of generators is obtained asxn−dyn−d multiplied by

(8) x0yd, . . . , xdy0, ux1yd, . . . , uxdy1, . . . , udxdyd.

We omit the xn−dyn−d throughout our analysis.

One easily shows that

ν

(2t+2B

j

){= α(B) j = 2t+1B

> α(B) 0 < |2t+1B − j| < 2t+1.

Here and throughout ν(−) denotes the exponent of 2 in an integer. Wewish to show that if t < e and d = 3 · 2t − 2, then 2e+t−1xd/2yd/2 +2e+tf(x, y) �= 0 in Gd, where f(x, y) is a polynomial of degree d in xand y.

In the reduced matrix for Pe we omit all columns and rows not ofthe form 3 · 2i − 3, 0 ≤ i ≤ t. Omitting columns amounts to takinga quotient, and when a column(generator) is omitted the row(relation)with its leading entry can be omitted, too. The resulting matrix ispresented below, where the various polynomials q are mostly distinct.


Table 4

0 3 9 3 · 23 − 3 · · · 3 · 2t−1 − 3 3 · 2t − 3

0 2e 2e−2p2 2e−3q 2e−4q 2e−tq 2e−t−1q

3 2e−1 2e−3p2(x2) 2e−4q 2e−tq 2e−t−1q

9 2e−2 2e−4p2(x4) 2e−tq 2e−t−1q

3 · 23 − 3 2e−3 2e−tq 2e−t−1q

.

.

.

...

3 · 2t−1 − 3 2e−t+1 2e−t−1p2(x2t−1)

3 · 2t − 3 2e−t

We temporarily ignore the polynomials q and the polynomial f(x, y).The first few relevant relations in the corresponding numerical matrixare xd/2yd/2 times the following polynomials. We omit writing powersof u; they equal the degree of the written polynomial.

2e + 2e−2(xy2 + x2y)

2e−1xy2 + 2e−3(x3y6 + x5y4)

2e−1x2y + 2e−3(x4y5 + x6y3)

2e−2x3y6 + 2e−4(x7y14 + x11y10)

2e−2x4y5 + 2e−4(x8y13 + x12y9)

2e−2x5y4 + 2e−4(x9y12 + x13y8)

2e−2x6y3 + 2e−4(x10y11 + x14y7).

From these relations, we obtain

2e+t−1 ∼ −2e+t−3(xy2 + x2y)(9)

∼ 2e+t−5(x3y6 + x4y5 + x5y4 + x6y3)

∼ −2e+t−714∑i=7

xiy21−i

∼ · · ·

∼ ±2e−t−12t+1−2∑i=2t−1

xiy3·2t−3−i

= ±2e−t−1(x3·2t−1−2y3·2

t−1−1 + x3·2t−1−1y3·2

t−1−2)

�= 0,

since maximum exponents are 3 · 2t−1 − 1. That the last line is nonzerofollows from the reduced form of the matrix Me of relations.

Now we incorporate the polynomials q in the above matrix. Wedenote by mi a monomial or sum of monomials of degree 3 · 2i − 3, in xand y. At the first step of the above reduction sequence, we would have

14 Donald M. Davis

an additional∑t

i=2 2t+e−i−2mi. At the second step, we add

(10)

t∑i=3

2t+e−i−3m′i.

We can incorporate the first monomials for i ≥ 3 into the second, andwe replace 2t+e−4m2 by

∑i≥3 2

t+e−i−3m′′i and incorporate these into

(10). The third step adds∑t

i=4 2t+e−i−4m′′′

i . We incorporate (10) intothis for i > 3, while the term in (10) with i = 3 is equivalent to a sumwhich can also be incorporated. Continuing, we end with

t∑i=t

2t+e−i−tm(t)t = 2e−tm

(t)t = 0,

so the q’s contribute nothing.

We easily see that incorporating 2e+tf(x, y) also contributes nothing,since

2e+tm ∼ 2e+t−2m1 ∼ 2e+t−4m2 ∼ · · · ∼ 2e+t−2tmt = 0.

The proof of (7) is very similar. We want to show (x−y)(2B−1)2t �= 0in G3·2t−2 if α(B) = e+ t and 1 ≤ t < e. For (B− 2)2t < j < (B+1)2t,we have

ν

((2B − 1)2t

j

){= α(B)− 1 if j = (B − 1)2t or B · 2t

> α(B)− 1 other j.

We have factored out xn−dyn−d with n−d = 2tB−2t+1+1. Our orderedset of generators is again (8), and our class now, mod higher 2-powers,is 2e+t−1(x2

t−1y2t+1−1 + x2

t+1−1y2t−1). Utilizing the relations similarly

to (9), we end with

±2e+t−1(x2t−1y2

t+1−1 + x2t+1−1y2

t−1)2t+1−2∑i=2t−1

xiy3·2t−3−i

= ±2e+t−1(x3·2t−3y3·2

t−2 + x3·2t−2y3·2

t−3) �= 0,

since xd+1 = 0 = yd+1 (after factoring out xn−dyn−d). �


Proof of (5). The proof is by induction on t. If t = 1, the theoremfollows from (6) with m = 4B + 1 if m ≡ 1 mod 4, and from (7) withm = 2B if m is even. If m ≡ 3 mod 4, then α(m+ 1) ≤ α(m)− 1 = e,so the result follows from the case t = 0 for n = m+ 1.

Now we assume that the result has been proved for all t′ < t. If mis odd, then α(m− 1) = e+ t− 1, so using the induction hypothesis inthe middle step,

b(m+ 2t − 1, e) ≥ b(m− 1 + 2t−1 − 1, e) ≥ 2(m− 1)− 2t−1 ≥ 2m− 2t.

If ν(m+ 2t) = k with 1 ≤ k ≤ t− 2, then, noting that ν(m) = k, too,

α(m− 2k) = (t+ e)− 2k + ν(m · · · (m− 2k + 1))

= t+ e− 2k + (2k − 1) = t+ e− 1.

Therefore

b(m+2t − 1, e) ≥ b(m− 2k +2t−1 − 1, e) ≥ 2(m− 2k)− 2t−1 ≥ 2m− 2t.

If ν(m) ≥ t, let m = 2tB with α(B) = α(m) = t + e. By (7), weobtain b(m+2t − 1, e) ≥ 2m− 2t, as desired. If m = 2t−1 +2t+1B withα(B) = t+ e− 1, then (6) is exactly the desired result.

Finally, if m = 3 · 2t−1 + 2t+1A with α(A) = t+ e− 2, then

α(m+ 2t−1) = α(A+ 1)

= α(A) + 1− ν(A+ 1)

= e+ v with v < t.

Thus, by the induction hypothesis,

b(m+2t − 1, e) ≥ b(m+2t−1 +2v − 1, e) ≥ 2(m+2t−1)− 2v ≥ 2m− 2t.

�


In this section we prove Theorem 2.1. Because it is a fairly complicatedrow reduction, we accompany the proof with diagrams of the matrix atseveral stages of the reduction. Although the proof of Theorem 2.2 inSection 5 is a complete proof and subsumes the much-longer proof for

16 Donald M. Davis

e = 6, we feel that the more explicit example renders the general proofmore comprehensible, or perhaps unnecessary.

If M is a Toeplitz matrix corresponding to a polynomial p(x) as de-scribed in the preceding section, then the Toeplitz matrix correspondingto the polynomial (1 + αx+ βx2)p(x) is obtained from M by adding αtimes each row to the one below it and β times each row to the row 2 be-low it. This illustrates how row operations on the matrix of polynomialscorrespond to row operations on the partitioned matrix of numbers.

Our matrices now refer to the case e = 6. The initial partitionedmatrix for Gd could be considered as the matrix of numbers associatedto the following matrix of polynomials, which has d + 1 columns and2(d+ 1) rows.

(11)

64(642

)x

(643

)x2

(644

)x3 · · ·

64(642

) (643

) (644

)· · ·

0 64(642

)x

(643

)x2 · · ·

0 64(642

) (643

)· · ·

0 0 64(642

)x · · ·

0 0 64(642

)· · ·

...

The first two row blocks of the associated matrices of numbers haved+ 1 rows of numbers, the next two d rows, etc., while the sizes of thecolumn blocks are d+1, d, . . .. The first (resp. second) (resp. third) rowblock corresponds to the first (resp. second) (resp. first) set of relationsin (3) with i+ j = d (resp. d) (resp. d− 1).

Note that if the first two rows and the first column of (11) aredeleted, we obtain exactly the initial matrix for Gd−1. We may assumethat the matrix for Gd−1 has already been reduced, to Qd−1. Thus wemay obtain the reduced form for Gd by taking Qd−1, placing a columnof 0’s in front of it and the top two rows of (11) above that, and thenreducing. By the nature of the matrix (11), the restriction of the reducedform Qd to its first d columns will be Qd−1.


This is an interesting property. Let Qd denote the reduced form ofthe polynomial matrix for Gd. Remove its last column, put a column of0’s in front, put the top two rows of (11) above this, and reduce. Theresult will be the original matrix, Qd. We will prove that the matrixdescribed in Theorem 2.1 is correct by removing its last column (theone with the 1 at the bottom), preceding the matrix by a column of 0’sand this by the first two rows of (11), and seeing that after reducing,we obtain the original matrix Q94. Because of the initial shifting, eachcolumn is determined by the column which precedes it, together with thereduction steps, which justifies the method of starting with the putativeanswer, shifted. This seems to be a rather remarkable proof. However,the reduction is far from being a simple matter.

Now we describe the steps in the reduction. We begin with the puta-tive answer pushed one unit to the right and two units down, precededby the first two rows of (11) and a column of 0’s. We often write Ri andCj for row i and column j.

Step 0: Subtract R0 from R1, then divide R1 by (1− x), and thensubtract xR1 from R0. These rows become64 0 −

(643

)x −

(644

)xp2 −

(645

)xp3 · · · −

(6464

)xp62 0 · · ·

0(642

) (643

)p2

(644

)p3

(645

)p4 · · ·

(6464

)p63 0 · · ·

Divide R1 by 63, which is the unit part of(642

). We now have, in R0

and R1,

Pi,j =

64 i = j = 0

32 i = j = 1

0 i+ j = 1

uj26−ν(j+1)xpj−1 i = 0, 2 ≤ j ≤ 63

u′j26−ν(j+1)pj i = 1, 2 ≤ j ≤ 63

0 0 ≤ i ≤ 1, 64 ≤ j ≤ 94,

where uj is the odd factor of −(

64j+1

), and u′j ≡ uj mod 64.

Our goal is to reduce this matrix so that the first nonzero entry(which we often call the “leading entry”) in Ri is

18 Donald M. Davis

(12)

64 in C0 i = 0

32 in C1 i = 1

16 in C4 i = 2

32 in Ci−1 3 ≤ i ≤ 4

8 in C10 i = 5

16 in Ci−1 6 ≤ i ≤ 10

4 in C22 i = 11

8 in Ci−1 12 ≤ i ≤ 22

2 in C46 i = 23

4 in Ci−1 24 ≤ i ≤ 46

1 in C94 i = 47

2 in Ci−1 48 ≤ i ≤ 94.

The above entries for i = 0, 1, 2, 5, 11, 23, and 47 will be the onlynonzero entry in their columns. Then we rearrange rows. For i = 2, 5,11, 23, and 47, Ri moves to position 2i. For other values of i > 2, Ri

moves to position i − 1. Then we are finished. The entries Pi,i will beas stated in Theorem 2.1, and the matrix will be upper triangular withnonzero entries above the diagonal less 2-divisible than the diagonalentry in their column.

Table 5 depicts the first 22 columns of the matrix at the end ofStep 0, except that we omit writing the odd factors in rows 0 and 1.

Table 5

0 1 2 3 4 5 6 7 8 9 10 11 12

0 64 0 64x 16xp2 64xp3 32xp4 64xp5 8xp6 64xp7 32xp8 64xp9 16xp10 64xp11

1 32 64p2 16p3 64p4 32p5 64p6 8p7 64p8 32p9 64p10 16p11 64p12

2 64 0 0 16xp2 0 0 0 8xp6 0 0 0 0

3 32 0 16p3 0 0 0 8p7 0 0 0 0

4 32 0 0 0 0 0 8x2p2(x2) 8xp6 0 0

5 32 0 0 0 0 0 8x2p2(x2) 0 0

6 16 0 0 0 8p3(x2) 8xp2(x

3) 0 0

7 16 0 0 0 8p3(x2) 0 0

8 16 0 0 0 0 0

9 16 0 0 0 0

10 16 0 0 0

11 16 0 0

12 8 0

13 8


13 14 15 16 17 18 19 20 21

0 32xp12 64xp13 4xp14 64xp15 32xp16 64xp17 16xp18 64xp19 32xp20

1 32p13 64p14 4p15 64p16 32p17 64p18 16p19 64p20 32p21

2 0 0 0 4xp14 0 0 0 0 4q

3 0 0 0 4p15 0 0 0 0 4q

4 0 0 0 0 4x2p6(x2) 4xp3(x4)p6 0 4q 4q

5 0 0 0 0 0 4x2p6(x2) 0 0 4q

6 0 0 0 0 4p7(x2) 4xp2(x3)p3(x4) 0 4q 4q

7 0 0 0 0 0 4p7(x2) 0 0 4q

8 0 0 0 0 0 0 4x4p2(x4) 0 4q

9 0 0 0 0 0 0 0 4x4p2(x4) 0

10 0 0 0 0 0 0 0 0 4x4p2(x4)

11 0 0 0 0 0 0 0 0 0

12 0 0 0 0 0 0 4p3(x4) 0 4q

13 0 0 0 0 0 0 0 4p3(x4) 0

14 8 0 0 0 0 0 0 0 4p3(x4)

15 8 0 0 0 0 0 0 0

16 8 0 0 0 0 0 0

17 8 0 0 0 0 0

18 8 0 0 0 0

19 8 0 0 0

20 8 0 0

21 8 0

22 8

Although it is just a simple shift, it will be useful to have for refer-ence, in Tables 6 and 7 below, the shifted versions of Tables 2 and 3.These are the relevant portions of the matrix at the outset of the reduc-tion. The shifted version of part b of Theorem 2.1 can be mostly seenin Table 5.

Table 6

19 20 21 22 35 36 37 38 67 68 69 70

2 0 0 4q 4q

3 0 0 4q 4q

4 0 4q 4q 4q

5 0 0 4q 4q B · 1

2p3(x8) B · 1

4p7(x8)

6 0 4q 4q 4q

7 0 0 4q 4q

8 4x4p2(x4) 0 4q 4q

9 0�

��

0 4q

10 0 0�

��

0

11 0 0 0��

12 4p3(x4) 0 4q 4q 2p7(x4) p15(x4)

13 0�

��

0 4q�

��

��

14 0 0�

��

0�

��

��

15 0 0 0��

��

20 Donald M. Davis

Table 7

39 40 41 42 43 44 45 46

2 0 0 2q 2q 2q 2q 2q 2q

3 0 0 2q 2q 2q 2q 2q 2q

4 0 2q 2q 2q 2q 2q 2q 2q

5 0 0 2q 2q 2q 2q 2q 2q

6 0 2q 2q 2q 2q 2q 2q 2q

7 0 0 2q 2q 2q 2q 2q 2q

8 0 0 2q 2q 2q 2q 2q 2q

9 0 0 0 2q 2q 2q 2q 2q

10 0 0 0 0 2q 2q 2q 2q

11 0 0 0 0 0 2q 2q 2q

12 0 0 2q 2q 2q 2q 2q 2q

13 0 0 0 2q 2q 2q 2q 2q

14 0 0 0 0 2q 2q 2q 2q

15 0 0 0 0 0 2q 2q 2q

16 2x8p2(x8) 0 0 0 2q 0 2q 2q

17 0��

0 0 0 2q 0 2q

18 0 0��

0 0 0 2q 0

19 0 0 0��

0 0 0 2q

20 0 0 0 0��

0 0 0

21 0 0 0 0 0��

0 0

22 0 0 0 0 0 0��

0

23 0 0 0 0 0 0 0��

24 2p3(x8) 0 0 0 2q 0 2q 2q

25 0��

0 0 0 2q 0 2q

26 0 0��

0 0 0 2q 0

27 0 0 0��

0 0 0 2q

28 0 0 0 0��

0 0 0

29 0 0 0 0 0��

0 0

30 0 0 0 0 0 0��

0

31 0 0 0 0 0 0 0��

At any stage of the reduction, let Ri denote Ri with its leadingentry changed to 0. The first nonzero entry of Ri at the outset occurs


in column

(13)

i+ 5 4 ≤ i ≤ 5

i+ 3 6 ≤ i ≤ 7

i+ 11 8 ≤ i ≤ 11

i+ 7 12 ≤ i ≤ 15

i+ 23 16 ≤ i ≤ 23

i+ 15 24 ≤ i ≤ 31

i+ 47 32 ≤ i ≤ 47

i+ 31 48 ≤ i ≤ 63.

For i ≥ 64, Ri has no nonzero elements.

The relationship between the three parts of Table 6 and the similarrelationship that columns 71 to 78 are mostly 1

2p3(x16) times Table 7 will

be very important. We call it a “proportionality” relation. We extendit to also include that in rows 4, 5, and 6 we have C18/C10 = 1

2p3(x4),

C34/C10 = 14p7(x

4), and C66/C10 = 18p15(x

4), and similarly in row 4,columns 9, 17, 33, and 65. When we perform row operations involvingthese rows, these relationships continue to hold. Rows 12–15 and 24–31, where the proportionality relationship does not hold, will not beinvolved in row operations, since the columns in which their leadingentry occurs have all 0’s above the leading entry. (Although rows 0and 1 are initially nonzero in these columns, clearing out R0 and R1,as in Step 1 below, is a 2-step process, and so Ri for 12 ≤ i ≤ 15 or24 ≤ i ≤ 31 will not be combining into R0 or R1, either.)

In Steps 3, 6, 9, and 12, we will divide rows 2, 5, 11, and 23 by x,x2, x4, and x8. It will be important that the entire rows are divisibleby these powers of x. We keep track of bounds for the x-divisibilityof the unspecified polynomials in Table 6 and 7 and in columns 79 to94. We postpone this analysis until all the reduction steps have beenoutlined. Similarly to the proportionality considerations just discussed,divisibility bounds are preserved when we add a multiple of one row toanother, in that the x-exponent of Pi,j + cPi′,j is ≥ the minimum ofthat of Pi,j and Pi′,j . The rows, 3, 6–7, 12–15, 24–31, and 48–63, whereentries not divisible by x occur will not be used to modify other rows.

Now we begin an attempt to remove most of the binomial coefficientsfrom R0 and R1.

22 Donald M. Davis

Step 1. The goal is to add multiples of lower rows to R0 and R1 toreduce them to

0 1 2 3 4 5 6 7 · · · 15 · · · 31 · · · 63 · · ·0 64 0 0 16xp2 0 0 0 8xp6 0 4xp14 0 2xp30 0 xp62 0

1 0 32 0 16p3 0 0 0 8p7 0 4p15 0 2p31 0 p63 0

with each 0 referring to all intervening columns. However, we will beforced to bring up some additional entries. We claim that, after Step1, the nonzero entries P1,j , in addition to those in columns 2t − 1 listed

just above, are combinations of various Rj with j ≥ 5 and j not in[6, 8] ∪ [12, 16] ∪ [24, 32] ∪ [48, 64]. Row 0 is similar but has an extrapower of x, since this is true at the outset. Rows 0 and 1 will thus havethe requisite proportionality and x-divisibility relations.

It will be useful to note that since at the outset all entries in Ri

for i ≥ 2 are a multiple of 12 times the leading entry at the bottom of

their column, then, using (13), 2Ri can be killed (reduced to all 0’s)by subtracting multiples of lower rows if i ≥ 32. For example, nonzeroentries of R32 occur only in Cj with j ≥ 79. If the entry in (32, j) is a

polynomial q, then subtracting qRj+1 from 2R32 kills the entry in Cj

without changing anything else, since Rj+1 = 0 for such j.

Similarly 4Ri can be killed in two steps if i ≥ 16, and 8Ri can bekilled if i ≥ 8. We can use this observation to kill the entries in R0 andR1 in many columns.

For example, if 32 ≤ j ≤ 46, then the numerical coefficient in P0,j

and P1,j is 0 mod 8, while there is a leading 4 in (j +1, j). Subtractingmultiples of 2Rj+1 from R0 and R1 kills the entries in (0, j) and (1, j)

while bringing up multiples of 2Rj+1. This can be killed by the obser-vation of the previous two paragraphs. This method works to eliminatethe entries in R0 and R1 in columns 12, 14, 16–18, 20–22, 24–30, and32–62. (Initial entries in R0 and R1 in columns > 63 were all 0.) Since−(642t

)≡ 26−t mod 213−2t for 2 ≤ t ≤ 6, the entries in R0 and R1 in

columns 3, 7, 15, 31, and 63 can be changed to their desired values withpure 2-power coefficients by similar steps.

For C23, we subtract even multiples of R24 from R0 and R1 to killthe entries. This brings into R0 and R1 multiples of 4 in some columns39 to 46 and even entries in some columns ≥ 71. The latter entriescan be cancelled from below, while cancelling multiples of 4 in Cj for


39 ≤ j ≤ 46 brings up multiples of Rj+1. A very similar argument andsimilar conclusion works for removal of entries in (0, 19) and (1, 19).

Now we consider C11. We subtract multiples of 2R12 to kill theentries in R0 and R1. This brings up multiples of 8 in C19, C21, andC22, 4 in Cj for 35 ≤ j ≤ 46, and 2 in some columns > 64, the latterof which can be cancelled from below. We kill the earlier elements withmultiples of Rj+1, leaving a combination of the various Rj+1

Column 9 is eliminated similarly, giving multiples of R21, R37, andsome others, while columns 8, 10, and 13 are, in a sense, easier sincetheir binomial coefficients are 4 times the number at the bottom of theircolumn, rather than 2. For example, to kill the entry in (1, 13), we firstsubtract a multiple of 4R14. This contains a 16q in C21, which is killedby a multiple of 2R22. This brings up a 2q′ in R45, the killing of whichbrings up a multiple of R46.

To kill the entry in (1, 5), we subtract a multiple of 2R6, which hasentries in Cj for many values of j ≥ 9. We can cancel each of theseby subtracting a multiple of Rj+1, accounting for the contributions to

R1 of multiples of many Rk with k �∈ [6, 8] ∪ [12, 16] ∪ [24, 32] ∪ [48, 64].Killing the entries in C4 and C6 is similar.

Finally, to kill the entry in (1, 2), we subtract a multiple of 2R3.This brings up entries in columns 4, 8, 16, 32, 64, and others, the killingof which brings up combinations of R5, R9, R17, and R33, as allowed.

Step 2. Subtract 2R1 from R2 to remove the 64 in P2,1. This bringsentries into R2 in columns

(14) j ∈ {3, 7, 10, 15, 18, 20-22, 31, 34, 36-38, 40-46}

and others with j ≥ 63. The entry brought into Cj has numerical coeffi-cient equal to Pj+1,j . These are then killed by subtracting correspondingmultiples of Rj+1, which brings up into R2 corresponding multiples of

Rj+1 for j as in (14). From R4, this will place q = 8x2p2(x2)p3 in C9,

12p3(x

4)q in C17,14p7(x

4)q in C33, and18p15(x

4)q in C65. This extendsthe proportionality property of columns 9, 17, 33, 65 to include also row2.

Now R2 has 16xp2 as its leading entry, in column 4.

24 Donald M. Davis

Step 3. Divide R2 by xp2. Dividing by a polynomial p of the form1+

∑αix

i, such as p2, is not a problem. If M is a Toeplitz matrix corre-sponding to a polynomial q, then the Toeplitz matrix corresponding toq/p is obtained from M by performing the row operations correspondingto finitely many of the terms of the power series 1/p. Dividing by x ismore worrisome, and is the reason for much of our work. In this step itis not a big problem, but later, when we have to divide by x4 and x8,more care is required, which will be handled in Theorem 4.2 after allsteps have been described.

We have the important relation

(15) p2t/p2 = pt(x2),

which implies that the entries in P2,j for j = 8, 16, 32, and 64 arenow 8p3(x

2), 4p7(x2), 2p15(x

2), and p31(x2). The relation (15) and its

variants will be used frequently without comment. In C9, we obtain

8xp2(x

2)p3p2

= 8xp2(x3) + 16x3/p2.

We use R10 to cancel the second term, at the expense of bringing upmultiples of x3R10 into R2. This satisfies proportionality properties,which continue to hold.

Step 4. Subtract p3R2 from R3 to change P3,4 to 0. Since

(16) p2k+1 − p3pk(x2) = −x2pk−1(x

2),

we obtain −8x2p2(x2) in P3,8, −4x2p6(x

2) in P3,16, and similar expres-sions in C32 and C64. We can change the minus to a plus by addinga multiple of R9, R17, etc. This brings up multiples of R9, R17, etc.,into R3, but these maintain proportionality and x-divisibility proper-ties. Note that x-divisibility keeps changing. For example, in Step 3,that of R2 was decreased by 1, and now all that we can say is thatthe x-divisibility of R3 is at least the minimum of that of R2 and itsprevious value for R3. But this will be handled later.

For the convenience of the reader, we list here columns 0 through 10at this stage of the reduction. Some of the specific polynomials are notvery important, and will later just be called q.


0 1 2 3 4 5 6 7 8 9 10

0 64 0 0 16xp2 0 0 0 8xp6 0 0 8x3p2p2(x3)

1 32 0 16p3 0 0 0 8p7 0 0 8x2p2(x2)p2(x

3)

2 0 0 0 16 0 0 0 8p3(x2) 8xp2(x

3) 8p3p3(x2)

3 32 0 0 0 0 0 8x2p2(x2) 8xp6 8p3(x

4)

4 32 0 0 0 0 0 8x2p2(x2) 8xp6

5 32 0 0 0 0 0 8x2p2(x2)

6 16 0 0 0 8p3(x2) 8xp2(x

3)

7 16 0 0 0 8p3(x2)

8 16 0 0 0

9 16 0 0

10 16 0

11 16

Step 5. Subtract 2R2 from R5 to remove the leading entry in R5. IfP2,j = q for j > 4, then adding qRj+1 to R5 will cancel the subtracted

entry, at the expense of adding qRj+1 to R5. So R5 gets multiples of

Rj+1 for many values of j in the intervals [8, 10], [16, 22], and [32, 46].The rows that we don’t want to bring up are 12–15, 24–31, etc., whichcontain the lower diagonals in Tables 6 and 7, where neither propor-tionality nor x-divisibility holds.

Now the leading entry of R5 is 8x2p2(x2) in C10.

Step 6. Divide R5 by x2p2(x2). We need to know that all entries

in R5 are divisible by x2. In Theorem 4.2, we will show that this is truefor columns 19–22, 35–46, and 67–94. The only other nonzero entries inR5 are those in columns 10, 18, 34, and 66 with which it started. SeeTable 5. The first nonzero entries in R5 after dividing are 8 in C10 and4p3(x

4) in C18.

Step 7. Subtract multiples of R5 from rows 0, 1, 2, 3, 4, 6, and7 to clear out C10 in these rows. Because it had been the case thatPi,18/Pi,10 = 1

2p3(x4) for 0 ≤ i ≤ 6, we will now have Pi,18 = 0 for

i ∈ {0, 1, 2, 3, 4, 6}. Also, by (16), P7,18 = 4(p7(x2) − p3(x

2)p3(x4)) =

−4x4p2(x4). We can change the minus to a plus by adding x4p2(x

4)R19.Similarly, the only nonzero entries in column 34 (resp. 66) (except forPj+1,j) are 2p7(x

4) (resp. p15(x4)) in R5, and 2x4p6(x

4) (resp. x4p14(x4))

in R7. This illustrates why the proportionality relations are important.

Step 8. Subtract 2R5 from R11 to remove the leading entry in R11.Similarly to Step 5, if P5,j = q for j > 10, then adding qRj+1 to R11 will

26 Donald M. Davis

cancel the subtracted entry, at the expense of adding qRj+1 to R11. So

R11 gets multiples of Rj+1 for many values of j in the intervals [18, 22]and [34, 46].

The first 23 columns now are as below.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140 64 0 0 16xp2 0 0 0 8xp6 0 0 0 0 0 0 01 32 0 16p3 0 0 0 8p7 0 0 0 0 0 0 0

2 0 0 0 16 0 0 0 8p3(x2) 8q0 0 0 0 0 0

3 32 0 0 0 0 0 8x2p2(x2) 8q1 0 0 0 0 0

4 32 0 0 0 0 0 8x2p2(x2) 0 0 0 0 05 0 0 0 0 0 0 8 0 0 0 0

6 16 0 0 0 8p3(x2) 0 0 0 0 07 16 0 0 0 0 0 0 0 08 16 0 0 0 0 0 0 09 16 0 0 0 0 0 0

10 16 0 0 0 0 011 0 0 0 0 012 8 0 0 013 8 0 014 8 015 8

15 16 17 18 19 20 21 22

0 4xp14 0 0 0 0 4q 4q 4q

1 4p15 0 0 0 0 4q 4q 4q

2 0 4p7(x2) 8q0p3(x4) 0 4q 4q 4q 4q

3 0 4x2p6(x2) 8q1p3(x4) 0 4q 4q 4q 4q

4 0 0 4x2p6(x2) 0 0 4q 4q 4q

5 0 0 0 4p3(x4) 0 4q 4q 4q

6 0 0 4p7(x2) 0 0 4q 4q 4q

7 0 0 0 4x4p2(x4) 0 4q 4q 4q

8 0 0 0 0 4x4p2(x4) 0 4q 4q

9 0 0 0 0 0 4x4p2(x4) 0 4q

10 0 0 0 0 0 0 4x4p2(x4) 0

11 0 0 0 0 0 0 0 4x4p2(x4)

12 0 0 0 0 4p3(x4) 0 4q 4q

13 0 0 0 0 0 4p3(x4) 0 4q

14 0 0 0 0 0 0 4p3(x4) 0

15 0 0 0 0 0 0 0 4p3(x4)

16 8 0 0 0 0 0 0 0

17 8 0 0 0 0 0 0

18 8 0 0 0 0 0

19 8 0 0 0 0

20 8 0 0 0

21 8 0 0

22 8 0

23 8

In addition, we have, at this stage of the reduction:

a. 4 in Pj+1,j for 23 ≤ j ≤ 46, and 2 in Pj+1,j for 47 ≤ j ≤ 94. Otherthan that:

b. 0 in columns 23 to 30 and 47 to 62.

c. A pattern resembling that of columns 15 to 18 in columns 31 to34 and 63 to 66.

d. Columns 35 to 38 (resp. 67 to 70) are 12p3(x

8) (resp. 14p7(x

8))times columns 19 to 22, except that corresponding to the 4p3(x

4)in rows 12 to 15 we have 2p7(x

4) (resp. p15(x4)).


e. Columns 39 to 46 resemble Table 7. Columns 71 to 78 are 12p3(x

16)times these, except for the diagonal near the bottom, which isp7(x

8).

f. Columns 79 to 94 have a form similar to that of columns 39 to 46.

g. The x-divisibility in columns 19–22, 39–46, and 79–94 will be de-scribed in Theorem 4.2 and its proof.

Step 9. Divide R11 by x4p2(x4). We will show in Theorem 4.2 that

all entries in R11 are divisible by x4. The leading entry in row 11 is nowa 4 in C22.

Step 10. Subtract multiples of R11 from rows 0 to 10 and 12 to15 to clear out their entries in C22. Similarly to Step 7, we now havethat Pi,38 = 0 except for P11,38 = 2p3(x

8), P15,38 = 2x8p2(x8), and

P39,38 = 4, with a similar situation in C70. In particular, P15,70 =x8p6(x

8) = 12p3(x

16)P15,38.

Step 11. Subtract 2R11 from R23, and, similarly to Steps 5 and8, kill entries subtracted from P23,j for j > 22 by adding multiples of

Rj+1, thus bringing up these multiples of Rj+1. The smallest such j is38, due to the entry in (11, 38) described in the previous step.

Step 12. Now the leading entry of R23 is 2x8p2(x8) in C46. (This

can be seen using (13) and that there have been no other changes toR23 in columns less than 62.) Divide R23 by x8p2(x

8). We will showlater that all entries in R23 are divisible by x8 at this stage.

Step 13. Subtract multiples of R23 from rows 0 to 22 and 24 to 31to make their entries in C46 equal to 0. Similarly to Step 10, this willcause Pi,78 = 0 except for P23,78 = p3(x

16), P31,78 = x16p2(x16), and

P79,78 = 2.

Step 14. Subtract 2R23 from R47. This will add multiples of 2to R47 in some columns j ≥ 78. These can be removed, without anyother effect, by subtracting a multiple of Rj+1. Now R47 has leadingentry x16p2(x

16) in C94. Divide R47 by x16p2(x16), and then subtract

multiples of R47 from the others to clear out C94.

Step 15. We are now in the situation described in the paragraphcontaining (12). Rearrange rows as specified there, and we are done.

28 Donald M. Davis

It remains to show that Steps 3, 6, 9, and 12 above could actually becarried out, by showing that there was sufficient divisibility by x. Thiswill follow from Theorem 4.2.

Definition 4.1. Let ∆(0) = 3 and ∆(1) = 2. For i ≥ 2, let b(i)denote the largest integer ≤ i of the form 2t − 1 or 3 · 2t − 1, and let∆(i) = i− b(i).

For example, the values of ∆(i) for 2 ≤ i ≤ 17 are as in the followingtable.

i 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

∆(i) 0 0 1 0 1 0 1 2 3 0 1 2 3 0 1 2

Theorem 4.2. Let ν(i, j) denote the exponent of x in Pi,j at any stageof the reduction from the end of Step 1 to the end of Step 14. Then

• If 19 ≤ j ≤ 22 and 0 ≤ i ≤ j − 8, then ν(i, j) ≥ 22− j +∆(i).



Since this applies to any stage of the reduction, it says that all x-exponents in these columns are nonnegative at the end of Steps 3, 6, 9,and 12, which means that there was enough x-divisibility to perform thestep. The divisibility of other columns in rows 2, 5, 11, and 23 at Steps3, 6, 9, and 12 is easily checked, mostly following from proportionality.

Proof. We give the proof for 79 ≤ j ≤ 94. The proof for the smallerranges is basically the same. The proof is by induction on j. By Theo-rem 2.1(e) shifted, at the outset ν(32, 79) = 16, while ν(i, 79) = ∞ fori �= 32 and i ≤ 47. If j ≥ 80, we assume the result is known for j − 1.With the rearranging and shifting, we start with, for i ≥ 2,

ν(i, j) =

νE(i− 1, j − 1) i �∈ {2, 3, 6, 12, 24, 48}νE(

12 i− 1, j − 1) i ∈ {6, 12, 24, 48}

νE(i− 2, j − 1) i ∈ {2, 3},

where νE(−,−) refers to the value of ν at the end of Step 14. By theinduction hypothesis, this is

≥

94− j + 1 +∆(i− 1) i �∈ {2, 3, 6, 12, 24, 48}94− j + 1 +∆(12 i− 1) = 94− j + 1 +∆(i− 1) i ∈ {6, 12, 24, 48}94− j + 1 + 5− i i ∈ {2, 3}.


Let µ(i, j) denote a lower bound for ν(i, j)− (94− j). At the outset, wehave, for all i ≥ 4 and j ≥ 79,

µ(i, j) ≥ ∆(i− 1) + 1,

while µ(2, j) ≥ 4 and µ(3, j) ≥ 3.

We will go through the steps of the reduction and see how µ changes.We can dispense with j as part of the notation. We will now call it µ(i).To emphasize that µ is changing, we will let µk denote the value ofµ after Step k. We have µ0(i) ≥ ∆(i − 1) + 1 for i ≥ 4, µ0(2) ≥ 4and µ0(3) ≥ 3. Although it is possible that actual divisibility couldincrease after a step (by having terms of smallest exponent cancel), ourlower bounds, being just bounds, cannot see this. Thus we always haveµk+1(i) ≤ µk(i), so we wish to prove that µ14(i) ≥ ∆(i).

Step 1 sets

µ1(1) ≥ min(µ0(5), µ0(9), µ0(17), µ0(33), µ0(10),(17)

µ0(18), µ0(34), µ0(20), µ0(36), µ0(40)) = 2

and µ1(0) = µ1(1) + 1 ≥ 3. Of course, µ1(i) = µ0(i) for i > 1, sinceStep 1 is only changing R0 and R1. In asserting (17), it is relevant thatthe various Ri which affect R1 do not include i = 2t or 3 ·2t, since thoseare the only i for which µ0(i) = 1.

Step 2 sets

µ2(2) ≥ min(µ1(2), µ1(4), µ1(8), µ1(16), µ1(32)) ≥ 1.

Other rows that affect R2 would contribute exponents at least this large.Step 3 subtracts 1 from µ2(2), so now µ3(2) ≥ 0. Step 4 sets

µ4(3) ≥ min(µ3(3), µ3(2)) ≥ 0.

We have µ4(5) = µ0(5) ≥ 2. Step 5 does not change this estimate,i.e., µ5(5) ≥ 2, because at Step 5, R5 is not affected by any of the rows,i = 2t with t ≥ 1 or i = 3 · 2t with t ≥ 0, for which µ4(i) < 2. This isdue to the fact that, for these values of i, Rk is 0 in Ci−1 throughout thereduction for all k ≥ 2. Step 6 subtracts 2 from µ(5), so now µ6(5) ≥ 0.

For Step 7, we need to know the x-exponents of the entries in C10

at this stage of the reduction. These exponents in row i will be 3, 2, 0,0, 1, 1, 0 for i = 0, 1, 2, 3, 4, 6, and 7. These can be seen in the table at

30 Donald M. Davis

the end of Step 4, or by noting that the entries in rows 4, 6, and 7 willbe unchanged from their values in Table 5, while R1 got x2 from R5 atStep 1, R2 got x from R4 at Step 2, then changed to x0 at Step 3, whileR3 then got x0 at Step 4. For these values of i, we obtain that µ7(i)is ≥ the minimum of µ6(i) and the exponent listed above. It turns outthat the only change is µ7(7) ≥ 0. Our bounds now for i from 0 to 7are 3, 2, 0, 0, 1, 0, 1, 0.

Since µ7(11) ≥ 4 and µ7(i) ≥ 4 for 19 ≤ i ≤ 23 and 35 ≤ i ≤ 47,we obtain µ8(11) ≥ 4, and then µ9(11) ≥ 0. For Step 10, we need toknow exponent bounds in C22 at this stage of the reduction, becauseit is these multiples of R11 that are being subtracted from the row inquestion. For i < 15, they will be the same as the µ-values that weare computing here, because the same steps apply. However, we haveµ10(15) = 0 due to the 4p3(x

4)-entry in P15,22. Our exponent boundsµ10(i) now for i from 0 to 15 are 3, 2, 0, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3,0.

Since µ10(23) ≥ 8 and µ10(i) ≥ 8 for 39 ≤ i ≤ 47, we obtainµ11(23) ≥ 8, and then µ12(23) ≥ 0. For Step 13, we need to knowexponent bounds in C46 at this stage of the reduction, because it is thesemultiples of R23 that are being subtracted from the row in question. Fori < 31, they will be the same as the µ-values that we are computinghere, because the same steps apply. However, we have µ13(31) = 0 dueto the 2p3(x

8)-entry in P31,46.

In Step 14, we obtain µ14(47) = 0, with no other changes to µ. Ourfinal values for µ14(i) are 0 for i = 2, 3, 5, 7, 11, 15, 23, 31, and 47, andincreasing in increments of 1 from one of these to the next. This equals∆(i), as claimed. �


In this section, we prove Theorem 2.2 by defining a sequence of matricesN0, . . . , Ne−1 at various stages of the reduction, and then show that Ns

reduces to Ns+1. After its rows are rearranged, Ne−1 will become thematrix described in Theorem 2.2. We explain in Theorem 5.2 how N0

is obtained from Ne−1. Comparing with the case e = 6, N0 through N4

are the matrix after Steps 1, 4, 7, 10, and 13, respectively, while N5 isthe matrix at the end of Step 14 except that P95,94 has not yet beencleared out.


In the following, lg(−) denotes [log2(−)], and δi,j is the usual Kro-necker symbol. We continue to suppress e from the notation.

Definition 5.1. For 0 ≤ s ≤ e− 1, Ns is a matrix with rows numberedfrom 0 to 3 · 2e−1 − 1, and columns from 0 to 3 · 2e−1 − 2 satisfying

a. Its leading entries are

• 2e in (0, 0) and 2e−1 in (1, 1);

• for 0 ≤ k ≤ e− 1, 2e−k in (i, i− 1) for

3 · 2k−1 ≤ i ≤ 3 · 2k −

{2 k ≤ s− 1

1 k ≥ s;

• for 1 ≤ � ≤ s, 2e−�−1 in (3 · 2�−1 − 1, 3 · 2� − 2).

b. For 2 ≤ �+ 2 ≤ t ≤ e, it has

• 2e−tx2�p2t−�−2(x

2�) in (2�+1+m−1− δ�+m,0, 2t+2�−2+m)

for

0 ≤ m ≤ 2� − 1 � < s

0 ≤ m ≤ 2� + 0 � = s

1 ≤ m ≤ 2� + 0 � > s;

• 2e−tp2t−�−1(x2�) in (3 · 2� +m− 1, 2t + 2� − 2 +m) for

1 ≤ m ≤ 2� +

{−1 � < s

0 � ≥ s,

and in (�3 · 2�−1� − 1, 2t + 2� − 2) if � ≤ s.

c. Except for the leading entries described in (a),

• all entries in Cj are 0 for 3 · 2k − 1 ≤ j ≤ 4 · 2k − 2, k ≥ 0,as are those in C3·2k−2 if k < s, while the only additionalnonzero entry in C3·2s−2 is 2e−s−1 in row 3 · 2s−1 − 1;

• if t ≥ 2 and j = 2t + d with −1 ≤ d ≤ 2t−1 − 2, then Pi,j = 0if i ≥ d+ 2lg(d+1.5)+1 + 2;

• if k ≥ 0 and i = 3 · 2k − 1, then Pi,j = 0 for i ≤ j < 2i.

32 Donald M. Davis

d. For 3 ≤ t < u ≤ e, 2t−2 − 1 ≤ d ≤ 2t−1 − 2, and i ≤ d+ 2t−1,

Pi,2u+d = 12u−t p2u−t+1−1(x

2t−1)Pi,2t+d.

This is also true for d = 2t−2 − 2 if s ≥ t − 2, except in row3 · 2t−3 − 1.

e. If 2 ≤ t ≤ e− 1, 3 · 2t − 2t−1 − 1 ≤ j ≤ 3 · 2t − 2, and i ≤ j − 2t,then Pi,j is divisible by xν with ν = 3 · 2t − 2− j + ηs(i), where

η0(i) =

3− i 0 ≤ i ≤ 1

6− i 2 ≤ i ≤ 3

i− c(i) + 1 i ≥ 4,

with c(i) the largest integer ≤ i of the form 2v or 3 · 2v, and

ηs(i) =

{0 if i+ 1 = 3 · 2v or 4 · 2v for 0 ≤ v < s

η0(i) otherwise.

Theorem 2.2 is an immediate consequence of the following result,together with the discussion preceding Step 0 of Section 4.

Theorem 5.2. Let Ns denote the matrices of Definition 5.1.

1. After subtracting 2R3·2e−2−1 from R3·2e−1−1 and then rearrangingrows, Ne−1 satisfies the properties of Theorem 2.2. Call this re-arranged matrix Q. The rearranging is that for i = 3 · 2t − 1 with0 ≤ t ≤ e − 2, Ri moves to position 2i, while for other values ofi > 2, Ri moves to position i− 1.

2. Delete the last column of Q, precede this by a column of 0’s, andprecede this by the following two rows.

0 1 2 3 2e − 1 2e 3 · 2e−1 − 2

0 2e(2e

2

)x

(2e

3

)x2

(2e

4

)x3 · · · x2e−1 0 . . . 0

1 2e(2e

2

) (2e

3

) (2e

4

)· · · 1 0 . . . 0

Then perform the 2e-analogues of Steps 0 and 1 of Section 4. Theresult is the matrix N0.

3. For 0 ≤ s ≤ e− 2, the matrix Ns reduces to Ns+1.


Proof. Part 1 is straightforward but tedious and mostly omitted. As anexample of the comparison, the final case of the second • of Definition5.1(b), after rearranging and changing t to T , says

P3·2�−2,2T+2�−2 = 2e−T p2T−�−1(x2�).

With � = s and T = s + t + 1, this becomes the case i = 3 · 2s − 2 ofTheorem 2.2(i).

Next we address Part 2. After shifting and performing Step 0 ofSection 4, we will have the 2e analogue of Table 5, in which we recallthat odd factors were not written. It is easy but tedious to verify thateverything except rows 0 and 1 will be as stated for N0. For example,the first • of Definition 5.1(b) with its s = 0, and t replaced by Tbecomes

P2�+1+m−1,2T+2�−2+m = 2e−Tx2�p2T−�−2(x

2�) for 1 ≤ m ≤ 2�

for � > 0. With � = s and t = T − s, this matches with part ii ofTheorem 2.2 shifted 2 down and 1 to the right.

Part e of Definition 5.1 for Part 2 is somewhat delicate. We hadηe−1(i) = 0 for i = 2, 3, 5, 7, 11, 15,. . ., i.e. i = 2t − 1 or 3 · 2t − 1,with ηe−1 increasing by 1’s between these values of i. The rearrangingdone in Part 1 puts these 0’s in i = 4, 2, 10, 6, 22, 14,. . ., i.e. i = 2t − 2or 3 · 2t − 2, with η again increasing by 1’s between these values of i.Shifting these down by 2, as is done in Part 2, puts the 0’s in 2t and3 · 2t, starting with i = 4, but we add 1 to the η values because of theshift of columns. For example, column 21 had ν ≥ 1 + η, but this nowapplies to column 22, where it is interpreted as 0 + (η + 1). The valuesof η0(2) and η0(3) are 1 greater than ηe−1(0) and ηe−1(1), respectively.These values are all as claimed of η0(i) for i ≥ 2.

We kill the terms in R0 and R1 except for those in columns of theform 2t − 1 by the method of Step 1 of Section 4. For example, if j isof the form 3 · 2t − 1 or 5 · 2t − 1, t ≥ 0, then the 2-exponent in R0 andR1 is 1 greater than that in Rj+1, which is a leading entry. We subtract

multiples of 2Rj+1 to kill the terms. This brings up multiples of 2Rj+1.If this is nonzero in Ck, the term brought up can be killed by subtractinga multiple of Rk+1. This brings up multiples of Rk+1. Because columns11–15, 23–31, etc., i.e. those j satisfying 3 · 2t− 1 ≤ j ≤ 4 · 2t− 1, are 0,we will not bring up Ri for i from 12–16, 24–32, etc., and these are the

34 Donald M. Davis

only rows which contain entries which do not satisfy the proportionalityand x-divisibility conditions stated in d and e of Definition 5.1, and theonly rows that will have η(i) < 2. Thus we will obtain η0(1) ≥ 2, andη0(0) ≥ 3 since R0 has an extra factor of x as compared to R1.

Similar reasoning applies to columns j not of the form 3 · 2t − 1 or5 · 2t − 1. If also j �= 2t − 1, then the 2-exponent in R0 and R1 willexceed that in Rj+1 by more than 1. We can use an even multiple atone of the two steps of the previous paragraph, or can break it up intomore steps, which will make the rows eventually brought up have largervalues of i, but, either way, we will not be bringing up the bad rowssuch as 12–16, etc., and so all the properties will be transferred to R0

and R1. Changing the terms −(2e

2t

)in C2t−1 to 2e−t is accomplished

similarly, using that these differ by a multiple of 2e−t+2, while the entryin (2t, 2t − 1) has 2-exponent e− t+ 1.

There are three steps to the reduction in Part 3, analogous to Steps5, 6, and 7 in Section 4. Note that the only nonzero entries of Ns inC3·2s−2 are 2e−s−1 in R3·2s−1−1, and 2e−s in R3·2s−1, and the secondnonzero entry in R3·2s−1 is 2e−s−2x2

sp2(x

2s) in C3·2s+1−2. The first

step is to subtract 2R3·2s−1−1 from R3·2s−1. If R3·2s−1−1 has q �= 0 inCj , then the −2q brought into R3·2s−1 can be killed by adding qRj+1.The net effect is to remove the leading entry of R3·2s−1, making the2e−s−2x2

sp2(x

2s) in C3·2s+1−2 its new leading entry, and to bring into

this row various qRj+1 for which P3·2s−1−1,j �= 0. By (c), such j must

satisfy j > 2s+2 + 1, and then nonzero entries in Rj only occur incolumns > 2s+3+1. This extends the first • in (c) to include also k = s,which is needed for Ns+1.

We must also consider the effect of these changes on η(3 · 2s − 1).We had ηs(3 · 2s − 1) = η0(3 · 2s − 1) = 2s. It follows from (c) that noneof the j’s appearing above can satisfy 3 · 2t − 1 ≤ j ≤ 3 · 2t + 2s − 3 or4 · 2t − 1 ≤ j ≤ 4 · 2t + 2s − 3, t ≥ s, which are the only values havingηs(j + 1) < 2s. Thus η(3 · 2s − 1) does not change at this step.

The second step divides R3·2s−1 by x2sp2(x

2s). This can be donebecause ηs(3 · 2s − 1) ≥ 2s. The dividing changes η(3 · 2s − 1) to 0,which is consistent with the claim for ηs+1(3 ·2s−1). This step changesP3·2s−1,2u+2s+1−2 from 2e−ux2

sp2u−s−2(x

2s) to 2e−up2u−s−1−1(x2s+1

) foru ≥ s+ 2. It removes the entry in the first • of (b) with = s, m = 2s,t = u and adds the final entry in the second • of (b) with = s+1 andt = u. Now C3·2s+1−2 has


• 2e−s−2 in row 3 · 2s − 1;

• 2e−s−2p3(x2s) in row 2s+2 − 1;

• multiples of 2e−s−2 in rows 0 through 2s+2 − 1;

• a leading 2e−s−1 in row 3 · 2s+1 − 1;

• other entries 0.

Now we subtract multiples of row 3 ·2s−1 from all other rows exceptrow 3 · 2s+1− 1 to make them 0 in column 3 · 2s+1− 2. By property (d),this will zero all entries in column 2u + 2s+1 − 2, u > s + 2, except inrows 3 · 2s − 1, 2s+2 − 1, and 2u + 2s+1 − 1. For u > s+ 2, the entry in(2s+2 − 1, 2u + 2s+1 − 2) is changed from 2e−up2u−s−1(x

2s) to

2e−u(p2u−s−1(x2s)− p3(x

2s)p2u−s−1−1(x2s+1

))

= −2e−ux2s+1

p2u−s−1−2(x2s+1

).

The minus here can be changed to plus by modifying by a multipleof row 2u + 2s+1 − 1, which will not affect the properties such as (d)and (e). Property (d) will now hold in Ns+1 for proportionality outof C2s+3+2s+1−2, to the extent claimed there. This change removes theentry of the second • of (b) with � = s, m = 2s, and t = u and replacesit by the entry of the first • with � = s+ 1, m = 0, and t = u.

Finally we consider the effect of this step on x-divisibility. If j is asin (e) with t > s+ 1, and i ≤ 2s+2 − 1 and i �= 3 · 2s − 1, then the newvalue of Pi,j will equal

P oldi,j −

Pi,3·2s+1−2

2e−s−2· P3·2s−1,j .

The old Pi,j is divisible by x3·2t−2−j+ηs(i). Also, Pi,3·2s+1−2 is divisible

by xηs(i) if i ≤ 2s+2−2, and by x0 if i = 2s+2−1. (Note that (e) did notapply in this latter case due to the condition there which here wouldsay i ≤ j − 2s+1.) We now have P3·2s−1,j divisible by x3·2

t−2−j sinceη(3 · 2s − 1) became 0 at the previous substep. Thus the x-divisibilityof Pi,j does not decrease except when i = 2s+2 − 1, where it changes to0, consistent with ηs+1(2

s+2 − 1) = 0. �

36 Donald M. Davis

6 An easily-checked proof for e ≤ 5

In this section, we give an easily-checked proof of Theorem 1.3 for e ≤ 5.Its discovery used the reduced form for M4 described in Section 2, anda Mathematica calculation by Gonzalez for the M5 analogue. However,checking its validity only requires elementary verifications.

It is proved in [5, Proposition 4.1] that Theorem 1.3 would followfrom showing that

(18) 2e−ku3·2k−1−3[0, 0] �= 0 in Me for 1 ≤ k ≤ e.

For e ≤ 5, (18) is an immediate consequence of the following, which isthe main result of this section.

Theorem 6.1. For e ≥ 1 and 1 ≤ k ≤ min(e, 5), there is a homomor-

phism φk,e : Me → Z/2k+e−1 sending 2e−ku3·2k−1−3[0, 0] nontrivially.

The homomorphism φk,e is nonzero only on the component of Me ingrading 2(3 ·2k−1−3). The component of Me in grading 2d is generatedby the same monomials ud−i−j [i, j] for any e, but the relations depend

on e. We will give an explicit formula for φk,e(u3·2k−1−3−i−j [i, j]) ∈ Z for

i, j ≥ 0, which is independent of e. Thus we usually call it just φk. Wewill prove that φk applied to a relation (3) in Me is divisible by 2k+e−1.

Since part of our formula is φk(u3·2k−1−3[0, 0]) = 22k−2 and hence

φk(2e−ku3·2

k−1−3[0, 0]) = 2k+e−2 �= 0 ∈ Z/2k+e−1,

Theorem 6.1 will follow. The hope was to see a pattern in the formulasfor φk that might extend to all k, but they seem a bit too delicate forthat.

Since the exponent of u in u3·2k−1−3−i−j [i, j] is determined by k, i,

and j, we do not list it. We write φk(i, j) for φk(u3·2k−1−3−i−j [i, j]), and

will sometimes omit the subscript k. We have φ1(0, 0) = 1, and the onlyrelation in grading 0 in Me is 2e[0, 0], which handles the case k = 1.

Here are the lists of values of φk(i, j) when k = 2 and k = 3.

[4 | 0, 0 | 2, 2, 2 | 0, 1, 1, 0],

[16 | 0, 0 | 0, 0, 0 | 0, 0, 0, 0 | 8, 0, 8, 0, 8 | 0, 8, 0, 0, 8, 0 | 0, 0, 4, 0, 4, 0, 0 |0, 4, 4, 4, 4, 4, 4, 0 | 0, 0, 6, 6, 4, 6, 6, 0, 0 | 0, 0, 0,−1,−1,−1,−1, 0, 0, 0]


Our functions always satisfy φ(i, j) = φ(j, i). The first line says thatthe nonzero values of φ2 are φ2(0, 0) = 4, φ2(2, 0) = φ2(1, 1) = 2, andφ2(2, 1) = 1, and their flips. The next pair of lines says, for example,that φ3(0, 0) = 16 and

φ3(i, 6− i) =

{4 i = 2, 4

0 i = 0, 1, 3, 5, 6.

Before we list the formulas for φ4 and φ5, we discuss the verifica-tion that φ3,e : Me → Z/2e+2 is well-defined for all e ≥ 3. This one issimple enough that it can be (and was) done by hand. We first con-sider the case e = 3. The coefficients

(81

), . . .

(88

)in (3) are of the form

8, 4α, 8α′, 2α′′, 8β, 4β′, 8, 1, where the α’s are 3 mod 4, and the β’s odd.There are 55 relations after symmetry is taken into account, but only13 of them contain any term for which ν(

(8

�+1

)φ(i−�, j)) < 5. The most

delicate is the case i = 5, j = 4, in which we have

8φ(5, 4) + 4αφ(4, 4) + 8α′φ(3, 4) + 2α′′φ(2, 4) + 8βφ(1, 4) + 4β′φ(0, 4)

= 8 · 1− 4α · 4 + 8α′ · 4− 2α′′ · 4 + 8β · 8 + 4β′ · 8≡ 8 + 16 + 0− 24 + 0 + 0 ≡ 0 (mod 32).

If e > 3, then it is as if the binomial coefficients are multiplied by2e−3. Their odd factors change, but where it matters, the odd factorsare still 3 mod 4. So φ applied to each relation is divisible by 2e−3 · 32.Terms with

(2e

9

)and

(2e

10

)also appear, but they are multiplied by φ(0, 0)

or φ(0, 1), and so yield multiples of 2e+2. This establishes the well-definedness of φ3,e, and that of φ2,e is much easier.

Next we list values of φ4(i, j) in rows of fixed i + j for which thereare some nonzero values. We precede the row by the value of i+ j. Forexample, the third listed row says that

φ4(i, 10− i) =

{32 i = 2, 8

0 i = 0, 1, 3, 4, 5, 6, 7, 9, 10.

0 : 64

8 : 32, 0, 0, 0, 32, 0, 0, 0, 32

10 : 0, 0, 32, 0, 0, 0, 0, 0, 32, 0, 0

12 : 0, 0, 0, 0, 16, 0, 0, 0, 16, 0, 0, 0, 0

14 : 0, 0, 16, 0, 16, 0, 16, 0, 16, 0, 16, 0, 16, 0, 0

15 : 0, 0, 0, 0, 0, 16, 16, 0, 0, 16, 16, 0, 0, 0, 0, 0

38 Donald M. Davis

16 : 0, 0, 0, 0, 8, 0, 8, 0, 16, 0, 8, 0, 8, 0, 0, 0, 0

17 : 0, 16, 0, 16, 16, 8, 0, 16,−8,−8, 16, 0, 8, 16, 16, 0, 16, 0

18 : 0, 0, 0, 0, 8, 8, 4, 8,−4, 0,−4, 8, 4, 8, 8, 0, 0, 0, 0

19 : 0, 8, 8, 0, 0,−4,−4,−4, 4, 8, 8, 4,−4,−4,−4, 0, 0, 8, 8, 0

20 : 0, 0,−4,−4,−4, 0,−6,−6,−4, 2, 4, 2,−4,−6,−6, 0,−4,−4,−4, 0, 0

21 : 0, 0, 0, 6, 6, 6, 0, 3, 3, 1,−1,−1, 1, 3, 3, 0, 6, 6, 6, 0, 0, 0.

These numbers were discovered using Table 1. Because of the way thatthey were obtained, it better be the case that they send all relations to 0,at least when e = 4. The beauty is that despite the hard work that wentinto obtaining them, once we have them, it is a simple computer checkto verify that they work. It is just a matter of reading these numbersφ4(i, j) into the computer and then having the computer check that

i∑�=0

(16�+1

)φ4(i− �, j) ≡ 0 (mod 128) for 0 ≤ i ≤ 21, 0 ≤ j ≤ 21− i.

Now we can prove by induction on e that if e > 4, then

i∑�=0

(2e

�+1

)φ4(i− �, j) ≡ 0 (mod 2e+3) for 0 ≤ i ≤ 21, 0 ≤ j ≤ 21− i.

It is easy to prove that, for 1 < � < 2e+1,

(19) ν((2e+1

�

)− 2

(2e

�

)) = 2e+ 1− [log2(�− 1)]− ν(�).

The induction argument follows from this and the values of φ4(−) listedabove. Indeed, the induction step requires

(20) ν(φ4(i− �, j)) ≥ [log2(�)] + ν(�+ 1)− 1,

and since i + j ≤ 21, we have ν(φ4(i − �, j)) ≥ 1, 2, 3, 4, 5, 6 if � ≥ 1,2, 4, 6, 10, 14, respectively, from which (20) follows.

Our treatment for φ5 is similar. Because of the longer lists, we takeadvantage of symmetry, and only list φ5(i, j) for i ≤ j. As before, we listvalues of φ5(i, j) in rows of fixed i+ j for which there are some nonzero


values. We precede the row by the value of i + j. If i + j = 2t + 1(resp. 2t), the last entry listed is φ5(t, t+ 1) (resp. φ5(t, t)).

0 : 256

16 : 128, 0, 0, 0, 0, 0, 0, 0, 128

20 : 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0

24 : 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0

28 : 0, 0, 0, 0, 64, 0, 0, 0, 64, 0, 0, 0, 64, 0, 0

30 : 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 64, 0, 0, 0

32 : 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 32, 0, 0, 0, 64

33 : 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 64

34 : 0, 0, 64, 0, 0, 0, 64, 0, 64, 0,−32, 0, 0, 0, 64, 0, 32, 0

35 : 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 64, 64, 0, 64, 64, 0, 64, 0

36 : 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 32, 0, 16, 0, 32, 0,−16, 0, 0

37 : 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 32, 32, 0, 0, 32, 0, 0

38 : 0, 0, 32, 0, 32, 0, 0, 0, 0, 0, 16, 0,−16, 32,−16, 0, 16, 32, 0, 0

39 : 0, 0, 0, 0, 0, 32, 32, 0, 0, 0, 0, 0, 32, 16, 16, 0, 0, 16, 16, 32

40 : 0, 0, 0, 0, 16, 0, 16, 0, 16, 0, 32, 32, 24, 0, 56, 32, 16, 32, 56, 32, 48

41 : 0, 32, 0, 32, 32, 48, 0, 0, 16, 48, 0, 48, 48, 56, 0, 16, 8, 24, 16, 16, 8

42 : 0, 0, 0, 0, 16, 16, 8, 16, 8, 16, 40, 0, 40, 8, 36, 8, 28, 16, 52, 8, 28, 48

43 : 0, 16, 16, 0, 0, 8, 8, 24, 8, 8, 8, 24, 0, 28, 28, 12, 4, 24, 8, 12, 12, 8

44 : 0, 0, 24, 24, 24, 0, 28, 28, 16, 4, 28, 24, 0, 0, 18, 2, 4, 26, 28, 2, 20, 2, 20

45 : 0, 0, 0, 12, 12, 12, 0, 10, 10, 14, 14, 4, 8, 10, 0, 15, 15, 5, 3, 15, 9, 1, 15

The computer checks that

i∑�=0

(2e

�+1

)φ5(i− �, j) ≡ 0 (mod 2e+4) for 0 ≤ i ≤ 45, 0 ≤ j ≤ 45− i

is true for e = 5. It is then proved for all e ≥ 5 by induction, using (19)as in the previous case.

Donald M. DavisDepartment of Mathematics,Bethlehem,PA 18015, USA,[email protected]

40 Donald M. Davis

References

[1] Astey L.; Geometric dimension of bundles over real projectivespaces, Quart.J. Math. Oxford 31 (1980), 139–155.

[2] Farber M., Topological complexity of motion planning, DiscreteComput. Geom. 29 (2003), 211–221.

[3] Gonzalez J., Connective K-theoretic Euler classes and nonimmer-sions of 2k-lens spaces, J. London Math. Soc. 63 (2001), 247–256.

[4] Gonzalez J., Topological robotics in lens spaces, Math. Proc. Cam-bridge Philos. Soc. 139 (2005), 469–485.

[5] Gonzalez J.; Velasco M., Wilson W., Biequivariant maps ofspheres and topological complexity of lens spaces, Comm. Con-temp. Math 15 (2013), 33.

[6] Gonzalez J.; Zarate L., BP-theoretic instabilities to the motion-planning problem in 4-torsion lens spaces, Osaka J. Math 43(2006), 581–596.


Geometric dimension of stable

vector bundles over spheres

Kee Yuen Lam Duane Randall

Abstract

We present a new method to determine the geometric dimen-sion of stable vector bundles over spheres, using a constructiveapproach. The basic tools include K-theory and representationtheory of Lie groups, and the use of spectral sequences is totallyavoided.

2010 Mathematics Subject Classification: Primary 55R50, 55R45.Keywords and phrases: stable vector bundles, geometric dimension, KO-cohomology, spinor representations, Adams maps, octonian multiplica-tion.

1 Introduction

Let X be a connected finite cell complex of dimension m and η a vectorbundle over X. In particular, the k-dimensional trivial vector bundle isdenoted by kε. Given η, one often seeks trivial sub-bundles of η⊕mε ofleast possible co-dimension q. This q is commonly called the geometricdimension of η, denoted q = gd(η). The definition, as formulated, is insuch a manner that

gd(η) = gd(η ⊕ ε) = gd(η ⊕ 2ε) = gd(η ⊕ 3ε) = . . .

holds, so that one can also speak of the geometric dimension gd(x) foran arbitrary element x in the Grothendieck cohomology group KO(X),see [1].

Bott’s periodicity theorem on the homotopy groups of the infiniteorthogonal group SO tells us that for X = Sm = the m-dimensional

41


Geometric dimension of stable

vector bundles over spheres

Kee Yuen Lam Duane Randall

Abstract

We present a new method to determine the geometric dimen-sion of stable vector bundles over spheres, using a constructiveapproach. The basic tools include K-theory and representationtheory of Lie groups, and the use of spectral sequences is totallyavoided.

2010 Mathematics Subject Classification: Primary 55R50, 55R45.Keywords and phrases: stable vector bundles, geometric dimension, KO-cohomology, spinor representations, Adams maps, octonian multiplica-tion.

1 Introduction

Let X be a connected finite cell complex of dimension m and η a vectorbundle over X. In particular, the k-dimensional trivial vector bundle isdenoted by kε. Given η, one often seeks trivial sub-bundles of η⊕mε ofleast possible co-dimension q. This q is commonly called the geometricdimension of η, denoted q = gd(η). The definition, as formulated, is insuch a manner that

gd(η) = gd(η ⊕ ε) = gd(η ⊕ 2ε) = gd(η ⊕ 3ε) = . . .

holds, so that one can also speak of the geometric dimension gd(x) foran arbitrary element x in the Grothendieck cohomology group KO(X),see [1].

Bott’s periodicity theorem on the homotopy groups of the infiniteorthogonal group SO tells us that for X = Sm = the m-dimensional

41

42 K. Y. Lam and D. Randall

sphere where m > 1, one has, in reduced cohomology,

KO(Sm) = [Sm, BSO] =

πm−1(SO) =

0 m ≡ 3, 5, 6 or 7 (mod 8),

Z m ≡ 0, 4 (mod 8),

Z/2Z m ≡ 1, 2 (mod 8).

An obvious question then is to determine gd(x) for x a generator ofany of the above nonzero groups. Here in fact one is trying to furtherpinpoint Bott’s result by asking: if x is interpreted as an essential mapin the diagram

SO(q) SO

Sm−1

inclusion

? x

what is the smallest q such that x can be homotopically compressedinto the finite orthogonal group SO(q)? The complete answer, obtainedthrough a period of 25 years, is due to Mark Mahowald and his co-workers [3, 5]. In these papers the principal tool is to use homotopyspectral sequences. The drawback of such an approach is that geo-metric features behind claimed results often become obscure, especiallywhen calculation steps or analysis of differentials in spectral sequencesare sometimes suppressed in favor of brevity. In this paper we aim atproviding a non-spectral sequence approach to determine gd(x) for all

x ∈ KO(Sm), re-obtaining Theorem 1.1 of [5]:

(I) For the generator x of KO(S8k+1) or KO(S8k+2) where k � 1,the geometric dimension gd(x) = 6.

(II) For any nonzero element y in KO(S4k), k � 5, the geometricdimention gd(y) = 2k + 1.

2 Spheres of dimension 8k + 1 and 8k + 2

Theorem 2.1. For k � 1, the generator x in KO(S8k+1) = Z/2Z hasgd(x) = 6.

Geometric Dimension 43

This is the second part of Theorem 1.1 of Davis-Mahowald [5], sum-marisable by a “compression diagram” below:

�� BSO(5) �� BSO(6) �� BSO(7) �� . . . �� BSO

S8k+1

No

��Yes

��

x(essential)

��

Alternatively, one can say that Bott’s generator in π8k(SO) = Z/2Z“originates” from the finite orthogonal group SO(6), but not fromSO(5). Our proof shall be accomplished in a number of geometric steps.

Step A. For k = 1, we construct a 6-dimensional vector bundle η overS9 that is stably non-trivial.

Let X = S9 ∪8 e10 be the Moore space obtained from S9 via attach-

ment of a 10-dimensional cell using a self-map of S9 of degree 8. Letc : X −→ S10 be the collapse map which pinches S9 to a point. Incomplex K-theory, the generator of K(S10) = Z can be represented bya C-vector bundle ω on S10 of C-dimension 5. It is well-known [4] thatin H10(S10;Z) = Z the 5th Chern class

c5(w) = (5− 1)! = 24

so that ω as a real vector bundle has Euler class 24. It follows that thepull-back vector bundle c!(ω) over X has zero Euler class, and is thussectionable.

As a result c!(ω) = ζ ⊕ εC for some complex vector bundle ζ over Xwith dimC ζ = 4 while εC is the trivial complex line bundle. Further-more, since the first Chern class c1(ζ) = 0, ζ can be regarded to havestructural group SU(4) rather than U(4).

Step B. In Lie group representation theory [6, Chapter 13], one has

�+6 : Spin(6) SU(4) U(4),≈

namely, the positive spinor representation�+6 of Spin(6) into U(4) sends

Spin(6) isomorphically onto the subgroup SU(4). It thus follows thatthere is a 6-dimensional spinor bundle η over X such that �+

6 (η) =ζ. i.e., such that ζ is the associated bundle of η via �+

6 . Also, from

construction one knows that y = ζ − 4εC is a generator of K(X) =K(S9 ∪8 e

10) = Z/8Z.


Step C. We now make the crucial claim that η restricted to S9 is stablynontrivial, in other words η|S9 represents the generator of KO(S9) =Z/2Z. Our argument is inspired by the methods in [1]. Recall fromrepresentation theory that the double covering of SO(6) by Spin(6) fitsinto a commutative diagram

Spin(6) SU(4) U(4)

SO(6) U(6)

�+6

λC2

inclusion

in which λC2 denotes second exterior power operation for complex vector

spaces. Going around clockwise one sees that the complexification of ηcan be computed via

η ⊗ C = λC2 (�+

6 (η)) = λC2 (y + 4εC)

= λC2 (y) + λC

1 (y)⊗C λC1 (4εC) + λC

2 (4εC)

= −16y + 4y + 6εC = 4y + 6εC

�= 0 in K(X)

Here the term −16y is due to the fact that on K(S10), λC2 operates as

multiplication by −25−1 = −16. If η were a stably trivial vector bundleon S9, then η⊕4ε would be trivialisable on S9, so that it can be regardedas a pullback, via c, of some 10-dimensional vector bundle defined overS10. But the complexification morphism

KO(S10) K(S10)⊗C

is a trivial homomorphism Z/2Z −→ Z, so η ⊗ C = (η ⊕ 4ε)⊗ C− 4εCwould be equal to 10εC − 4εC = 6εC, contradicting the computationabove.

Step D. On spheres S8k+1 with k > 1 we can obtain 6-dimensionalvector bundle by pulling back η via Adams maps A between mod 8Moore spaces [2], as in

S8k+1∪8e8k+2 A−→ S8k−7∪8e

8k−6 A−→ ...A−→ S17∪8e

18 A−→ S9∪8e10 = X

These spaces are 8j-fold suspensions of X with their K groups allequal to Z/8Z, mutually isomorphic [2] under the induced homomor-phisms A∗. We go on to point out, moreover, that for all k � 1,


KO(S8k+1) = Z/2Z = KO(S8k+2), so that KO(S8k+1 ∪8 e8k+2) equals

Z/2Z ⊕ Z/2Z, with generators u, v such that v restricts trivially to

KO(S8k+1) while u doesn’t. One needs to argue, further beyond Adams,

that these KO groups are mutually isomorphic under A∗ as well. Tothis end notice that the v generators are obtained from the generators ofthe Z/8Z groups of K theory by realification, i.e. by forgetting complexstructures. They thus do correspond under A∗. The pullback of the 6-dimensional vector bundle η on X to S8k+1 ∪8 e

8k+2 retains the crucialproperty of η pointed out in Step C, namely that it stably complexifiesinto the element of order 2 in K(S8k+1 ∪8 e

8k+2). This entails, againas in Step C, that the pullback of η is stably non-trivial on S8k+1. Itmust therefore represent an u generator. In particular the generator xof KO(S8k+1) now has a 6-dimensional vector bundle as representative,and gd(x) � 6 follows.

Step E. Finally we will rule out the possibility that gd(x) � 5, byestablishing

Proposition 2.2. Any 5-dimensional vector bundle ξ over S8k+1 mustbe stably trivial for all k � 1.

For the proof let us consider

�5 : Spin(5) Sp(2) U(4)≈ ,

the 5-dimensional spinor representation into U(4) taking up isomorphicimage Sp(2) inside U(4). From representation theory one knows thatthe (left) quaternionic 2-plane bundle �5(ξ) associated to ξ satisfies

µ2(�5(ξ)) ≈ ξ ⊕ ε

as real vector bundles, where µ2 is the functorial operation described asin [8, §4].

Briefly, like the second symmetric power on real vector spaces, µ2

is a functor which converts left vector spaces of dimension m over thequaternions into real vector spaces of dimension m(2m − 1). It enjoysthe property

µ2(V ⊕W ) = µ2(V )⊕ V ⊗W ⊕ µ2(W )

where ⊗ means taking tensor product over the quaternions after con-verting V into a right quaternionic vector space in the usual manner.


Since KSP (S8k+1) = 0 one has an isomorphism of symplectic vectorbundles

�5(ξ)⊕ nεH = (n+ 2)εH

for n sufficiently large. Taking n large as well as even and applying µ2,one gets, in KO(S8k+1),

µ2(�5(ξ))⊕�5(ξ)⊗H(nεH)⊕ µ2(nεH) = µ2 ((n+ 2)εH)

See [8]. The last two terms in this equation are, of course, trivial R-vector bundles. Since n, when even, annihilates KO(S8k+1), the middleterm on the left side is also R-trivial. Hence the equation implies thatµ2(�5(ξ)), and hence ξ itself, must be stably trivial. �

The case of S8k+2 follows easily, as in

Corollary 2.3. For k � 1 the generator x′ in KO(S8k+2) = Z/2Z hasgd(x′) = 6.

This is because the unique essential map

η : S8k+2 −→ S8k+1

is well-known to satisfy η!(x) = x′. Also, Proposition (2.2) on 5-dimensional vector bundles carries over from S8k+1 to S8k+2, with nochange whatsoever.

3 The 6-dimensional bundle η over S9

When octonions are regarded as pairs of quaternions, their multiplica-tion is well-known to be given by the formula

(a1, a2)× (b1, b2) −→ (a1b1 − b2a2, b2a1 + a2b1).

This provides a bilinear multiplication R8 × R8 −→ R8 free of zerodivisors. One can use the same formula to define multiplication of “bi-octonions”, by regarding a1, a2, b1, b2 themselves as octonions. Thisgives a bilinear “Cayley-Dickson” multiplication

µ : R16 × R16 −→ R16

which is long known to have zero divisors. However [7] some restrictionsof µ to “partial multiplications” will be zero-divisior free. Two such


restrictions are the µr and the µc below:

µr : R9 × R16 −→ R16

‖µc : R10 × R10 −→ R16

where for µr we take (a1, a2) only with a1 ∈ R, and for µc we take(a1, a2), (b1, b2) with a1 ∈ C, b1 ∈ C. These two zero-divisor free multi-plications are “compatible” in that they coincide over the mutual sub-domain R9 ×R10. Both multiplications are in fact “orthogonal”, in thesense that the norm of the product equals the product of the norms ofthe factors.

For a line λ through the origin in Rn write [λ] for the correspondingpoint of the real projective space RPn−1. Let η0 be the 6-dimensional

vector bundle over RP 9 obtained by setting up over [λ] the fiber η[λ]0 ,

given by the short exact sequence

0 −→ λ⊗(λ⊗ R10

) idλ⊗µc

−−−−−−→ λ⊗ R16 −−−−−−→ η[λ]0 −→ 0

where µc sends λ⊗�v ⊂ λ⊗R10 to µc(λ×−→v ). To see the geometric mean-ing of this sequence, note that λ⊗

(λ⊗ R10

)is canonically isomorphic to

R10. Exactness means that, over RP 9, the Whitney sum of 16 copies ofthe Hopf line bundle has 10 independent sections, with η0 as direct sumcomplement. Note that η0 isn’t stably trivial, as KO

(RP 9

)= Z/32Z,

with λ − ε as generator. When [λ] happens to be in RP 8, this exactsequence fits into a commutative diagram

λ⊗(λ⊗ R10

)λ⊗ R16 η

[λ]0

λ⊗(λ⊗ R16

)

idλ⊗µc

≈idλ⊗µr

This is to say that over RP 8, η0 is isomorphic to the 6-dimensionaltrivial bundle whose “constant” fiber is the normal bundle of λ⊗(λ⊗R10)inside λ⊗

(λ⊗ R16

), which is framed by the standard normal frame of

R10 inside R16. By pinching RP 8 to a point and identifying all{η[λ]0

}[λ]∈RP 8

into one vector space according to such framing, one obtains a 6-dimen-sional vector bundle over S9, denoted again by η0, which represents the


generator of KO(S9). Indeed it is not hard to show that this η0 furtherextends to a vector bundle η0 over S9 ∪8 e

10, but we’ll omit the details.Such η0 can provide a concrete alternative to the vector bundle η in §2.

4 Spheres of dimension 4k, k � 5

Theorem 4.1. For k � 5 any nonzero element x in KO(S4k) = Z hasgd(x) = 2k + 1.

This is the first part of Theorem (1.1) of Davis-Mahowald [5]. Itwas actually established much earlier in [3]. Very briefly, the complexi-

fication map KO(S4k)⊗C−−−→ K(S4k) is a monomorphism. If x is repre-

sented by a vector bundle ζ of R-dimension m then x �= 0 ⇒ x⊗C �= 0.Thus x ⊗ C has nontrivial Chern character, and so ζ ⊗ C must have anonzero 2kth Chern class. This already forces m � 2k. If m = 2k, thenζ ⊗ C ≈ ζ ⊕ ζ as real vector bundles and in terms of Euler classes χ

0 �= c2k (ζ ⊗ C) = χ4k (ζ ⊕ ζ) = χ2k(ζ) � χ2k(ζ) = 0

because cup product is taken in the ring H∗(S4k). This contradictionforces m � 2k + 1.

Finally, the fact that there exists indeed a (2k+1)-dimensional vec-

tor bundle which represents the generator x of KO(S4k) has been es-tablished quite early in [3]. This is again established in [5], through acomparison of the Postnikov tower of BSO with its connective covers[9]. We are not aware of any other existence proofs in the literature. Inany event, with the above theorems, the geometric dimension of nonzeroelements in the KO theory of an arbitrary sphere of dimensions differ-ent from 1,2,4,8,12 and 16 is now completely determined. There is littledifficulty in handling these exceptional dimensions case by case. Indeedthe answer to all nontrivial cases are already tabulated at the end of [5].

Postscript: The content of this article was presented by the firstauthor at the 2008 meeting of the Sociedad Matematica Mexicana inGuanajuato, in a special session on algebraic topology dedicated to Pro-fessor Sam Gitler. At that time the authors were unaware of the paperby M. Cadek and M. Crabb entitled ”G−structures on spheres”, pub-lished in Proc. London Math. Soc. (3)93(2006), 791-816. In an appendixto their paper, Cadek and Crabb included a self-contained proof of the


Davis-Mahowald Theorem 1.1 in [5]. Their proof and ours, while inde-pendently arrived at, overlap to some extent. In our treatment moreemphasis was placed on explicit constructions, such as the algebraicdescription of the 6−dimensional vector bundle η over S9. Also, as aconclusive step, we provided an original proof that any 5−dimensionalvector bundle over a sphere of dimension at least 9 must be stably triv-ial. Our approach is elementary throughout, using standard K−theory.Such an approach, we believe, is much in line with Sam Gitler’s math-ematical style.

AcknowledgementThe authors are grateful to the editors of Morfismos for an invita-

tion to publish this write-up of the 2008 talk. They wish to record theirsincere appreciation to Sam Gitler’s long term friendship and encour-agement to both of them.

The second author is partially supported by a Distinguished Profes-sorship at Loyola University.

Kee Yuen LamDepartment of Mathematics,University of British Columbia,Vancouver, B.C.,V6T 1Z2, Canada,[email protected]

Duane RandallDepartment of Mathematics,Loyola University,New Orleans,LA 70118, USA,[email protected]

References

[1] Adams J. F.; Geometric dimension of bundles over RPn, Interna-tional Conference on Prospects in Mathematics, Kyoto University(1973), 1–17.

[2] Adams J. F.; On the groups J(X), IV. Topology 5 (1966), 21–71.

[3] Barratt M. G., Mahowald M.; The metastable homotopy of O(n),Bull. Amer. Math. Soc. 70 (1964), 758–760.

[4] Bott R.; Stable homotopy of classical groups, Ann. of Math. 70(1959), 313–337.

[5] Davis D., Mahowald M.; The SO(n)- of origin, Forum Math. 1(1989), 239–250.

[6] Husemoller D.; Fiber bundles, 2nd edition, Springer-Verlag, NewYork (1975).


[7] Lam K.Y.; Construction of non-singular bilinear maps, Topology6 (1967), 423-426.

[8] Lam K.Y.; A formula for the tangent bundle of flag manifolds andrelated manifolds, Tran. Amer. Math. Soc. 213 (1975), 305–314.

[9] Stong R.; Determination of H∗ (BO (k,∞)) and H∗ (BU (k,∞)),Tran. Amer. Math. Soc. 107 (1963), 526–544.


The equivariant cohomology rings of regular

nilpotent Hessenberg varieties in Lie type A:

Research Announcement

Hiraku Abe, Megumi Harada, Tatsuya Horiguchi,and Mikiya Masuda

Dedicated to the memory of Samuel Gitler (1933-2014).

Abstract

Let n be a fixed positive integer and h : {1, 2, ..., n} → {1, 2, ..., n}a Hessenberg function. The main result of this manuscript is togive a systematic method for producing an explicit presentationby generators and relations of the equivariant and ordinary coho-mology rings (with Q coefficients) of any regular nilpotent Hes-senberg variety Hess(h) in type A. Specifically, we give an explicitalgorithm, depending only on the Hessenberg function h, whichproduces the n defining relations {fh(j),j}nj=1 in the equivariantcohomology ring. Our result generalizes known results: for thecase h = (2, 3, 4, . . . , n, n), which corresponds to the Peterson va-riety Petn, we recover the presentation of H∗

S(Petn) given previ-ously by Fukukawa, Harada, and Masuda. Moreover, in the caseh = (n, n, . . . , n), for which the corresponding regular nilpotentHessenberg variety is the full flag variety F�ags(Cn), we can ex-plicitly relate the generators of our ideal with those in the usualBorel presentation of the cohomology ring of F�ags(Cn). Theproof of our main theorem includes an argument that the restric-tion homomorphismH∗

T (F�ags(Cn)) → H∗S(Hess(h)) is surjective.

In this research announcement, we briefly recount the context andstate our results; we also give a sketch of our proofs and concludewith a brief discussion of open questions. A manuscript containingmore details and full proofs is forthcoming.

2010 Mathematics Subject Classification: 55N91, 14N15.Keywords and phrases: Equivariant cohomology, Hessenberg varieties,flag varieties.

51


The equivariant cohomology rings of regular

nilpotent Hessenberg varieties in Lie type A:

Research Announcement

Hiraku Abe, Megumi Harada, Tatsuya Horiguchi,and Mikiya Masuda

Dedicated to the memory of Samuel Gitler (1933-2014).

Abstract

Let n be a fixed positive integer and h : {1, 2, ..., n} → {1, 2, ..., n}a Hessenberg function. The main result of this manuscript is togive a systematic method for producing an explicit presentationby generators and relations of the equivariant and ordinary coho-mology rings (with Q coefficients) of any regular nilpotent Hes-senberg variety Hess(h) in type A. Specifically, we give an explicitalgorithm, depending only on the Hessenberg function h, whichproduces the n defining relations {fh(j),j}nj=1 in the equivariantcohomology ring. Our result generalizes known results: for thecase h = (2, 3, 4, . . . , n, n), which corresponds to the Peterson va-riety Petn, we recover the presentation of H∗

S(Petn) given previ-ously by Fukukawa, Harada, and Masuda. Moreover, in the caseh = (n, n, . . . , n), for which the corresponding regular nilpotentHessenberg variety is the full flag variety F�ags(Cn), we can ex-plicitly relate the generators of our ideal with those in the usualBorel presentation of the cohomology ring of F�ags(Cn). Theproof of our main theorem includes an argument that the restric-tion homomorphismH∗

T (F�ags(Cn)) → H∗S(Hess(h)) is surjective.

In this research announcement, we briefly recount the context andstate our results; we also give a sketch of our proofs and concludewith a brief discussion of open questions. A manuscript containingmore details and full proofs is forthcoming.

2010 Mathematics Subject Classification: 55N91, 14N15.Keywords and phrases: Equivariant cohomology, Hessenberg varieties,flag varieties.

51

52 Abe, Harada, Horiguchi, and Masuda

1 Introduction

This paper is a research announcement and is a contribution to the vol-ume dedicated to the illustrious career of Samuel Gitler. A manuscriptcontaining full details is in preparation [1].

Hessenberg varieties (in type A) are subvarieties of the full flag va-riety F�ags(Cn) of nested sequences of subspaces in Cn. Their geome-try and (equivariant) topology have been studied extensively since thelate 1980s [6, 8, 7]. This subject lies at the intersection of, and makesconnections between, many research areas such as: geometric represen-tation theory [26, 14], combinatorics [12, 23], and algebraic geometryand topology [5, 20]. Hessenberg varieties also arise in the study of thequantum cohomology of the flag variety [22, 25].

The (equivariant) cohomology rings of Hessenberg varieties has beenactively studied in recent years. For instance, Brion and Carrell showedan isomorphism between the equivariant cohomology ring of a regularnilpotent Hessenberg variety with the affine coordinate ring of a certainaffine curve [5]. In the special case of Peterson varieties Petn (in type A),the second author and Tymoczko provided an explicit set of generatorsforH∗

S(Petn) and also proved a Schubert-calculus-type “Monk formula”,thus giving a presentation of H∗

S(Petn) via generators and relations [16].Using this Monk formula, Bayegan and the second author derived a“Giambelli formula” [3] for H∗

S(Petn) which then yields a simplificationof the original presentation given in [16]. Drellich has generalized theresults in [16] and [3] to Peterson varieties in all Lie types [10]. Inanother direction, descriptions of the equivariant cohomology rings ofSpringer varieties and regular nilpotent Hessenberg varieties in typeA have been studied by Dewitt and the second author [9], the thirdauthor [18], the first and third authors [2], and Bayegan and the secondauthor [4]. However, it has been an open question to give a general andsystematic description of the equivariant cohomology rings of all regularnilpotent Hessenberg varieties [19, Introduction, page 2], to which ourresults provide an answer (in Lie type A).

Finally, we mention that, as a stepping stone to our main result,we can additionally prove a fact (cf. Section 4) which seems to bewell-known by experts but for which we did not find an explicit proofin the literature: namely, that the natural restriction homomorphismH∗

T (F�ags(Cn)) → H∗S(Hess(h)) is surjective when Hess(h) is a regular

nilpotent Hessenberg variety (of type A).

Cohomology rings of Hessenberg varieties 53

2 Background on Hessenberg varieties

In this section we briefly recall the terminology required to understandthe statements of our main results; in particular we recall the definitionof a regular nilpotent Hessenberg variety, denoted Hess(h), along witha natural S1-action on it. In this manuscript we only discuss the Lietype A case (i.e. the GL(n,C) case). We also record some observationsregarding the S1-fixed points of Hess(h), which will be important inlater sections.

By the flag variety we mean the homogeneous space GL(n,C)/Bwhich may also be identified with

F�ags(Cn) := {V• = ({0} ⊆ V1 ⊆ · · ·Vn−1 ⊆ Vn = Cn) | dimC(Vi) = i}.

A Hessenberg function is a function h : {1, 2, . . . , n} → {1, 2, . . . , n}satisfying h(i) ≥ i for all 1 ≤ i ≤ n and h(i + 1) ≥ h(i) for all 1 ≤i < n. We frequently denote a Hessenberg function by listing its valuesin sequence, h = (h(1), h(2), . . . , h(n) = n). Let N : Cn → Cn be alinear operator. The Hessenberg variety (associated to N and h)Hess(N, h) is defined as the following subvariety of F�ags(Cn):

(1) Hess(N, h) := {V• ∈ F�ags(Cn) | NVi ⊆ Vh(i) for all i = 1, . . . , n}

⊆ F�ags(Cn).

If N is nilpotent, we say Hess(N, h) is a nilpotent Hessenberg vari-ety, and if N is a principal nilpotent operator then Hess(N, h) is calleda regular nilpotent Hessenberg variety. In this manuscript we re-strict to the regular nilpotent case, and as such we denote Hess(N, h)simply as Hess(h) where N is understood to be the standard principalnilpotent operator, i.e. N has one Jordan block with eigenvalue 0.

Next recall that the following standard torus

(2) T =

g1g2

. . .

gn

| gi ∈ C∗ (i = 1, 2, . . . n)

acts on the flag variety Flags(Cn) by left multiplication. However, thisT -action does not preserve the subvariety Hess(h) in general. This prob-lem can be rectified by considering instead the action of the following


circle subgroup S of T , which does preserve Hess(h) ([17, Lemma 5.1]):

(3) S :=

gg2

. . .

gn

| g ∈ C∗

.

(Indeed it can be checked that S−1NS = gN which implies that Spreserves Hess(h).) Recall that the T -fixed points Flags(Cn)T of theflag variety Flags(Cn) can be identified with the permutation groupSn on n letters. More concretely, it is straightforward to see that theT -fixed points are the set

{(〈ew(1)〉 ⊂ 〈ew(1), ew(2)〉 ⊂ · · · ⊂ 〈ew(1), ew(2), ..., ew(n)〉 = Cn) | w ∈ Sn}

where e1, e2, . . . , en denote the standard basis of Cn.

It is known that for a regular nilpotent Hessenberg variety Hess(h)we have

Hess(h)S = Hess(h) ∩ (Flags(Cn))T

so we may view Hess(h)S as a subset of Sn.

3 Statement of the main theorem

In this section we state the main result of this paper. We first recallsome notation and terminology. Let Ei denote the subbundle of thetrivial vector bundle Flags(Cn) × Cn over Flags(Cn) whose fiber ata flag V• is just Vi. We denote the T -equivariant first Chern class ofthe line bundle Ei/Ei−1 by τi ∈ H2

T (Flags(Cn)). Let Ci denote theone dimensional representation of T through the map T → C∗ given bydiag(g1, . . . , gn) �→ gi. In addition we denote the first Chern class of theline bundle ET×TCi over BT by ti ∈ H2(BT ). It is well-known that thet1, . . . , tn generate H∗(BT ) as a ring and are algebraically independent,so we may identify H∗(BT ) with the polynomial ring Q[t1, . . . , tn] asrings. Furthermore, it is known that H∗

T (Flags(Cn)) is generated asa ring by the elements τ1, . . . , τn, t1, . . . , tn. Indeed, by sending xi toτi and the ti to ti we obtain that H∗

T (Flags(Cn)) is isomorphic to thequotient

Q[x1, . . . , xn, t1, . . . , tn]/ (ei(x1, . . . , xn)− ei(t1, . . . , tn) | 1 ≤ i ≤ n).


Here the ei denote the degree-i elementary symmetric polynomials inthe relevant variables. In particular, since the odd cohomology of theflag variety Flags(Cn) vanishes, we additionally obtain the following:

(4) H∗(Flags(Cn)) ∼= Q[x1, . . . , xn]/(ei(x1, . . . , xn) | 1 ≤ i ≤ n).

As mentioned in Section 2, in this manuscript we focus on a particularcircle subgroup S of the usual maximal torus T . For this subgroup S,we denote the first Chern class of the line bundle ES ×S C over BS byt ∈ H2(BS), where by C we mean the standard one-dimensional repre-sentation of S through the map S → C∗ given by diag(g, g2, . . . , gn) �→g. Analogous to the identification H∗(BT ) ∼= Q[t1, . . . , tn], we may alsoidentify H∗(BS) with Q[t] as rings.

Consider the restricion homomorphism

(5) H∗T (F�ags(Cn)) → H∗

S(Hess(h)).

Let τi denote the image of τi under (5). We next analyze some algebraicrelations satisfied by the τi. For this purpose, we now introduce somepolynomials fi,j = fi,j(x1, . . . , xn, t) ∈ Q[x1, . . . , xn, t].

First we define

(6) pi :=

i∑k=1

(xk − kt) (1 ≤ i ≤ n).

For convenience we also set p0 := 0 by definition. Let (i, j) be a pair ofnatural numbers satisfying n ≥ i ≥ j ≥ 1. These polynomials should bevisualized as being associated to the (i, j)-th spot in an n × n matrix.Note that by assumption on the indices, we only define the fi,j for entriesin the lower-triangular part of the matrix, i.e. the part at or below thediagonal. The definition of the fi,j is inductive, beginning with the casewhen i = j, i.e. the two indices are equal. In this case we make thefollowing definition:

(7) fj,j := pj (1 ≤ j ≤ n).

Now we proceed inductively for the rest of the fi,j as follows: for (i, j)with n ≥ i > j ≥ 1 we define:

(8) fi,j := fi−1,j−1 +(xj − xi − t

)fi−1,j .


Again for convenience we define f∗,0 := 0 for any ∗. Informally, we mayvisualize each fi,j as being associated to the lower-triangular (i, j)-thentry in an n× n matrix, as follows:

(9)

f1,1 0 · · · · · · 0f2,1 f2,2 0 · · ·

f3,1 f3,2 f3,3. . .

...fn,1 fn,2 · · · fn,n

To make the discussion more concrete, we present an explicit exam-ple.

Example 1. Suppose n = 4. Then the fi,j have the following form.fi,i = pi (1 ≤ i ≤ 4)f2,1 = (x1 − x2 − t)p1f3,2 = (x1 − x2 − t)p1 + (x2 − x3 − t)p2f4,3 = (x1 − x2 − t)p1 + (x2 − x3 − t)p2 + (x3 − x4 − t)p3f3,1 = (x1 − x3 − t)(x1 − x2 − t)p1f4,2 = (x1−x3− t)(x1−x2− t)p1+(x2−x4− t){(x1−x2− t)p1+(x2−x3 − t)p2}f4,1 = (x1 − x4 − t)(x1 − x3 − t)(x1 − x2 − t)p1

For general n, the polynomials fi,j for each (i, j)-th entry in thematrix (9) above can also be expressed in a closed formula in terms ofcertain polynomials ∆i,j for i ≥ j which are determined inductively,starting on the main diagonal. As for the fi,j , we think of ∆i,j fori ≥ j as being associated to the (i, j)-th box in an n × n matrix. Inwhat follows, for 0 < k ≤ n− 1, we refer to the lower-triangular matrixentries in the (i, j)-th spots where i−j = k as the k-th lower diagonal.(Equivalently, the k-th lower diagonal is the “usual” diagonal of thelower-left (n− k)× (n− k) submatrix.) The usual diagonal is the 0-thlower diagonal in this terminology. We now define the ∆i,j as follows.

1. First place the linear polynomial xi− it in the i-th entry along the0-th lower (i.e. main) diagonal, so ∆i,i := xi − it.

2. Suppose that ∆i,j for the (k − 1)-st lower diagonal have alreadybeen defined. Let (i, j) be on the k-th lower diagonal, so i−j = k.Define

∆i,j :=

(j∑

�=1

∆i−j+�−1,�

)(xj − xi − t).


In words, this means the following. Suppose k = i − j > 0. Then ∆i,j

is the product of (xj − xi − t) with the sum of the entries in the boxeswhich are in the “diagonal immediately above the (i, j) box” (i.e. theboxes which are in the (k−1)-st lower diagonal), but we omit any boxesto the right of the (i, j) box (i.e. in columns j + 1 or higher). Finally,the polynomial fi,j is obtained by taking the sum of the entries in the(i, j)-th box and any boxes “to its left” in the same lower diagonal.More precisely,

(10) fi,j =

j∑k=1

∆i−j+k,k.

We are now ready to state our main result.

Theorem 3.1. Let n be a positive integer and h : {1, 2, . . . , n} →{1, 2, . . . , n} a Hessenberg function. Let Hess(h) ⊂ F�ags(Cn) denotethe corresponding regular nilpotent Hessenberg variety equipped with thecircle S-action described above. Then the restriction map

H∗T (F�ags(Cn)) → H∗

S(Hess(h))

is surjective. Moreover, there is an isomorphism of Q[t]-algebras

H∗S(Hess(h))

∼= Q[x1, . . . , xn, t]/I(h)

sending xi to τi and t to t and we identify H∗(BS) = Q[t]. Here theideal I(h) is defined by

(11) I(h) := (fh(j),j | 1 ≤ j ≤ n).

We can also describe the ideal I(h) defined in (11) as follows. AnyHessenberg function h : {1, 2, . . . , n} → {1, 2, . . . , n} determines a sub-space of the vector space M(n×n,C) of matrices as follows: an (i, j)-thentry is required to be 0 if i > h(j). If we represent a Hessenberg func-tion h by listing its values (h(1), h(2), · · · , h(n)), then the Hessenbergsubspace can be described in words as follows: the first column (start-ing from the left) is allowed h(1) non-zero entries (starting from thetop), the second column is allowed h(2) non-zero entries, et cetera. For


example, if h = (3, 3, 4, 5, 7, 7, 7) then the Hessenberg subspace is

� � � � � � �� 0 0 � � � � �0 0 0 � � � �0 0 0 0 � � �0 0 0 0 � � �

⊆ M(7× 7,C).

Then, using the association of the polynomials fi,j with the (i, j)-thentry of the matrix (9), the ideal I(h) can be described as being “gen-erated by the fi,j in the boxes at the bottom of each column in theHessenberg space”. For instance, in the h = (3, 3, 4, 5, 7, 7, 7) exampleabove, the generators are {f3,1, f3,2, f4,3, f5,4, f7,5, f7,6, f7,7}.

Our main result generalizes previous known results.

Remark 1. Consider the special case h = (2, 3, . . . , n, n). In this casethe corresponding regular nilpotent Hessenberg variety has been well-studied and it is called aPeterson variety Petn (of type A). Our resultabove is a generalization of the result in [11] which gives a presentationof H∗

S(Petn). Indeed, for 1 ≤ j ≤ n−1, we obtain from (8) and (6) that

fj+1,j = fj,j−1 + (xj − xj+1 − t)fj,j

= fj,j−1 + (−pj−1 + 2pj − pj+1 − 2t)pj

and since fn,n = pn we have

H∗S(Petn) ∼= Q[x1, . . . , xn, t]

/(fj,j−1 + (−pj−1 + 2pj − pj+1 − 2t)pj , pn | 1 ≤ j ≤ n− 1

)

= Q[x1, . . . , xn, t]

/((−pj−1 + 2pj − pj+1 − 2t)pj , pn | 1 ≤ j ≤ n− 1

)∼= Q[p1, . . . , pn−1, t]

/((−pj−1 + 2pj − pj+1 − 2t)pj | 1 ≤ j ≤ n− 1

)

which agrees with [11]. (Note that we take by convention p0 = pn = 0.)

The main theorem above also immediately yields a computation ofthe ordinary cohomology ring. Indeed, since the odd degree cohomologygroups of Hess(h) vanish [29], by setting t = 0 we obtain the ordinary


cohomology. Let fi,j := fi,j(x, t = 0) denote the polynomials in thevariables xi obtained by setting t = 0. A computation then shows that

fi,j =

j∑k=1

xk

i∏�=j+1

(xk − x�).

(For the case i = j we take by convention∏i

�=j+1(xk − x�) = 1.) Wehave the following.

Corollary 3.2. Let the notation be as above. There is a ring isomor-phism

H∗(Hess(h)) ∼= Q[x1, . . . , xn]/I(h)

where I(h) :=(fh(j),j | 1 ≤ j ≤ n

).

Remark 2. Consider the special case h = (n, n, . . . , n). In this casethe condition in (1) is vacuous and the associated regular nilpotentHessenberg variety is the full flag variety F�ags(Cn). In this case we canexplicitly relate the generators fh(j)=n,j of our ideal I(h) = I(n, n, . . . , n)with the power sums pr(x) = pr(x1, . . . , xn) :=

∑nk=1 x

rk, thus relating

our presentation with the usual Borel presentation as in (4), see e.g.[13]. More explicitly, for r be an integer, 1 ≤ r ≤ n, define

qr(x) = qr(x1, . . . , xn) :=n+1−r∑k=1

xk

n∏�=n+2−r

(xk − x�).

Note that by definition qr(x) = fn,n+1−r so these are the generators ofI(n, n, . . . , n). The polynomials qr(x) and the power sums pr(x) canthen be shown to satisfy the relations

(12) qr(x) =r−1∑i=0

(−1)iei(xn+2−r, . . . , xn)pr−i(x).

Remark 3. In the usual Borel presentation of H∗(F�ags(Cn)), theideal I of relations is taken to be generated by the elementary sym-metric polynomials. The power sums pr generate this ideal I when weconsider the cohomology with Q coefficients, but this is not true withZ coefficients. Thus our main Theorem 3.1 does not hold with Z coeffi-cients in the case when h = (n, n, . . . , n), suggesting that there is somesubtlety in the relationship between the choice of coefficients and thechoice of generators of the ideal I(h).


4 Sketch of the proof of the main theorem

We now sketch the outline of the proof of the main result (Theorem 3.1)above. As a first step, we show that the elements τi satisfy the relationsfh(j),j = fh(j),j(τ1, . . . , τn, t) = 0. The main technique of this part of theproof is (equivariant) localization, i.e. the injection

(13) H∗S(Hess(h)) → H∗

S(Hess(h)S).

Specifically, we show that the restriction fh(j),j(w) of each fh(j),j to an

S-fixed point w ∈ Hess(h)S is equal to 0; by the injectivity of (13) thisthen implies that fh(j),j = 0 as desired. This part of the argumentis rather long and requires a technical inductive argument based on aparticular choice of total ordering on Hess(h)S which refines a certainnatural partial order on Hessenberg functions. Once we show fh(j),j = 0for all j, we obtain a well-defined ring homomorphism which sends xito τi and t to t:

(14) ϕh : Q[x1, . . . , xn, t]/(fh(j),j | 1 ≤ j ≤ n) → H∗S(Hess(h)).

We then show that the two sides of (14) have identical Hilbert series.This part of the argument is rather straightforward, following the tech-niques used in e.g. [11].

The next key step in our proof of Theorem 3.1 relies on the followingtwo key ideas: firstly, we use our knowledge of the special case wherethe Hessenberg function h is h = (n, n, . . . , n), for which the associatedregular nilpotent Hessenberg variety is the full flag variety F�ags(Cn),and secondly, we consider localizations of the rings in question withrespect to R := Q[t]\{0}. For the following, for h = (n, n, . . . , n) we letH := Hess(h = (n, n, . . . , n)) = F�ags(Cn) denote the full flag varietyand let I denote the associated ideal I(n, n, . . . , n). In this case weknow that the map ϕ := ϕ(n,n,...,n) is surjective since the Chern classesτi are known to generate the cohomology ring of F�ags(Cn). Since theHilbert series of both sides are identical, we then know that ϕ is anisomorphism.

The following commutative diagram is crucial for the remainder ofthe argument.

R−1(Q[x1, . . . , xn, t]/I

) R−1ϕ−−−−−→∼=

R−1H∗S(H) −−−−−→

∼=R−1H∗

S(HS)

�surj

��surj

R−1(Q[x1, . . . , xn, t]/I(h)

) R−1ϕh−−−−−→ R−1H∗S(Hess(h)) −−−−−→

∼=R−1H∗

S(Hess(h)S).


The horizontal arrows in the right-hand square are isomorphisms bythe localization theorem. Since ϕ is an isomorphism, so is R−1ϕ. Therightmost and leftmost vertical arrows are easily seen to be surjective,implying that R−1ϕh is also surjective. A comparison of Hilbert seriesshows that R−1ϕh is an isomorphism. Finally, to complete the proof weconsider the commutative diagram

Q[x1, . . . , xn, t]/I(h)ϕh−−−−→ H∗

S(Hess(h))�inj

�inj

R−1Q[x1, . . . , xn, t]/I(h)R−1ϕh−−−−→∼=

R−1H∗S(Hess(h))

for which it is straightforward to see that the vertical arrows are injec-tions. From this it follows that ϕh is an injection, and once again acomparison of Hilbert series shows that ϕh is in fact an isomorphism.

5 Open questions

We outline a sample of possible directions for future work.

• In [24], Mbirika and Tymoczko suggest a possible presentationof the cohomology rings of regular nilpotent Hessenberg varieties.Using our presentation, we can show that the Mbirika-Tymoczkoring is not isomorphic to H∗(Hess(h)) in the special case of Pe-terson varieties for n − 1 ≥ 2, i.e. when h(i) = i + 1, 1 ≤ i < nand n ≥ 3. (However, they do have the same Betti numbers.) Inthe case n = 4, we have also checked explicitly for the Hessen-berg functions h = (2, 4, 4, 4), h = (3, 3, 4, 4), and h = (3, 4, 4, 4)that the relevant rings are not isomorphic. It would be of inter-est to understand the relationship between the two rings in somegenerality.

• In [15], the last three authors give a presentation of the (equivari-ant) cohomology rings of Peterson varieties for general Lie typein a pleasant uniform way, using entries in the Cartan matrix. Itwould be interesting to give a similar uniform description of thecohomology rings of regular nilpotent Hessenberg varieties for allLie types.


• In the case of the Peterson variety (in type A), a basis for theS-equivariant cohomology ring was found by the second authorand Tymoczko in [16]. In the general regular nilpotent case, andfollowing ideas of the second author and Tymoczko [17], it wouldbe of interest to construct similar additive bases for H∗

S(Hess(h)).Additive bases with suitable geometric or combinatorial proper-ties could lead to an interesting ‘Schubert calculus’ on regularnilpotent Hessenberg varieties.

• Fix a Hessenberg function h and let S : Cn → Cn be a reg-ular semisimple linear operator, i.e. a diagonalizable operatorwith distinct eigenvalues. There is a natural Weyl group actionon the cohomology ring H∗(Hess(S, h)) of the regular semisim-ple Hessenberg variety corresponding to h (cf. for instance [30,p. 381] and also [28]). Let H∗(Hess(S, h))W denote the ring ofW -invariants where W denotes the Weyl group. It turns out thatthere exists a surjective ring homomorphism H∗(Hess(N, h)) →H∗(Hess(S, h))W which is an isomorphism in the special case ofthe Peterson variety. (Historically this line of thought goes backto Klyachko’s 1985 paper [21].) In an ongoing project, we areinvestigating properties of this ring homomorphism for generalHessenberg functions h.

Hiraku AbeAdvanced Mathematical Institute,Osaka City University,Sumiyoshi-ku,Japan, Osaka 558-8585,[email protected]

Megumi HaradaDepartment of Mathematicsand Statistics,McMaster University1280 Main Street West, Hamilton,Ontario L8S4K1, [email protected]

Tatsuya HoriguchiDepartment of Mathematics,Osaka City University,Sumiyoshi-ku,Osaka 558-8585, [email protected]

Mikiya MasudaDepartment of Mathematics,Osaka City University,Sumiyoshi-ku,Osaka 558-8585, [email protected]

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