Morfismos, Vol 18, No 2, 2014 by Morfismos, Department of Mathematics, Cinvestav

VOLUMEN 18 NÚMERO 2 JULIO A DICIEMBRE 2014 ISSN: 1870-6525

Chief Editors - Editores Generales • Isidoro Gitler • Jesu ´s Gonza´lez

Guest Editors - Editores Invitados • Ernesto Lupercio • Miguel A. Xicot´encatl

Associate Editors - Editores Asociados • Ruy Fabila • Ismael Hernańdez • Onésimo Herna ńdez-Lerma • Héctor Jasso Fuentes • Sadok Kallel • Miguel Maldonado • Carlos Pacheco • Enrique Ram´ırez de Arellano • Enrique Reyes • Dai Tamaki • Enrique Torres Giese

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VOLUMEN 18 NÚMERO 2 JULIO A DICIEMBRE DE 2014 ISSN: 1870-6525

Morfismos, Volumen 18, Nu ´mero 2, julio a diciembre 2014, es una publicacioń semestral editada por el Centro de Investigacioń y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav), a través del Departamento de Matema ´ticas. Av. Instituto Politécnico Nacional No. 2508, Col. San Pedro Zacatenco, Delegacio ń Gustavo A. Madero, C.P. 07360, D.F., Tel. 55-57473800, www.cinvestav.mx, morfismos@math.cinvestav.mx, Editores Generales: Drs. Isidoro Gitler y Jesu ´s Gonza ´lez Espino Barros. Reserva de Derechos No. 04-2012-011011542900-102, ISSN: 1870-6525, ambos otorgados por el Instituto Nacional del Derecho de Autor. Certificado de Licitud de T´ıtulo No. 14729, Certificado de Licitud de Contenido No. 12302, ambos otorgados por la Comisio ń Calificadora de Publicaciones y Revistas Ilustradas de la Secretar´ıa de Gobernacio ń. Impreso por el Departamento de Matema´ticas del Cinvestav, Avenida Instituto Politécnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, México, D.F. Este nu ´mero se termino´ de imprimir en marzo de 2015 con un tiraje de 50 ejemplares. Las opiniones expresadas por los autores no necesariamente reflejan la postura de los editores de la publicacio ń . Queda estrictamente prohibida la reproduccioń total o parcial de los contenidos e ima ´genes de la publicacio ń, sin previa autorizacioń del Cinvestav.

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In memory of Samuel Gitler, a brilliant mathematician and an extraordinary human being. Samuel guidance will be much missed.

A la memoria de Samuel Gitler, un matem´atico genial y, adem´as, un ser humano excepcional. Extra˜ naremos profundamente las ense˜ nanzas de Samuel.

Samuel Gitler (July 14, 1933 – September 9, 2014) was one of the foremost thinkers in the history of Mexican mathematics. He graduated under Norman Steenrod with a thesis in algebraic topology from Princeton University in 1960 and, throughout the XX century, he published famous works on the applications of homotopy theory to various geometric problems, most famously the immersion of manifolds into euclidian spaces problem. As a monument to his relentless creativity he continued to produce beautiful mathematics into the XXI century with his work on toric topology, a field in which he became a leader during the last stretch of his career. This volume represents the proceedings of the conference SAM80 that, to celebrate 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 the world (on the next page you can find the list of speakers) converged in Mexico City to discuss the state of the art of algebraic topology on various fields of interest to Gitler. Professor Gitler didn’t miss a beat and was active asking questions 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 results regarding the topological complexity of 2-torsion lens spaces using a very delicate linear algebra argument. The second contribution, by Kee Lam and Duane Randall, avoids using spectral sequences in a truly elegant method for computing the geometric dimension of stable vector bundles over spheres using only K-theory. Finally, the third contribution is an announcement of an important method for producing an explicit presentation by generators and relations of the equivariant and ordinary cohomology rings (with rational coefficients) of any 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 international character of the contributors are a testament to Sam’s influence in modern mathematics. Ernesto Lupercio Miguel Xicot´encatl Mexico City, April 2015.

Sam’s 80 A 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 Mark and Sam (and Martin) 11:40 – 12:00 COFFEE BREAK 12:00 – 12:50 Fred Cohen (University of Rochester) An excursion into moment-angle complexes, polyhedral products, 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 17:10 – 17:30 COFFEE BREAK 17:30 – 18:20 Soren Galatius (Stanford University) Manifolds and moduli spaces

September 26 (Thursday) 10:00 – 10:50 Jesus ´ Gonz´alez (Cinvestav-IPN) Topological robotics 10:50 – 11:40 Martin Bendersky (Hunter College, CUNY) Structure of The Polyhedral Product and Related Spaces 11:40 – 12:00 COFFEE BREAK 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 17:10 – 17:30 COFFEE BREAK 17:30 – 18:20 Soren Galatius (Stanford University) The method of scanning

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 construction of toric manifolds and applications involving various operations on simplicial complexes 11:40 – 12:00 COFFEE BREAK 12:00 – 12:50 Taras Panov (Moscow State University) Complex geometry of moment-angle manifolds 12:50 – 13:40 Alberto Verjovsky (UNAM) Poincar´e theory for compact abelian one-dimensional solenoidal 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 17:10 – 17:30 COFFEE BREAK 17:30 – 18:20 Soren Galatius (Stanford University) Homological stability for moduli spaces

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 in Lie type A: Research Announcement Hiraku Abe, Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda . . . . 51

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

Topological complexity of 2-torsion lens spaces and ku-(co)homology Topological complexity of 2-torsion lens spaces and ku-(co)homology Donald M. Davis

Donald M. Davis Abstract

We use ku-cohomology to determine lower bounds for the topologAbstract ical complexity of mod-2e lens spaces. In the process, we give an ∞ e ∞ e (L (2 ))⊗ )), almost-complete determination of ku ∗ ku∗ kufor ∗ (Lthe(2 We use ku-cohomology to determine lower bounds topologelez about the annihilator ideal of the proving a conjecture of Gonz´ a ical complexity of mod-2 lens spaces. In the process, we give an ∞ erow reduction∞of e bottom class. Our proof involves an elaborate almost-complete determination of ku ∗ (L (2 ))⊗ku∗ ku∗ (L (2 )), presentation matrices of arbitrary size. proving a conjecture of Gonz´ alez about the annihilator ideal of the bottom class. Our proof involves an elaborate row reduction of 2000 Mathematics Subject Classification: 55M30, 55N15. presentation matrices of arbitrary size.

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

Main Theorems

The1determination of the topological complexity of topological spaces Main Theorems has been much studied since its introduction by Farber in [2]. The The determination the topological complexity ofXtopological spaces (normalized) topologicalofcomplexity, TC(X), of a space is 1 less than beennumber much studied its ofintroduction Farber in [2]. The the has smallest of open since subsets X × X over by which the fibration topological TC(X), of a space X is 1 less P X(normalized) → X × X, which sends acomplexity, path σ to (σ(0), σ(1)), has a section. Seethan the smallest of open subsetsofofthis X ×concept, X over which the fibration [4] and [5] for annumber expanded discussion especially as it P Xto →lens X ×spaces. X, which sends a path σ to (σ(0), σ(1)), has a section. See relates 2n+1 [4] and fordenote an expanded discussion of this concept,t-torsion especially Let L [5](t) the standard (2n + 1)-dimensional lensas it relates to lens spaces. space, and let b(n, e), as defined in [5], denote the smallest integer k such Let exists L2n+1 (t) denote the standard (2n + 1)-dimensional t-torsion lens that there a map space, and let b(n, e), as defined in [5], denote the smallest integer k such (2e ) × L2n+1 (2e ) → L2k+1 (2e ) (1) that there exists L2n+1 a map ∞ e homotopic to a(2restriction of the which 2n+1L e (2 ) is e (2 )× L2n+1 (2ee) → L2k+1 ) (1) when followed Linto ∞ e H-space multiplication of L (2 ) = BZ/2 . In [4], it is proved that ∞ (2e ) is homotopic to a restriction of the which when followed into L2n+1 e TC(L ≤ 2b(n, e) [4], + 1.it is proved that 2b(n, e) ≤ ∞ e . In ) =))BZ/2 H-space multiplication of L (2e(2 2b(n, e) ≤ TC(L12n+1 (2e )) ≤ 2b(n, e) + 1. 1

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 binary expansion 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 complexity, which might be considered our main result. Corollary 1.2. If e ≥ 2 and e ≤ α(m) < 2e, then TC(L2m+2

α(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 bounds for 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 the same flavor as our theorem. Their conjecture depends on conjectures about BP ∗ (L2n+1 (2e ) × L2n+1 (2e )), while our theorem depends on a theorem 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), obtaining a contradiction under appropriate choice of parameters. Our main ingredient 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 that its 2e -series satisfies e

[2 ](x) =

2 2e i

ui−1 xi .

i=1

It is proved in [3, Proposition 3.1] that (2) kuev (L2n (2e ) × L2n (2e )) = ku∗ [x, y]/(xn+1 , y n+1 , [2e ](x), [2e ](y)), where |x| = |y| = 2. One of our main accomplishments is to give a more useful description of ku4n−2d (L2n (2e ) × L2n (2e )).

Topological Complexity

On the other hand, ku-homology, ku∗ (L2e ), of the infinite-dimensional lens 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−1 L2e ) ⊗ku∗ ku∗ (Σ−1 L2e ). 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

2e +1

u [i, j − ], i, j ≥ 0.

The desuspending was just for notational convenience. Note that the component of Me in grading 2d, which we denote by Gd , is isomorphic to ku4n−2d (L2n (2e ) × L2n (2e )) under the correspondence uk [i, j] ↔ uk xn−i y n−j . Much of our work goes into an almost-complete description of Me . The result is described in Section 2. In [5, Theorem 2.1], it is proved that the ideal Ie := (2e , 2e−1 u, 2e−2 u3·2−2 , 2e−3 u3·2

2 −2

, . . . , 21 u3·2

e−2 −2

, u3·2

e−1 −2

)

annihilates the bottom class [0, 0] of Me , and in [5, Conjecture 2.1] it is conjectured that Ie is precisely the annihilator ideal of [0, 0] in Me . One of our main theorems is that this conjecture is true. Theorem 1.3. For e ≥ 1, the annihilator ideal of [0, 0] in Me is precisely Ie . This is immediate from our description of Me in Section 2. See the remark preceding Theorem 2.2. After describing Me in Section 2, we use this description in Section 3 to prove Theorem 1.1. In Section 4, we prove our result for M6 , and in Section 5, we explain how this proof generalizes to arbitrary Me .

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 Jes´ us Gonz´alez for suggesting this problem, some guidance as to method, and for providing some computer results which were very helpful for finding a general proof.

Description of Me

Our approach to describing Me is via an associated matrix Pe of polynomials, which we row reduce. The row-reduced form of Pe is quite complicated, and involves some polynomials which are not completely determined. That is why we call our description “almost complete.” In this section, we approach the description of Me in three steps. First we give an introduction to our method, define the polynomial matrices Pe , and give in Table 1, without proof, the reduced form of P4 , obtained without a computer. The result for P4 is not used in our general proof, but provides a useful example for comparison. Jes´ us Gonz´alez obtained an equivalent result using Mathematica. Next we give in Theorem 2.1 an almost-complete description of the reduced form of P6 . This incorporates all aspects of the general reduced Pe , but is still describable in a moderately tractable way. Finally we give in Theorem 2.2 the general result for Pe , which involves a plethora of indices. Let Gd denote the component of Me in grading 2d. Our ordered set of 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 columns in a block Mi,j correspond to monomials uj [−, −]. We will use polynomials to represent the submatrices Mi,j . A polynomial or power series p(x) = α0 + α1 x + α2 x2 + · · · corresponds to a Toeplitz matrix (of any size) with (j + k, j) entry equal to αk . Thus the matrix is

Topological Complexity



 α 1  α 2   α 3   α4  .. .

α0 α1

0 α0

α2 .. . .. .

α1 .. . .. .



..  .    ..  .  ..  .  .. .

We define Pe to be the polynomial matrix associated to the partitioned presentation matrix of Me corresponding to the generators (4) and relations (3). In (11) we depict P6 . We let xn − 1 pn (x) = = 1 + x + · · · + xn−1 . x−1 We will display a single upper-triangular matrix of polynomials, whose restriction 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 form of 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 0      8 0 0 0 0  .  4 0 0 0     4 0 0     4 0  4

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

16I8

          

0 8I7

0 M0,3 0 M1,3 8I6 0 8I5

0 0 0 0 4I4

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

0 0 0 0 0 4I3

 0 M0,7 0 M1,7   0 0   0 0  , 0 0   0 0   0  4I2 4I1

Donald M. Davis

M0,3

 0 4  4  0 = 0  0  0 0

M1,3

 4 4  4  = 0 0  0 0

0 0 4 4 0 0 0 0 0 4 4 4 0 0 0

0 0 0 4 4 0 0 0 0 0 4 4 4 0 0

0 0 0 0 4 4 0 0 0 0 0 4 4 4 0

 0 0  0  0 , 0  4  4 0

 0 0  0  0 , 4  4 4

M0,7

  0 2   2   2  = 2 ,   2   2 0

M1,7

  2 2   2    = 2 . 2   2 2

The precise reduced form of P4 is as in Table 1 below. We do not offer a proof here, but can prove it by the methods of Section 4. We often write pk instead of pk (x). The abelian group that the associated matrix of numbers presents has 276 generators and 276 relations. This associated matrix of numbers is almost, but not quite, in Hermite form. For example, the polynomial in position (2, 17) contains terms such as 2x5 , and so the associated matrix of numbers will have some 2’s sitting far above 2’s at the bottom of the column. For the matrix to be Hermite, all nonzero entries above a 2 at the bottom should be 1’s. We could obtain such a polynomial in position (2, 17) by subtracting (x5 + x6 + x9 + x10 ) times row 17 from row 2. We have chosen not to do this here because it will be important to our reduction that the first three nonzero entries in column 17 are 1 4 2 p3 (x ) times the corresponding entries of column 9. By restricting to G1 , the 8 in position (1, 1) shows that 8u[0, 0] = 0 in M4 . Similarly, by restriction to G4 , the 4 in position (4, 4) implies that 4u4 [0, 0] = 0. We also obtain 2u10 [0, 0] = 0 and u22 [0, 0] = 0 from the matrix. The Hermite form of the associated matrix of numbers implies that 8[0, 0], 4u3 [0, 0], 2u9 [0, 0], and u21 [0, 0] are all nonzero, and this implies Theorem 1.3 when e = 4. Next we describe the reduced form of P6 . We let Pi,j denote the entry in row i and column j, where the numbering of each starts with 0.

Topological Complexity

Throughout the paper, the same notation Pi,j will be used for entries in 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

1 0 8

2 0 0 8

3 4xp2 4p3 0 8

4 0 0 0 0 4

5 0 0 0 0 0 4

6 0 0 0 0 0 0 4

7 2xp6 2p7 0 0 0 0 0 4

8 0 0 2x2 p2 (x2 ) 0 2p3 (x2 ) 0 0 0 4

9 0 0 2xp6 2x2 p2 (x2 ) 2xp2 (x3 ) 2p3 (x2 ) 0 0 0 4

10 0 0 0 0 0 0 0 0 0 0 2

11 0 0 0 0 0 0 0 0 0 0 0 2

12 0 0 0 0 0 0 0 0 0 0 0 0 2

13 0 0 0 0 0 0 0 0 0 0 0 0 0 2

14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2

15 xp14 p15 0 0 0 0 0 0 0 0 0 0 0 0 0 2

x7 p4 (x2 )

x5 p2 (x2 )p4 (x3 )

x 6 p8

x4 p2 (x12 )

xp6 p3 (x4 )

x3 p3 p2 (x2 )p2 (x7 )

x4 p4 p2 (x7 )

x2 p6 (x2 )

x5 p2 (x3 )p2 (x4 )

0 4

xp2 (x )p3 (x ) 2

x p2 (x )p2 (x )

x p2 (x )p2 (x ) 5

xp2 p2 (x16 ) + x8 p2 (x3 ) 6

x4 p2 (x7 )p4 9

0 0

xp2 (x )(1 + x p3 + x )

p7 (x )

x p2 p2 (x )

x p2 (x )p2 (x )

x4 p2 (x4 )

x2 p6 (x2 )

x5 p2 p2 (x4 )

x4 p2 (x4 )

x2 p6 (x2 )

0 0

x4 p2 (x4 )

p3 (x4 )

x2 p2 (x6 ) 4

p3 (x )

x p2 (x )

p3 (x4 )

0 4

p3 (x )

18 19

0 1

Theorem 2.1. The reduced form with diagonal entries   64      32      16 Pi,i = 8    4      2     1

16 0 0 x2 p6 (x2 ) 0 p7 (x2 ) 0 0 0 0 0 0 0 0 0 0 0 2

of the matrix P6 is upper-triangular i=0 1≤i≤3 4≤i≤9 10 ≤ i ≤ 21 22 ≤ i ≤ 45 46 ≤ i ≤ 93 i = 94.

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, and 94. b. The nonzero entries in columns 3, 7–9, 15–17, 31–33, and 63–65 are as below.

16xp2

8xp6

16p3

8p7

4x2 p6 (x2 )

4xp6 p3 (x4 )

4p7 (x2 )

4xp2 (x3 )p3 (x4 )

4x2 p6 (x2 )

8x p2 (x ) 8p3 (x2 )

8xp2 (x3 ) 8p3 (x2 )

2xp30

2p31

4p7 (x2 )

x2 p30 (x2 )

xp6 p15 (x4 )

p31 (x2 )

xp2 (x3 )p15 (x4 )

xp62 p63 2x2 p14 (x2 )

2xp6 p7 (x4 ) 2x2 p14 (x2 )

3 5

8xp6 2

4p15 8x2 p2 (x2 )

15 4xp14

2p15 (x2 )

2xp2 (x3 )p7 (x4 ) 2p15 (x2 )

x2 p30 (x2 ) p31 (x2 )

c. The nonzero entries in columns 18–21, 34–37, and 66–69 are as in Table 2. Here B refers to everything in the 18–21 block except the 4p3 (x4 )-diagonal near the bottom. The •s along a diagonal refer to the entry at the beginning of the diagonal. Each letter q refers to a polynomial. These polynomials are, for the most part, distinct. The meaning of the diagram is that, except for the diagonal near the bottom, each entry in the middle portion equals 12 p3 (x8 ) times the corresponding entry in the left portion, and similarly for the right portion, as indicated. More formally, for 18 ≤ j ≤ 21 and i < j − 8, Pi,j+16 = 12 p3 (x8 ) · Pi,j and Pi,j+48 = 14 p7 (x8 ) · Pi,j . d. Similarly, the nonzero elements in columns 38 to 45 (other than Pi,i ) are as in Table 3. If C denotes all the entries except the 2p3 (x8 )-diagonal near the bottom, then columns 70 to 77 are filled

Topological Complexity

exactly with C 路 12 p3 (x16 ) together with a p7 (x8 )-diagonal going down from (22, 70).

p15 (x4 )

2p7 (x4 ) 0

4p3 (x4 )

4q 0 4q 0 4q 0 0 0 0

0 7

4q 0 4q 0 4q 0 0 0 0 4x p2 (x ) 6

4q 4q

4q 4q 4q

4q 4q 0

4q 4q

4q 0

4q 4q

20 19

B 路 12 p3 (x8 )

B 路 14 p7 (x8 )

Table 2

e. Finally, columns 78 to 93 have a form very similar to Table 3 with q instead of 2q and rows going from 0 to 61. The lower two diagonals are x16 p2 (x16 ) coming down from (30, 78) and p3 (x16 ) coming down from (46, 78), and these are the only non-leading nonzero entries in column 78.

Donald M. Davis

Table 3 38

2x8 p2 (x8 )

19 20 21

0 0 0 0 0 0 0

2p3 (x )

27 28 29

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 2q

0 0 0 0 0 0 0 0 0 0

Now we state the general theorem, of which Theorem 1.3 is an immediate consequence, since the first occurrence of 2e− along the diagonal occurs in (3 · 2 −1 − 2, 3 · 2 −1 − 2).

Topological Complexity

Theorem 2.2. Let Pi,j denote the entries in the reduced polynomial matrix 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, s

Pi,i+2s+1 (2t −1) = 2e−1−s−t p2t+1 −1 (x2 ). ii. For 0 ≤ s ≤ e − 1 and 2 · 2s − 2 ≤ i < 3 · 2s − 2, Pi,i = 2e−s and, for 2 ≤ t ≤ e − s, s

Pi,i+2s (2t −1) = 2e−s−t x2 p2t −2 (x2 ). iii. For 3 ≤ t ≤ e and 2t + 2t−2 − 2 ≤ j ≤ 2t + 2t−1 − 3, there are possibly nonzero entries Pi,j = 2e−t qi,j for 0 ≤ i < j − 2t−1 , and also, for 1 ≤ v ≤ e − t, Pi,j+2t (2v −1) = 2e−t−v p2v+1 −1 (x2

t−1

)qi,j .

This generalizes Table 1 and Theorem 2.1. Note that some of the entries 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 smallest value of i.

Proof of Theorem 1.1

In this section, we prove Theorem 1.1 by proving the equivalent statement (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 first prove the following cases of (5) and then will show that all other cases follow by naturality. Theorem 3.1. For 1 ≤ t < e, (6)

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

and (7)

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

Donald M. Davis

These are the cases m = 2α(B)−e (4B + 1) and m = 2α(B)−e B of Theorem 1.1 or (5). Proof. We focus on (6), and then discuss the minor changes required for (7). Let n = 3 · 2t−1 − 1 + 2t+1 B and suppose there is a map L2n (2e ) × L2n (2e ) → L2

t+3 B−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)2

t+2 B

= 0 ∈ ku∗ (L2n (2e ) × L2n (2e )).

The result (6) will follow from showing that (x − y)2

t+2 B

= 0 ∈ ku2(2n−d) (L2n (2e ) × L2n (2e ))

with n = 3 · 2t−1 − 1 + 2t+1 B and d = 3 · 2t − 2. This group is isomorphic to the component group Gd for Me whose presentation matrix was described in Section 2. The ordered set of generators is obtained as xn−d y n−d multiplied by (8)

x0 y d , . . . , xd y 0 , ux1 y d , . . . , uxd y 1 , . . . , ud xd y d .

We omit the xn−d y n−d throughout our analysis. One easily shows that t+2 = α(B) j = 2t+1 B 2 B ν j > α(B) 0 < |2t+1 B − j| < 2t+1 . Here and throughout ν(−) denotes the exponent of 2 in an integer. We wish to show that if t < e and d = 3 · 2t − 2, then 2e+t−1 xd/2 y d/2 + 2e+t f (x, y) = 0 in Gd , where f (x, y) is a polynomial of degree d in x and y. In the reduced matrix for Pe we omit all columns and rows not of the form 3 · 2i − 3, 0 ≤ i ≤ t. Omitting columns amounts to taking a quotient, and when a column(generator) is omitted the row(relation) with its leading entry can be omitted, too. The resulting matrix is presented below, where the various polynomials q are mostly distinct.

Topological Complexity

Table 4 0 3

2eâ&#x2C6;&#x2019;2 p2 2eâ&#x2C6;&#x2019;1

2eâ&#x2C6;&#x2019;3 q 2eâ&#x2C6;&#x2019;3 p2 (x2 ) 2eâ&#x2C6;&#x2019;2

9 3 Âˇ 23 â&#x2C6;&#x2019; 3

3 Âˇ 23 â&#x2C6;&#x2019; 3 2eâ&#x2C6;&#x2019;4 q

3 Âˇ 2tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 3 2eâ&#x2C6;&#x2019;t q

ÂˇÂˇÂˇ

3 Âˇ 2t â&#x2C6;&#x2019; 3

2eâ&#x2C6;&#x2019;4 q

2eâ&#x2C6;&#x2019;t q

2eâ&#x2C6;&#x2019;tâ&#x2C6;&#x2019;1 q 2eâ&#x2C6;&#x2019;tâ&#x2C6;&#x2019;1 q

2eâ&#x2C6;&#x2019;4 p2 (x4 ) 2eâ&#x2C6;&#x2019;3

2eâ&#x2C6;&#x2019;t q 2eâ&#x2C6;&#x2019;t q

2eâ&#x2C6;&#x2019;tâ&#x2C6;&#x2019;1 q

. . .

2eâ&#x2C6;&#x2019;tâ&#x2C6;&#x2019;1 q

3 Âˇ 2tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 3 3 Âˇ 2t â&#x2C6;&#x2019; 3

2eâ&#x2C6;&#x2019;t+1

2eâ&#x2C6;&#x2019;tâ&#x2C6;&#x2019;1 p2 (x2 2eâ&#x2C6;&#x2019;t

tâ&#x2C6;&#x2019;1

)

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

2eâ&#x2C6;&#x2019;2 (xy 2 + x2 y) 2eâ&#x2C6;&#x2019;1 xy 2

2eâ&#x2C6;&#x2019;3 (x3 y 6 + x5 y 4 )

2eâ&#x2C6;&#x2019;1 x2 y

2eâ&#x2C6;&#x2019;3 (x4 y 5 + x6 y 3 ) 2eâ&#x2C6;&#x2019;2 x3 y 6

2eâ&#x2C6;&#x2019;4 (x7 y 14 + x11 y 10 )

2eâ&#x2C6;&#x2019;2 x4 y 5

2eâ&#x2C6;&#x2019;4 (x8 y 13 + x12 y 9 )

2eâ&#x2C6;&#x2019;2 x5 y 4

2eâ&#x2C6;&#x2019;4 (x9 y 12 + x13 y 8 )

2eâ&#x2C6;&#x2019;2 x6 y 3

2eâ&#x2C6;&#x2019;4 (x10 y 11 + x14 y 7 ).

From these relations, we obtain (9)

2e+tâ&#x2C6;&#x2019;1 â&#x2C6;ź â&#x2C6;&#x2019;2e+tâ&#x2C6;&#x2019;3 (xy 2 + x2 y)

â&#x2C6;ź 2e+tâ&#x2C6;&#x2019;5 (x3 y 6 + x4 y 5 + x5 y 4 + x6 y 3 ) 14 e+tâ&#x2C6;&#x2019;7 â&#x2C6;ź â&#x2C6;&#x2019;2 xi y 21â&#x2C6;&#x2019;i i=7

â&#x2C6;ź ÂˇÂˇÂˇ

eâ&#x2C6;&#x2019;tâ&#x2C6;&#x2019;1

â&#x2C6;ź Âą2

eâ&#x2C6;&#x2019;tâ&#x2C6;&#x2019;1

= Âą2

2t+1 â&#x2C6;&#x2019;2

i=2t â&#x2C6;&#x2019;1

(x3Âˇ2

xi y 3Âˇ2

tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2

t â&#x2C6;&#x2019;3â&#x2C6;&#x2019;i

y 3Âˇ2

tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

+ x3Âˇ2

tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

y 3Âˇ2

tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2

)

= 0,

since maximum exponents are 3 Âˇ 2tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 1. That the last line is nonzero follows from the reduced form of the matrix Me of relations. Now we incorporate the polynomials q in the above matrix. We denote by mi a monomial or sum of monomials of degree 3 Âˇ 2i â&#x2C6;&#x2019; 3, in x and y. At the first step of the above reduction sequence, we would have

Donald M. Davis

an additional

i=2 2

t+eâ&#x2C6;&#x2019;iâ&#x2C6;&#x2019;2 m

(10)

At the second step, we add

2t+eâ&#x2C6;&#x2019;iâ&#x2C6;&#x2019;3 m i .

i=3

We can incorporate the first monomials for i â&#x2030;Ľ 3 into the second, and we replace 2t+eâ&#x2C6;&#x2019;4 m2 by iâ&#x2030;Ľ3 2t+eâ&#x2C6;&#x2019;iâ&#x2C6;&#x2019;3 m i and incorporate these into (10). The third step adds ti=4 2t+eâ&#x2C6;&#x2019;iâ&#x2C6;&#x2019;4 m i . We incorporate (10) into this for i > 3, while the term in (10) with i = 3 is equivalent to a sum which can also be incorporated. Continuing, we end with t

(t)

2t+eâ&#x2C6;&#x2019;iâ&#x2C6;&#x2019;t mt = 2eâ&#x2C6;&#x2019;t mt = 0,

i=t

so the qâ&#x20AC;&#x2122;s contribute nothing. We easily see that incorporating 2e+t f (x, y) also contributes nothing, since 2e+t m â&#x2C6;ź 2e+tâ&#x2C6;&#x2019;2 m1 â&#x2C6;ź 2e+tâ&#x2C6;&#x2019;4 m2 â&#x2C6;ź Âˇ Âˇ Âˇ â&#x2C6;ź 2e+tâ&#x2C6;&#x2019;2t mt = 0. t

The proof of (7) is very similar. We want to show (xâ&#x2C6;&#x2019;y)(2Bâ&#x2C6;&#x2019;1)2 = 0 in G3Âˇ2t â&#x2C6;&#x2019;2 if Îą(B) = e + t and 1 â&#x2030;¤ t < e. For (B â&#x2C6;&#x2019; 2)2t < j < (B + 1)2t , we have = Îą(B) â&#x2C6;&#x2019; 1 if j = (B â&#x2C6;&#x2019; 1)2t or B Âˇ 2t (2B â&#x2C6;&#x2019; 1)2t Î˝ j > Îą(B) â&#x2C6;&#x2019; 1 other j. We have factored out xnâ&#x2C6;&#x2019;d y nâ&#x2C6;&#x2019;d with nâ&#x2C6;&#x2019;d = 2t B â&#x2C6;&#x2019;2t+1 +1. Our ordered set of generators is again (8), and our class now, mod higher 2-powers, t t+1 t+1 t is 2e+tâ&#x2C6;&#x2019;1 (x2 â&#x2C6;&#x2019;1 y 2 â&#x2C6;&#x2019;1 + x2 â&#x2C6;&#x2019;1 y 2 â&#x2C6;&#x2019;1 ). Utilizing the relations similarly to (9), we end with e+tâ&#x2C6;&#x2019;1

Âą2

e+tâ&#x2C6;&#x2019;1

= Âą2

2t â&#x2C6;&#x2019;1 2t+1 â&#x2C6;&#x2019;1

3Âˇ2t â&#x2C6;&#x2019;3 3Âˇ2t â&#x2C6;&#x2019;2

2t+1 â&#x2C6;&#x2019;1 2t â&#x2C6;&#x2019;1

)

2t+1 â&#x2C6;&#x2019;2

xi y 3Âˇ2

t â&#x2C6;&#x2019;3â&#x2C6;&#x2019;i

i=2t â&#x2C6;&#x2019;1

3Âˇ2t â&#x2C6;&#x2019;2 3Âˇ2t â&#x2C6;&#x2019;3

) = 0,

since xd+1 = 0 = y d+1 (after factoring out xnâ&#x2C6;&#x2019;d y nâ&#x2C6;&#x2019;d ).

Topological Complexity

Proof of (5). The proof is by induction on t. If t = 1, the theorem follows from (6) with m = 4B + 1 if m ≡ 1 mod 4, and from (7) with m = 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 m is odd, then α(m − 1) = e + t − 1, so using the induction hypothesis in the 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 = 2t B with α(B) = α(m) = t + e. By (7), we obtain b(m + 2t − 1, e) ≥ 2m − 2t , as desired. If m = 2t−1 + 2t+1 B with α(B) = t + e − 1, then (6) is exactly the desired result. Finally, if m = 3 · 2t−1 + 2t+1 A 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 .

Proof of Theorem 2.1

In this section we prove Theorem 2.1. Because it is a fairly complicated row reduction, we accompany the proof with diagrams of the matrix at several stages of the reduction. Although the proof of Theorem 2.2 in Section 5 is a complete proof and subsumes the much-longer proof for

Donald M. Davis

e = 6, we feel that the more explicit example renders the general proof more comprehensible, or perhaps unnecessary. If M is a Toeplitz matrix corresponding to a polynomial p(x) as described in the preceding section, then the Toeplitz matrix corresponding to the polynomial (1 + Îąx + Î˛x2 )p(x) is obtained from M by adding Îą times each row to the one below it and Î˛ times each row to the row 2 below it. This illustrates how row operations on the matrix of polynomials correspond to row operations on the partitioned matrix of numbers. Our matrices now refer to the case e = 6. The initial partitioned matrix for Gd could be considered as the matrix of numbers associated to the following matrix of polynomials, which has d + 1 columns and 2(d + 1) rows. ďŁŤ

(11)

ďŁŹ64 ďŁŹ ďŁŹ64 ďŁŹ ďŁŹ ďŁŹ0 ďŁŹ ďŁŹ ďŁŹ0 ďŁŹ ďŁŹ ďŁŹ0 ďŁŹ ďŁŹ ďŁŹ0 ďŁŹ ďŁ

2 x 64 2

64 3

2 x 64 2

64 .. .

64 4

2 3 x 64 3 64 2 x 64 2

ďŁś Âˇ Âˇ ÂˇďŁˇ ďŁˇ Âˇ Âˇ ÂˇďŁˇ ďŁˇ ďŁˇ Âˇ Âˇ ÂˇďŁˇ ďŁˇ ďŁˇ Âˇ Âˇ ÂˇďŁˇ ďŁˇ ďŁˇ Âˇ Âˇ ÂˇďŁˇ ďŁˇ ďŁˇ Âˇ Âˇ ÂˇďŁˇ ďŁˇ ďŁ¸

The first two row blocks of the associated matrices of numbers have d + 1 rows of numbers, the next two d rows, etc., while the sizes of the column blocks are d + 1, d, . . .. The first (resp. second) (resp. third) row block corresponds to the first (resp. second) (resp. first) set of relations in (3) with i + j = d (resp. d) (resp. d â&#x2C6;&#x2019; 1). Note that if the first two rows and the first column of (11) are deleted, we obtain exactly the initial matrix for Gdâ&#x2C6;&#x2019;1 . We may assume that the matrix for Gdâ&#x2C6;&#x2019;1 has already been reduced, to Qdâ&#x2C6;&#x2019;1 . Thus we may obtain the reduced form for Gd by taking Qdâ&#x2C6;&#x2019;1 , placing a column of 0â&#x20AC;&#x2122;s in front of it and the top two rows of (11) above that, and then reducing. By the nature of the matrix (11), the restriction of the reduced form Qd to its first d columns will be Qdâ&#x2C6;&#x2019;1 .

Topological Complexity

This is an interesting property. Let Qd denote the reduced form of the polynomial matrix for Gd . Remove its last column, put a column of 0’s in front, put the top two rows of (11) above this, and reduce. The result will be the original matrix, Qd . We will prove that the matrix described in Theorem 2.1 is correct by removing its last column (the one with the 1 at the bottom), preceding the matrix by a column of 0’s and this by the first two rows of (11), and seeing that after reducing, we obtain the original matrix Q94 . Because of the initial shifting, each column is determined by the column which precedes it, together with the reduction steps, which justifies the method of starting with the putative answer, 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 putative answer pushed one unit to the right and two units down, preceded by the first two rows of (11) and a column of 0’s. We often write Ri and Cj for row i and column j. Step 0: Subtract R0 from R1 , then divide R1 by (1 − x), and then subtract xR1 from R0 . These rows become   64 64 64 − · · · − 0 · · · x − xp xp xp 64 0 − 64 2 3 62 3 4 5 64   64 64 64 64 64 0 ··· 0 ··· 2 3 p2 4 p3 5 p4 64 p63

Divide R1 by 63, which is the unit part of 64 2 . We now have, in R0 and R1 ,   i=j=0  64    32 i=j=1    0 i+j =1 Pi,j = 6−ν(j+1)  uj 2 xpj−1 i = 0, 2 ≤ j ≤ 63     6−ν(j+1)  uj 2 pj i = 1, 2 ≤ j ≤ 63    0 0 ≤ i ≤ 1, 64 ≤ j ≤ 94, where uj is the odd factor of −

64 j+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

Donald M. Davis

  64 in C0      32 in C1      16 in C4      32 in Ci−1      8 in C10    16 in C i−1  4 in C22      8 in Ci−1      2 in C46     4 in Ci−1     1 in C94    2 in C i−1

(12)

i=0 i=1 i=2 3≤i≤4 i=5 6 ≤ i ≤ 10 i = 11 12 ≤ i ≤ 22 i = 23 24 ≤ i ≤ 46 i = 47 48 ≤ i ≤ 94.

The above entries for i = 0, 1, 2, 5, 11, 23, and 47 will be the only nonzero 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 be as stated in Theorem 2.1, and the matrix will be upper triangular with nonzero entries above the diagonal less 2-divisible than the diagonal entry in their column. Table 5 depicts the first 22 columns of the matrix at the end of Step 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 13

0 64

1 0 32 64

2 64x 64p2 0 32

3 16xp2 16p3 0 0 32

4 64xp3 64p4 16xp2 16p3 0 32

5 32xp4 32p5 0 0 0 0 16

6 64xp5 64p6 0 0 0 0 0 16

7 8xp6 8p7 0 0 0 0 0 0 16

8 64xp7 64p8 8xp6 8p7 0 0 0 0 0 16

9 32xp8 32p9 0 0 8x2 p2 (x2 ) 0 8p3 (x2 ) 0 0 0 16

10 64xp9 64p10 0 0 8xp6 8x2 p2 (x2 ) 8xp2 (x3 ) 8p3 (x2 ) 0 0 0 16

11 16xp10 16p11 0 0 0 0 0 0 0 0 0 0 8

12 64xp11 64p12 0 0 0 0 0 0 0 0 0 0 0 8

Topological Complexity

64xp13

4xp14

64xp15

32xp16

64xp17

32xp20

64p14

4p15

32p17

64p18

64p20

32p21

64p16 4xp14 4p15

16xp18 16p19

64xp19

32xp12 32p13

4xp3 (x4 )p6 4x2 p6 (x2 ) 4xp2 (x3 )p3 (x4 ) 4p7 (x2 )

0 4x2 p6 (x2 ) 0 4p7 (x2 ) 0

4x4 p2 (x4 )

4p3 (x4 )

4p3 (x4 ) 0

4p3 (x4 )

Although it is just a simple shift, it will be useful to have for reference, 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 reduction. The shifted version of part b of Theorem 2.1 can be mostly seen in Table 5. Table 6 19

4x4 p2 (x4 )

4p3 (x4 )

0 4q 0 0 0 0 4q 0 4q 0 4q 0 0 0 0

B 路 12 p3 (x8 )

2p7 (x4 )

B 路 14 p7 (x8 )

p15 (x4 )

Donald M. Davis

Table 7 39

2x8 p2 (x8 )

0 0 0 0 0 0 0

2p3 (x8 )

29 30 31

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

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

Topological Complexity

in column

(13)

ďŁą ďŁ´ i+5 ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ i+3 ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ i + 11 ďŁ´ ďŁ´ ďŁ´ ďŁ˛i + 7 ďŁ´ i + 23 ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ i + 15 ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ i + 47 ďŁ´ ďŁ´ ďŁ´ ďŁłi + 31

4â&#x2030;¤iâ&#x2030;¤5 6â&#x2030;¤iâ&#x2030;¤7 8 â&#x2030;¤ i â&#x2030;¤ 11 12 â&#x2030;¤ i â&#x2030;¤ 15 16 â&#x2030;¤ i â&#x2030;¤ 23 24 â&#x2030;¤ i â&#x2030;¤ 31 32 â&#x2030;¤ i â&#x2030;¤ 47 48 â&#x2030;¤ i â&#x2030;¤ 63.

i has no nonzero elements. For i â&#x2030;Ľ 64, R

The relationship between the three parts of Table 6 and the similar relationship that columns 71 to 78 are mostly 21 p3 (x16 ) times Table 7 will be very important. We call it a â&#x20AC;&#x153;proportionalityâ&#x20AC;? relation. We extend it to also include that in rows 4, 5, and 6 we have C18 /C10 = 12 p3 (x4 ), C34 /C10 = 14 p7 (x4 ), and C66 /C10 = 18 p15 (x4 ), and similarly in row 4, columns 9, 17, 33, and 65. When we perform row operations involving these rows, these relationships continue to hold. Rows 12â&#x20AC;&#x201C;15 and 24â&#x20AC;&#x201C; 31, where the proportionality relationship does not hold, will not be involved in row operations, since the columns in which their leading entry occurs have all 0â&#x20AC;&#x2122;s above the leading entry. (Although rows 0 and 1 are initially nonzero in these columns, clearing out R0 and R1 , i for 12 â&#x2030;¤ i â&#x2030;¤ 15 or as in Step 1 below, is a 2-step process, and so R 24 â&#x2030;¤ i â&#x2030;¤ 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 divisible by these powers of x. We keep track of bounds for the x-divisibility of the unspecified polynomials in Table 6 and 7 and in columns 79 to 94. We postpone this analysis until all the reduction steps have been outlined. Similarly to the proportionality considerations just discussed, divisibility bounds are preserved when we add a multiple of one row to another, in that the x-exponent of Pi,j + cPi ,j is â&#x2030;Ľ the minimum of that of Pi,j and Pi ,j . The rows, 3, 6â&#x20AC;&#x201C;7, 12â&#x20AC;&#x201C;15, 24â&#x20AC;&#x201C;31, and 48â&#x20AC;&#x201C;63, where entries not divisible by x occur will not be used to modify other rows. Now we begin an attempt to remove most of the binomial coefficients from R0 and R1 .

Donald M. Davis

Step 1. The goal is to add multiples of lower rows to R0 and R1 to reduce them to 0

16xp2

8xp6

16p3

8p7

···

4xp14

4p15

···

2xp30

2p31

···

xp62

p63

··· 0 0

with each 0 referring to all intervening columns. However, we will be forced to bring up some additional entries. We claim that, after Step 1, the nonzero entries P1,j , in addition to those in columns 2t − 1 listed j with j ≥ 5 and j not in just above, are combinations of various R [6, 8] ∪ [12, 16] ∪ [24, 32] ∪ [48, 64]. Row 0 is similar but has an extra power of x, since this is true at the outset. Rows 0 and 1 will thus have the requisite proportionality and x-divisibility relations. i It will be useful to note that since at the outset all entries in R 1 for i ≥ 2 are a multiple of 2 times the leading entry at the bottom of i can be killed (reduced to all 0’s) their column, then, using (13), 2R by subtracting multiples of lower rows if i ≥ 32. For example, nonzero 32 occur only in Cj with j ≥ 79. If the entry in (32, j) is a entries of R 32 kills the entry in Cj polynomial q, then subtracting qRj+1 from 2R j+1 = 0 for such j. without changing anything else, since R i can be killed in two steps if i ≥ 16, and 8R i can be Similarly 4R killed if i ≥ 8. We can use this observation to kill the entries in R0 and R1 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). Subtracting multiples of 2Rj+1 from R0 and R1 kills the entries in (0, j) and (1, j) j+1 . This can be killed by the obserwhile bringing up multiples of 2R vation of the previous two paragraphs. This method works to eliminate the entries in R0 and R1 in columns 12, 14, 16–18, 20–22, 24–30, and 32–62. entries in R0 and R1 in columns > 63 were all 0.) Since 64 (Initial 6−t mod 213−2t for 2 ≤ t ≤ 6, the entries in R0 and R1 in − 2t ≡ 2 columns 3, 7, 15, 31, and 63 can be changed to their desired values with pure 2-power coefficients by similar steps. For C23 , we subtract even multiples of R24 from R0 and R1 to kill the entries. This brings into R0 and R1 multiples of 4 in some columns 39 to 46 and even entries in some columns ≥ 71. The latter entries can be cancelled from below, while cancelling multiples of 4 in Cj for

Topological Complexity

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

Now we consider C11 . We subtract multiples of 2R12 to kill the entries in R0 and R1 . This brings up multiples of 8 in C19 , C21 , and C22 , 4 in Cj for 35 ≤ j ≤ 46, and 2 in some columns > 64, the latter of which can be cancelled from below. We kill the earlier elements with j+1 multiples of Rj+1 , leaving a combination of the various R

21 , R 37 , and Column 9 is eliminated similarly, giving multiples of R some others, while columns 8, 10, and 13 are, in a sense, easier since their binomial coefficients are 4 times the number at the bottom of their column, rather than 2. For example, to kill the entry in (1, 13), we first subtract a multiple of 4R14 . This contains a 16q in C21 , which is killed by a multiple of 2R22 . This brings up a 2q in R45 , the killing of which 46 . brings up a multiple of R To kill the entry in (1, 5), we subtract a multiple of 2R6 , which has entries in Cj for many values of j ≥ 9. We can cancel each of these by subtracting a multiple of Rj+1 , accounting for the contributions to k with k ∈ [6, 8] ∪ [12, 16] ∪ [24, 32] ∪ [48, 64]. R1 of multiples of many R 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 killing 5 , R 9 , R 17 , and R 33 , as allowed. of which brings up combinations of R

Step 2. Subtract 2R1 from R2 to remove the 64 in P2,1 . This brings entries 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 coefficient equal to Pj+1,j . These are then killed by subtracting corresponding multiples of Rj+1 , which brings up into R2 corresponding multiples of 4 , this will place q = 8x2 p2 (x2 )p3 in C9 , j+1 for j as in (14). From R R 1 1 1 4 4 4 2 p3 (x )q in C17 , 4 p7 (x )q in C33 , and 8 p15 (x )q in C65 . This extends the proportionality property of columns 9, 17, 33, 65 to include also row 2. Now R2 has 16xp2 as its leading entry, in column 4.

Donald M. Davis

Step 3. Divide R2 by xp2 . Dividing by a polynomial p of the form 1+ αi xi , such as p2 , is not a problem. If M is a Toeplitz matrix corresponding to a polynomial q, then the Toeplitz matrix corresponding to q/p is obtained from M by performing the row operations corresponding to finitely many of the terms of the power series 1/p. Dividing by x is more worrisome, and is the reason for much of our work. In this step it is 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 all steps have been described. We have the important relation p2t /p2 = pt (x2 ),

(15)

which implies that the entries in P2,j for j = 8, 16, 32, and 64 are now 8p3 (x2 ), 4p7 (x2 ), 2p15 (x2 ), and p31 (x2 ). The relation (15) and its variants will be used frequently without comment. In C9 , we obtain 8x

p2 (x2 )p3 = 8xp2 (x3 ) + 16x3 /p2 . p2

We use R10 to cancel the second term, at the expense of bringing up 10 into R2 . This satisfies proportionality properties, multiples of x3 R which continue to hold. Step 4. Subtract p3 R2 from R3 to change P3,4 to 0. Since

(16)

p2k+1 − p3 pk (x2 ) = −x2 pk−1 (x2 ),

we obtain −8x2 p2 (x2 ) in P3,8 , −4x2 p6 (x2 ) in P3,16 , and similar expressions in C32 and C64 . We can change the minus to a plus by adding 9 , R 17 , etc., a multiple of R9 , R17 , etc. This brings up multiples of R into R3 , but these maintain proportionality and x-divisibility properties. 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 that the x-divisibility of R3 is at least the minimum of that of R2 and its previous value for R3 . But this will be handled later. For the convenience of the reader, we list here columns 0 through 10 at this stage of the reduction. Some of the specific polynomials are not very important, and will later just be called q.

Topological Complexity

0 1 2 3 4 5 6 7 8 9 10

16xp2

8xp6

8x3 p2 p2 (x3 )

16p3

8p7

0 32

0 0 32

16 0

0 0

8p3 (x ) 2

8x2 p2 (x2 )p2 (x3 )

0 2

8x p2 (x )

8xp2 (x )

8p3 p3 (x2 )

8xp6

8p3 (x4 )

8x p2 (x )

8xp6

8x2 p2 (x2 )

8p3 (x2 )

8xp2 (x3 )

8p3 (x2 )

0 16

Step 5. Subtract 2R2 from R5 to remove the leading entry in R5 . If P2,j = q for j > 4, then adding qRj+1 to R5 will cancel the subtracted j+1 to R5 . So R5 gets multiples of entry, at the expense of adding q R j+1 for many values of j in the intervals [8, 10], [16, 22], and [32, 46]. R The rows that we don’t want to bring up are 12–15, 24–31, etc., which contain the lower diagonals in Tables 6 and 7, where neither proportionality nor x-divisibility holds. Now the leading entry of R5 is 8x2 p2 (x2 ) in C10 . Step 6. Divide R5 by x2 p2 (x2 ). We need to know that all entries in R5 are divisible by x2 . In Theorem 4.2, we will show that this is true for columns 19–22, 35–46, and 67–94. The only other nonzero entries in R5 are those in columns 10, 18, 34, and 66 with which it started. See Table 5. The first nonzero entries in R5 after dividing are 8 in C10 and 4p3 (x4 ) in C18 . Step 7. Subtract multiples of R5 from rows 0, 1, 2, 3, 4, 6, and 7 to clear out C10 in these rows. Because it had been the case that Pi,18 /Pi,10 = 12 p3 (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 (x2 )p3 (x4 )) = −4x4 p2 (x4 ). We can change the minus to a plus by adding x4 p2 (x4 )R19 . Similarly, the only nonzero entries in column 34 (resp. 66) (except for Pj+1,j ) are 2p7 (x4 ) (resp. p15 (x4 )) in R5 , and 2x4 p6 (x4 ) (resp. x4 p14 (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

Donald M. Davis

j+1 to R11 . So cancel the subtracted entry, at the expense of adding q R j+1 for many values of j in the intervals [18, 22] R11 gets multiples of R 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 14 15

0 64

1 0 32 0

2 0 0 0 32

3 16xp2 16p3 0 0 32

4 0 0 16 0 0 0

5 0 0 0 0 0 0 16

0 1 2 3

15 4xp14 4p15 0 0

16 0 0 4p7 (x2 ) 4x2 p6 (x2 )

4 5 6 7

0 0 0 0

17 0 0 8q0 p3 (x4 ) 8q1 p3 (x4 ) 4x2 p6 (x2 ) 0 4p7 (x2 ) 0

8 9 10 11

0 0 0 0

12 13 14 15 16 17 18 19 20 21 22 23

0 0 0 0 8

0 0 0 0 0 8

0 0 0 0 0 0 8

6 0 0 0 0 0 0 0 16

7 8xp6 8p7 0 0 0 0 0 0 16

8 0 0 8p3 (x2 ) 8x2 p2 (x2 ) 0 0 0 0 0 16

9 0 0 8q0 8q1 8x2 p2 (x2 ) 0 8p3 (x2 ) 0 0 0 16

10 0 0 0 0 0 8 0 0 0 0 0 0

11 0 0 0 0 0 0 0 0 0 0 0 0 8

12 0 0 0 0 0 0 0 0 0 0 0 0 0 8

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8

14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8

18 0 0 0 0

19 0 0 4q 4q

20 4q 4q 4q 4q

21 4q 4q 4q 4q

22 4q 4q 4q 4q

0 4p3 (x4 ) 0 4x4 p2 (x4 ) 0 0 0 0

0 0 0 0

4q 4q 4q 4q

4x4 p2 (x4 ) 0 0 0 4p3 (x4 ) 0 0 0 0 0 0 0 8

0 4x4 p2 (x4 ) 0 0

4q 0 4x4 p2 (x4 ) 0

4q 4q 0 4x4 p2 (x4 )

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

4q 0 4p3 (x4 ) 0 0 0 0 0 0 0 8

4q 4q 0 4p3 (x4 ) 0 0 0 0 0 0 0 8

0 0 0 0 0 0 0 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. Other than 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 to 34 and 63 to 66. d. Columns 35 to 38 (resp. 67 to 70) are 12 p3 (x8 ) (resp. 14 p7 (x8 )) times columns 19 to 22, except that corresponding to the 4p3 (x4 ) in rows 12 to 15 we have 2p7 (x4 ) (resp. p15 (x4 )).

Topological Complexity

e. Columns 39 to 46 resemble Table 7. Columns 71 to 78 are 12 p3 (x16 ) times these, except for the diagonal near the bottom, which is p7 (x8 ). 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 described in Theorem 4.2 and its proof. Step 9. Divide R11 by x4 p2 (x4 ). We will show in Theorem 4.2 that all entries in R11 are divisible by x4 . The leading entry in row 11 is now a 4 in C22 . Step 10. Subtract multiples of R11 from rows 0 to 10 and 12 to 15 to clear out their entries in C22 . Similarly to Step 7, we now have that Pi,38 = 0 except for P11,38 = 2p3 (x8 ), P15,38 = 2x8 p2 (x8 ), and P39,38 = 4, with a similar situation in C70 . In particular, P15,70 = x8 p6 (x8 ) = 12 p3 (x16 )P15,38 . Step 11. Subtract 2R11 from R23 , and, similarly to Steps 5 and 8, kill entries subtracted from P23,j for j > 22 by adding multiples of j+1 . The smallest such j is Rj+1 , thus bringing up these multiples of R 38, due to the entry in (11, 38) described in the previous step. Step 12. Now the leading entry of R23 is 2x8 p2 (x8 ) in C46 . (This can be seen using (13) and that there have been no other changes to R23 in columns less than 62.) Divide R23 by x8 p2 (x8 ). We will show later 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 31 to make their entries in C46 equal to 0. Similarly to Step 10, this will cause Pi,78 = 0 except for P23,78 = p3 (x16 ), P31,78 = x16 p2 (x16 ), and P79,78 = 2. Step 14. Subtract 2R23 from R47 . This will add multiples of 2 to R47 in some columns j ≥ 78. These can be removed, without any other effect, by subtracting a multiple of Rj+1 . Now R47 has leading entry x16 p2 (x16 ) in C94 . Divide R47 by x16 p2 (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 paragraph containing (12). Rearrange rows as specified there, and we are done.

Donald M. Davis

It remains to show that Steps 3, 6, 9, and 12 above could actually be carried out, by showing that there was sufficient divisibility by x. This will 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 following table. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 i ∆(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 stage of 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). • If 39 ≤ j ≤ 46 and 0 ≤ i ≤ j − 16, then ν(i, j) ≥ 46 − j + ∆(i). • If 79 ≤ j ≤ 94 and 0 ≤ i ≤ j − 32, then ν(i, j) ≥ 94 − j + ∆(i). Since this applies to any stage of the reduction, it says that all xexponents 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 the step. The divisibility of other columns in rows 2, 5, 11, and 23 at Steps 3, 6, 9, and 12 is easily checked, mostly following from proportionality. Proof. We give the proof for 79 ≤ j ≤ 94. The proof for the smaller ranges is basically the same. The proof is by induction on j. By Theorem 2.1(e) shifted, at the outset ν(32, 79) = 16, while ν(i, 79) = ∞ for i = 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,   i ∈ {2, 3, 6, 12, 24, 48} νE (i − 1, j − 1) 1 ν(i, j) = νE ( 2 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 the induction hypothesis, this is   i ∈ {2, 3, 6, 12, 24, 48} 94 − j + 1 + ∆(i − 1) 1 ≥ 94 − j + 1 + ∆( 2 i − 1) = 94 − j + 1 + ∆(i − 1) i ∈ {6, 12, 24, 48}   94 − j + 1 + 5 − i i ∈ {2, 3}.

Topological Complexity

Let µ(i, j) denote a lower bound for ν(i, j) − (94 − j). At the outset, we have, 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 of µ after Step k. We have µ0 (i) ≥ ∆(i − 1) + 1 for i ≥ 4, µ0 (2) ≥ 4 and µ0 (3) ≥ 3. Although it is possible that actual divisibility could increase after a step (by having terms of smallest exponent cancel), our lower bounds, being just bounds, cannot see this. Thus we always have µk+1 (i) ≤ µk (i), so we wish to prove that µ14 (i) ≥ ∆(i). Step 1 sets (17)

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

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, since Step 1 is only changing R0 and R1 . In asserting (17), it is relevant that i which affect R1 do not include i = 2t or 3 · 2t , since those the various R are 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 is k is 0 in Ci−1 throughout the due to the fact that, for these values of i, R reduction 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

Donald M. Davis

the end of Step 4, or by noting that the entries in rows 4, 6, and 7 will 5 at be unchanged from their values in Table 5, while R1 got x2 from R 0 Step 1, R2 got x from R4 at Step 2, then changed to x at Step 3, while R3 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 out that the only change is µ7 (7) ≥ 0. Our bounds now for i from 0 to 7 are 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 to know exponent bounds in C22 at this stage of the reduction, because it is these multiples of R11 that are being subtracted from the row in question. For i < 15, they will be the same as the µ-values that we are computing here, because the same steps apply. However, we have µ10 (15) = 0 due to the 4p3 (x4 )-entry in P15,22 . Our exponent bounds µ10 (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 obtain µ11 (23) ≥ 8, and then µ12 (23) ≥ 0. For Step 13, we need to know exponent bounds in C46 at this stage of the reduction, because it is these multiples of R23 that are being subtracted from the row in question. For i < 31, they will be the same as the µ-values that we are computing here, because the same steps apply. However, we have µ13 (31) = 0 due to the 2p3 (x8 )-entry in P31,46 . In Step 14, we obtain µ14 (47) = 0, with no other changes to µ. Our final values for µ14 (i) are 0 for i = 2, 3, 5, 7, 11, 15, 23, 31, and 47, and increasing in increments of 1 from one of these to the next. This equals ∆(i), as claimed.

Proof of Theorem 2.2

In this section, we prove Theorem 2.2 by defining a sequence of matrices N0 , . . . , 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 the matrix 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 is the matrix at the end of Step 14 except that P95,94 has not yet been cleared out.

Topological Complexity

In the following, lg(−) denotes [log2 (−)], and δi,j is the usual Kronecker symbol. We continue to suppress e from the notation. Definition 5.1. For 0 ≤ s ≤ e − 1, Ns is a matrix with rows numbered from 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 1

k ≤s−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−t x2 p2t− −2 (x2 ) 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−t p2t− −1 (x2 ) in (3 · 2 + m − 1, 2t + 2 − 2 + m) for 1 ≤ m ≤ 2 +

−1 0

<s ≥ 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 additional nonzero 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 = 0 if i ≥ d + 2lg(d+1.5)+1 + 2; • if k ≥ 0 and i = 3 · 2k − 1, then Pi,j = 0 for i ≤ j < 2i.

Donald M. Davis

d. For 3 â&#x2030;¤ t < u â&#x2030;¤ e, 2tâ&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 1 â&#x2030;¤ d â&#x2030;¤ 2tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 2, and i â&#x2030;¤ d + 2tâ&#x2C6;&#x2019;1 , Pi,2u +d =

tâ&#x2C6;&#x2019;1 1 p (x2 )Pi,2t +d . 2uâ&#x2C6;&#x2019;t 2uâ&#x2C6;&#x2019;t+1 â&#x2C6;&#x2019;1

This is also true for d = 2tâ&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 2 if s â&#x2030;Ľ t â&#x2C6;&#x2019; 2, except in row 3 Âˇ 2tâ&#x2C6;&#x2019;3 â&#x2C6;&#x2019; 1. e. If 2 â&#x2030;¤ t â&#x2030;¤ e â&#x2C6;&#x2019; 1, 3 Âˇ 2t â&#x2C6;&#x2019; 2tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 1 â&#x2030;¤ j â&#x2030;¤ 3 Âˇ 2t â&#x2C6;&#x2019; 2, and i â&#x2030;¤ j â&#x2C6;&#x2019; 2t , then Pi,j is divisible by xÎ˝ with Î˝ = 3 Âˇ 2t â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019; j + Îˇs (i), where ďŁą ďŁ´ ďŁ˛3 â&#x2C6;&#x2019; i Îˇ0 (i) = 6 â&#x2C6;&#x2019; i ďŁ´ ďŁł i â&#x2C6;&#x2019; c(i) + 1

0â&#x2030;¤iâ&#x2030;¤1 2â&#x2030;¤iâ&#x2030;¤3 i â&#x2030;Ľ 4,

with c(i) the largest integer â&#x2030;¤ i of the form 2v or 3 Âˇ 2v , and 0 if i + 1 = 3 Âˇ 2v or 4 Âˇ 2v for 0 â&#x2030;¤ v < s Îˇs (i) = Îˇ0 (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 2R3Âˇ2eâ&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1 from R3Âˇ2eâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1 and then rearranging rows, Neâ&#x2C6;&#x2019;1 satisfies the properties of Theorem 2.2. Call this rearranged matrix Q. The rearranging is that for i = 3 Âˇ 2t â&#x2C6;&#x2019; 1 with 0 â&#x2030;¤ t â&#x2030;¤ e â&#x2C6;&#x2019; 2, Ri moves to position 2i, while for other values of i > 2, Ri moves to position i â&#x2C6;&#x2019; 1. 2. Delete the last column of Q, precede this by a column of 0â&#x20AC;&#x2122;s, and precede this by the following two rows. 0 0

1 2e

x 22e 2

2 2e

x2 32e 3

3 2 e

x3 42e 4

ÂˇÂˇÂˇ ÂˇÂˇÂˇ

2e â&#x2C6;&#x2019; 1

2e 0

...

â&#x2C6;&#x2019;1

3 Âˇ 2eâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 2 0

Then perform the 2e -analogues of Steps 0 and 1 of Section 4. The result is the matrix N0 . 3. For 0 â&#x2030;¤ s â&#x2030;¤ e â&#x2C6;&#x2019; 2, the matrix Ns reduces to Ns+1 .

Topological Complexity

Proof. Part 1 is straightforward but tedious and mostly omitted. As an example of the comparison, the final case of the second • of Definition 5.1(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 of Theorem 2.2(i). Next we address Part 2. After shifting and performing Step 0 of Section 4, we will have the 2e analogue of Table 5, in which we recall that odd factors were not written. It is easy but tedious to verify that everything 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 T becomes

P2 +1 +m−1,2T +2 −2+m = 2e−T x2 p2T − −2 (x2 ) for 1 ≤ m ≤ 2 for > 0. With = s and t = T − s, this matches with part ii of Theorem 2.2 shifted 2 down and 1 to the right. Part e of Definition 5.1 for Part 2 is somewhat delicate. We had ηe−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 rearranging done in Part 1 puts these 0’s in i = 4, 2, 10, 6, 22, 14,. . ., i.e. i = 2t − 2 or 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 and 3 · 2t , starting with i = 4, but we add 1 to the η values because of the shift of columns. For example, column 21 had ν ≥ 1 + η, but this now applies to column 22, where it is interpreted as 0 + (η + 1). The values of 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 the form 2t − 1 by the method of Step 1 of Section 4. For example, if j is of the form 3 · 2t − 1 or 5 · 2t − 1, t ≥ 0, then the 2-exponent in R0 and R1 is 1 greater than that in Rj+1 , which is a leading entry. We subtract j+1 . multiples of 2Rj+1 to kill the terms. This brings up multiples of 2R If this is nonzero in Ck , the term brought up can be killed by subtracting k+1 . Because columns a multiple of Rk+1 . This brings up multiples of R t 11–15, 23–31, etc., i.e. those j satisfying 3 · 2 − 1 ≤ j ≤ 4 · 2t − 1, are 0, i for i from 12–16, 24–32, etc., and these are the we will not bring up R

Donald M. Davis

only rows which contain entries which do not satisfy the proportionality and x-divisibility conditions stated in d and e of Definition 5.1, and the only rows that will have η(i) < 2. Thus we will obtain η0 (1) ≥ 2, and η0 (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 or 5 · 2t − 1. If also j = 2t − 1, then the 2-exponent in R0 and R1 will exceed that in Rj+1 by more than 1. We can use an even multiple at one of the two steps of the previous paragraph, or can break it up into more steps, which will make the rows eventually brought up have larger values of i, but, either way, we will not be bringing up the bad rows such as 12–16, etc., and so all the will be transferred to R0 2eproperties and R1 . Changing the terms − 2t in C2t −1 to 2e−t is accomplished similarly, using that these differ by a multiple of 2e−t+2 , while the entry in (2t , 2t − 1) has 2-exponent e − t + 1.

There are three steps to the reduction in Part 3, analogous to Steps 5, 6, and 7 in Section 4. Note that the only nonzero entries of Ns in C3·2s −2 are 2e−s−1 in R3·2s−1 −1 , and 2e−s in R3·2s −1 , and the second s s nonzero entry in R3·2s −1 is 2e−s−2 x2 p2 (x2 ) in C3·2s+1 −2 . The first 3·2s−1 −1 has q = 0 in step is to subtract 2R3·2s−1 −1 from R3·2s −1 . If R s Cj , then the −2q brought into R3·2 −1 can be killed by adding qRj+1 . The net effect is to remove the leading entry of R3·2s −1 , making the s s 2e−s−2 x2 p2 (x2 ) in C3·2s+1 −2 its new leading entry, and to bring into j+1 for which P3·2s−1 −1,j = 0. By (c), such j must this row various q R j only occur in satisfy j > 2s+2 + 1, and then nonzero entries in R s+3 columns > 2 + 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 none of the j’s appearing above can satisfy 3 · 2t − 1 ≤ j ≤ 3 · 2t + 2s − 3 or 4 · 2t − 1 ≤ j ≤ 4 · 2t + 2s − 3, t ≥ s, which are the only values having ηs (j + 1) < 2s . Thus η(3 · 2s − 1) does not change at this step. s

The second step divides R3·2s −1 by x2 p2 (x2 ). This can be done because 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 changes s s s+1 P3·2s −1,2u +2s+1 −2 from 2e−u x2 p2u−s −2 (x2 ) to 2e−u p2u−s−1 −1 (x2 ) for u ≥ 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 and t = u. Now C3·2s+1 −2 has

Topological Complexity

â&#x20AC;˘ 2eâ&#x2C6;&#x2019;sâ&#x2C6;&#x2019;2 in row 3 Âˇ 2s â&#x2C6;&#x2019; 1; s

â&#x20AC;˘ 2eâ&#x2C6;&#x2019;sâ&#x2C6;&#x2019;2 p3 (x2 ) in row 2s+2 â&#x2C6;&#x2019; 1; â&#x20AC;˘ multiples of 2eâ&#x2C6;&#x2019;sâ&#x2C6;&#x2019;2 in rows 0 through 2s+2 â&#x2C6;&#x2019; 1; â&#x20AC;˘ a leading 2eâ&#x2C6;&#x2019;sâ&#x2C6;&#x2019;1 in row 3 Âˇ 2s+1 â&#x2C6;&#x2019; 1; â&#x20AC;˘ other entries 0. Now we subtract multiples of row 3Âˇ2s â&#x2C6;&#x2019;1 from all other rows except row 3 Âˇ 2s+1 â&#x2C6;&#x2019; 1 to make them 0 in column 3 Âˇ 2s+1 â&#x2C6;&#x2019; 2. By property (d), this will zero all entries in column 2u + 2s+1 â&#x2C6;&#x2019; 2, u > s + 2, except in rows 3 Âˇ 2s â&#x2C6;&#x2019; 1, 2s+2 â&#x2C6;&#x2019; 1, and 2u + 2s+1 â&#x2C6;&#x2019; 1. For u > s + 2, the entry in s (2s+2 â&#x2C6;&#x2019; 1, 2u + 2s+1 â&#x2C6;&#x2019; 2) is changed from 2eâ&#x2C6;&#x2019;u p2uâ&#x2C6;&#x2019;s â&#x2C6;&#x2019;1 (x2 ) to s

2eâ&#x2C6;&#x2019;u (p2uâ&#x2C6;&#x2019;s â&#x2C6;&#x2019;1 (x2 ) â&#x2C6;&#x2019; p3 (x2 )p2uâ&#x2C6;&#x2019;sâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1 (x2

= â&#x2C6;&#x2019;2eâ&#x2C6;&#x2019;u x

2s+1

p2uâ&#x2C6;&#x2019;sâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2 (x

2s+1

s+1

))

The minus here can be changed to plus by modifying by a multiple of row 2u + 2s+1 â&#x2C6;&#x2019; 1, which will not affect the properties such as (d) and (e). Property (d) will now hold in Ns+1 for proportionality out of C2s+3 +2s+1 â&#x2C6;&#x2019;2 , to the extent claimed there. This change removes the entry of the second â&#x20AC;˘ of (b) with = s, m = 2s , and t = u and replaces it by the entry of the first â&#x20AC;˘ with = s + 1, m = 0, and t = u. Finally we consider the effect of this step on x-divisibility. If j is as in (e) with t > s + 1, and i â&#x2030;¤ 2s+2 â&#x2C6;&#x2019; 1 and i = 3 Âˇ 2s â&#x2C6;&#x2019; 1, then the new value of Pi,j will equal old Pi,j â&#x2C6;&#x2019;

Pi,3Âˇ2s+1 â&#x2C6;&#x2019;2 Âˇ P3Âˇ2s â&#x2C6;&#x2019;1,j . 2eâ&#x2C6;&#x2019;sâ&#x2C6;&#x2019;2 t

The old Pi,j is divisible by x3Âˇ2 â&#x2C6;&#x2019;2â&#x2C6;&#x2019;j+Îˇs (i) . Also, Pi,3Âˇ2s+1 â&#x2C6;&#x2019;2 is divisible by xÎˇs (i) if i â&#x2030;¤ 2s+2 â&#x2C6;&#x2019; 2, and by x0 if i = 2s+2 â&#x2C6;&#x2019; 1. (Note that (e) did not apply in this latter case due to the condition there which here would t say i â&#x2030;¤ j â&#x2C6;&#x2019; 2s+1 .) We now have P3Âˇ2s â&#x2C6;&#x2019;1,j divisible by x3Âˇ2 â&#x2C6;&#x2019;2â&#x2C6;&#x2019;j since Îˇ(3 Âˇ 2s â&#x2C6;&#x2019; 1) became 0 at the previous substep. Thus the x-divisibility of Pi,j does not decrease except when i = 2s+2 â&#x2C6;&#x2019; 1, where it changes to 0, consistent with Îˇs+1 (2s+2 â&#x2C6;&#x2019; 1) = 0.

Donald M. Davis

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, and a Mathematica calculation by Gonz´ alez 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 follow from showing that (18)

2e−k u3·2

k−1 −3

[0, 0] = 0 in Me for 1 ≤ k ≤ e.

For e ≤ 5, (18) is an immediate consequence of the following, which is the main result of this section. Theorem 6.1. For e ≥ 1 and 1 ≤ k ≤ min(e, 5), there is a homomork−1 phism φk,e : Me → Z/2k+e−1 sending 2e−k u3·2 −3 [0, 0] nontrivially. The homomorphism φk,e is nonzero only on the component of Me in grading 2(3 · 2k−1 − 3). The component of Me in grading 2d is generated by the same monomials ud−i−j [i, j] for any e, but the relations depend k−1 on e. We will give an explicit formula for φk,e (u3·2 −3−i−j [i, j]) ∈ Z for i, j ≥ 0, which is independent of e. Thus we usually call it just φk . We will prove that φk applied to a relation (3) in Me is divisible by 2k+e−1 . k−1 Since part of our formula is φk (u3·2 −3 [0, 0]) = 22k−2 and hence φk (2e−k u3·2

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 formulas for φk that might extend to all k, but they seem a bit too delicate for that. k−1

Since the exponent of u in u3·2 −3−i−j [i, j] is determined by k, i, k−1 and j, we do not list it. We write φk (i, j) for φk (u3·2 −3−i−j [i, j]), and will sometimes omit the subscript k. We have φ1 (0, 0) = 1, and the only relation 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]

Topological Complexity

Our functions always satisfy φ(i, j) = φ(j, i). The first line says that the nonzero values of φ2 are φ2 (0, 0) = 4, φ2 (2, 0) = φ2 (1, 1) = 2, and φ2 (2, 1) = 1, and their flips. The next pair of lines says, for example, that φ3 (0, 0) = 16 and 4 i = 2, 4 φ3 (i, 6 − i) = 0 i = 0, 1, 3, 5, 6. Before we list the formulas for φ4 and φ5 , we discuss the verification that φ3,e : Me → Z/2e+2 is well-defined for all e ≥ 3. This one is simple enough that it can be (and was) 8 done 8by hand. We first consider the case e = 3. The coefficients 1 , . . . 8 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 only 8 13 of them contain any term for which ν( +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 by 2e−3 . Their odd factors change, but where it matters, the odd factors are still 3 mod to each relation is divisible by 2e−3 · 32. 2e 4. So φ2eapplied Terms with 9 and 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 welldefinedness 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 there are some nonzero values. We precede the row by the value of i + j. For example, the third listed row says that 32 i = 2, 8 φ4 (i, 10 − i) = 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

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 that they 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 went into obtaining them, once we have them, it is a simple computer check to verify that they work. It is just a matter of reading these numbers φ 4 (i, j) into the computer and then having the computer check that i

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

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

φ4 (i − , j) ≡ 0

It is easy to prove that, for 1 < < 2e+1 , (19)

ν(

2e+1

−2

) = 2e + 1 − [log2 ( − 1)] − ν( ).

The induction argument follows from this and the values of φ4 (−) listed above. 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 take advantage of symmetry, and only list φ5 (i, j) for i ≤ j. As before, we list values of φ5 (i, j) in rows of fixed i + j for which there are some nonzero

Topological Complexity

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. Davis Department of Mathematics, Bethlehem, PA 18015, USA, dmd1@lehigh.edu

Donald M. Davis

References [1] Astey L.; Geometric dimension of bundles over real projective spaces, Quart.J. Math. Oxford 31 (1980), 139–155. [2] Farber M., Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221. [3] Gonz´ alez J., Connective K-theoretic Euler classes and nonimmersions of 2k -lens spaces, J. London Math. Soc. 63 (2001), 247–256. [4] Gonz´ alez J., Topological robotics in lens spaces, Math. Proc. Cambridge Philos. Soc. 139 (2005), 469–485. [5] Gonz´ alez J.; Velasco M., Wilson W., Biequivariant maps of spheres and topological complexity of lens spaces, Comm. Contemp. Math 15 (2013), 33. [6] Gonz´ alez J.; Z´ arate L., BP-theoretic instabilities to the motionplanning problem in 4-torsion lens spaces, Osaka J. Math 43 (2006), 581–596.

Morfismos, Vol. 18, No. 2, 2014, pp. 41â&#x20AC;&#x201C;50 Morfismos, Vol. 18, No. 2, 2014, pp. 41â&#x20AC;&#x201C;50

Geometric dimension of stable vector bundles over spheres Geometric dimension of stable bundlesDuane over Randall spheres Keevector Yuen Lam Kee Yuen Lam

Duane Randall

Abstract Abstract We present a new method to determine the geometric dimension of stable vector bundles over spheres, using a constructive approach. The basic tools include We present a new method to K-theory determineand the representation geometric dimentheory of Lie groups, and the use of spectral sequences totally sion of stable vector bundles over spheres, using a is constructive avoided. approach. The basic tools include K-theory and representation theory of Lie groups, and the use of spectral sequences is totally

2010 Mathematics Subject Classification: Primary 55R50, 55R45. avoided. Keywords and phrases: stable vector bundles, geometric dimension, KOcohomology, spinor representations, Adams maps, octonian 2010 Mathematics Subject Classification: Primary 55R50,multiplica55R45. tion. Keywords and phrases: stable vector bundles, geometric dimension, KOcohomology, spinor representations, Adams maps, octonian multiplication.

Introduction

Let1X beIntroduction a connected finite cell complex of dimension m and Îˇ a vector bundle over X. In particular, the k-dimensional trivial vector bundle is denoted Given Îˇ, one often trivial sub-bundles Îˇ â&#x160;&#x2022;Îˇ m of Let Xbybek . a connected finite cellseeks complex of dimension m of and a vector geometric least possible q. This is commonly called bundle overco-dimension X. In particular, theq k-dimensional trivialthe vector bundle is dimension denoted gd(Îˇ). definition, as formulated, is m in of denotedofbyÎˇ,k . Given qÎˇ,=one oftenThe seeks trivial sub-bundles of Îˇ â&#x160;&#x2022; such a manner least possiblethat co-dimension q. This q is commonly called the geometric dimension of Îˇ, denoted q = gd(Îˇ). The definition, as formulated, is in gd(Îˇ) =that gd(Îˇ â&#x160;&#x2022; ) = gd(Îˇ â&#x160;&#x2022; 2 ) = gd(Îˇ â&#x160;&#x2022; 3 ) = . . . such a manner holds, so thatgd(Îˇ) one can also â&#x160;&#x2022; speak theâ&#x160;&#x2022;geometric for = gd(Îˇ ) = of gd(Îˇ 2 ) = gd(Îˇdimension â&#x160;&#x2022; 3 ) = . .gd(x) . an arbitrary element x in the Grothendieck cohomology group KO(X), see holds, [1]. so that one can also speak of the geometric dimension gd(x) for an arbitrary element x in theonGrothendieck cohomology KO(X), Bottâ&#x20AC;&#x2122;s periodicity theorem the homotopy groups of group the infinite see [1]. m orthogonal group SO tells us that for X = S = the m-dimensional Bottâ&#x20AC;&#x2122;s periodicity theorem on the homotopy groups of the infinite m orthogonal group SO tells us that 41 for X = S = the m-dimensional 41

K. Y. Lam and D. Randall

sphere where m > 1, one has, in reduced cohomology, m ) = [S m , BSO] = KO(S

  0 πm−1 (SO) = Z   Z/2Z

m ≡ 3, 5, 6 or 7 (mod 8), m ≡ 0, 4 (mod 8), m ≡ 1, 2 (mod 8).

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

SO(q)

SO x

S m−1 what is the smallest q such that x can be homotopically compressed into the finite orthogonal group SO(q)? The complete answer, obtained through a period of 25 years, is due to Mark Mahowald and his coworkers [3, 5]. In these papers the principal tool is to use homotopy spectral sequences. The drawback of such an approach is that geometric features behind claimed results often become obscure, especially when calculation steps or analysis of differentials in spectral sequences are sometimes suppressed in favor of brevity. In this paper we aim at providing a non-spectral sequence approach to determine gd(x) for all m ), re-obtaining Theorem 1.1 of [5]: x ∈ KO(S 8k+1 ) or KO(S 8k+2 ) where k 1, (I) For the generator x of KO(S the geometric dimension gd(x) = 6.

4k ), k 5, the geometric (II) For any nonzero element y in KO(S dimention gd(y) = 2k + 1.

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

8k+1 ) = Z/2Z has Theorem 2.1. For k 1, the generator x in KO(S gd(x) = 6.

Geometric Dimension

This is the second part of Theorem 1.1 of Davis-Mahowald [5], summarisable by a â&#x20AC;&#x153;compression diagramâ&#x20AC;? below: BSO(5)

BSO(6)

BSO(7)

...

Yes

BSO

x(essential)

S 8k+1 Alternatively, one can say that Bottâ&#x20AC;&#x2122;s generator in Ď&#x20AC;8k (SO) = Z/2Z â&#x20AC;&#x153;originatesâ&#x20AC;? from the finite orthogonal group SO(6), but not from SO(5). Our proof shall be accomplished in a number of geometric steps. Step A. For k = 1, we construct a 6-dimensional vector bundle Îˇ over S 9 that is stably non-trivial. Let X = S 9 â&#x2C6;Ş8 e10 be the Moore space obtained from S 9 via attachment of a 10-dimensional cell using a self-map of S 9 of degree 8. Let c : X â&#x2C6;&#x2019;â&#x2020;&#x2019; S 10 be the collapse map which pinches S 9 to a point. In 10 ) = Z can be represented by complex K-theory, the generator of K(S a C-vector bundle Ď&#x2030; on S 10 of C-dimension 5. It is well-known [4] that in H 10 (S 10 ; Z) = Z the 5th Chern class c5 (w) = (5 â&#x2C6;&#x2019; 1)! = 24 so that Ď&#x2030; as a real vector bundle has Euler class 24. It follows that the pull-back vector bundle c! (Ď&#x2030;) over X has zero Euler class, and is thus sectionable. As a result c! (Ď&#x2030;) = Îś â&#x160;&#x2022; C for some complex vector bundle Îś over X with dimC Îś = 4 while C is the trivial complex line bundle. Furthermore, since the first Chern class c1 (Îś) = 0, Îś can be regarded to have structural group SU (4) rather than U (4). Step B. In Lie group representation theory [6, Chapter 13], one has + 6 : Spin(6)

â&#x2030;&#x2C6;

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 that there is a 6-dimensional spinor bundle Îˇ over X such that + 6 (Îˇ) = . Also, from Îś. i.e., such that Îś is the associated bundle of Îˇ via + 6 construction one knows that y = Îś â&#x2C6;&#x2019; 4 C is a generator of K(X) = 9 â&#x2C6;Ş8 e10 ) = Z/8Z. K(S

K. Y. Lam and D. Randall

Step C. We now make the crucial claim that Îˇ restricted to S 9 is stably 9) = nontrivial, in other words Îˇ|S 9 represents the generator of KO(S Z/2Z. Our argument is inspired by the methods in [1]. Recall from representation theory that the double covering of SO(6) by Spin(6) fits into a commutative diagram Spin(6)

+ 6

SU (4)

U (4) ÎťC 2

inclusion

SO(6)

U (6)

in which ÎťC 2 denotes second exterior power operation for complex vector spaces. Going around clockwise one sees that the complexification of Îˇ can be computed via + C Îˇ â&#x160;&#x2014; C = ÎťC 2 ( 6 (Îˇ)) = Îť2 (y + 4 C )

C C C = ÎťC 2 (y) + Îť1 (y) â&#x160;&#x2014;C Îť1 (4 C ) + Îť2 (4 C )

= â&#x2C6;&#x2019;16y + 4y + 6 C = 4y + 6 C Ë&#x153; = 0 in K(X)

Ë&#x153; 10 ), ÎťC operates as Here the term â&#x2C6;&#x2019;16y is due to the fact that on K(S 2 5â&#x2C6;&#x2019;1 = â&#x2C6;&#x2019;16. If Îˇ were a stably trivial vector bundle multiplication by â&#x2C6;&#x2019;2 on S 9 , then Îˇâ&#x160;&#x2022;4 would be trivialisable on S 9 , so that it can be regarded as a pullback, via c, of some 10-dimensional vector bundle defined over S 10 . But the complexification morphism 10 ) KO(S

â&#x160;&#x2014;C

10 ) K(S

is a trivial homomorphism Z/2Z â&#x2C6;&#x2019;â&#x2020;&#x2019; Z, so Îˇ â&#x160;&#x2014; C = (Îˇ â&#x160;&#x2022; 4 ) â&#x160;&#x2014; C â&#x2C6;&#x2019; 4 C would be equal to 10 C â&#x2C6;&#x2019; 4 C = 6 C , contradicting the computation above. Step D. On spheres S 8k+1 with k > 1 we can obtain 6-dimensional vector bundle by pulling back Îˇ via Adams maps A between mod 8 Moore spaces [2], as in A

S 8k+1 â&#x2C6;Ş8 e8k+2 â&#x2C6;&#x2019;â&#x2020;&#x2019; S 8kâ&#x2C6;&#x2019;7 â&#x2C6;Ş8 e8kâ&#x2C6;&#x2019;6 â&#x2C6;&#x2019;â&#x2020;&#x2019; ... â&#x2C6;&#x2019;â&#x2020;&#x2019; S 17 â&#x2C6;Ş8 e18 â&#x2C6;&#x2019;â&#x2020;&#x2019; S 9 â&#x2C6;Ş8 e10 = X

groups all These spaces are 8j-fold suspensions of X with their K equal to Z/8Z, mutually isomorphic [2] under the induced homomorphisms Aâ&#x2C6;&#x2014; . We go on to point out, moreover, that for all k 1,

Geometric Dimension

8k+2 ), so that KO(S 8k+1 ∪8 e8k+2 ) equals 8k+1 ) = Z/2Z = KO(S KO(S Z/2Z ⊕ Z/2Z, with generators u, v such that v restricts trivially to 8k+1 ) while u doesn’t. One needs to argue, further beyond Adams, KO(S groups are mutually isomorphic under A∗ as well. To that these KO this end notice that the v generators are obtained from the generators of theory by realification, i.e. by forgetting complex the Z/8Z groups of K structures. They thus do correspond under A∗ . The pullback of the 6dimensional vector bundle η on X to S 8k+1 ∪8 e8k+2 retains the crucial property of η pointed out in Step C, namely that it stably complexifies 8k+1 ∪8 e8k+2 ). This entails, again into the element of order 2 in K(S as in Step C, that the pullback of η is stably non-trivial on S 8k+1 . It must therefore represent an u generator. In particular the generator x 8k+1 ) now has a 6-dimensional vector bundle as representative, of KO(S and gd(x) 6 follows. Step E. Finally we will rule out the possibility that gd(x) 5, by establishing

Proposition 2.2. Any 5-dimensional vector bundle ξ over S 8k+1 must be 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 isomorphic image Sp(2) inside U (4). From representation theory one knows that the (left) quaternionic 2-plane bundle 5 (ξ) associated to ξ satisfies µ2 ( 5 (ξ)) ≈ ξ ⊕

as real vector bundles, where µ2 is the functorial operation described as in [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 the quaternions into real vector spaces of dimension m(2m − 1). It enjoys the property ˆ ⊕ µ2 (W ) µ2 (V ⊕ W ) = µ2 (V ) ⊕ V ⊗W

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

K. Y. Lam and D. Randall

(S 8k+1 ) = 0 one has an isomorphism of symplectic vector Since KSP bundles 5 (ξ) ⊕ n H = (n + 2) H for n sufficiently large. Taking n large as well as even and applying µ2 , one gets, in KO(S 8k+1 ), ˆ H (n H ) ⊕ µ2 (n H ) = µ2 ((n + 2) H ) µ2 ( 5 (ξ)) ⊕ 5 (ξ)⊗ See [8]. The last two terms in this equation are, of course, trivial R 8k+1 ), the middle vector bundles. Since n, when even, annihilates KO(S term on the left side is also R-trivial. Hence the equation implies that µ2 ( 5 (ξ)), and hence ξ itself, must be stably trivial. 8k+2 follows easily, as in The case of S 8k+2 ) = Z/2Z has Corollary 2.3. For k 1 the generator x in KO(S gd(x ) = 6. This is because the unique essential map

η : S 8k+2 −→ S 8k+1 is well-known to satisfy η ! (x) = x . Also, Proposition (2.2) on 5dimensional vector bundles carries over from S 8k+1 to S 8k+2 , with no change whatsoever.

The 6-dimensional bundle η over S 9

When octonions are regarded as pairs of quaternions, their multiplication is well-known to be given by the formula (a1 , a2 ) × (b1 , b2 ) −→ (a1 b1 − b2 a2 , b2 a1 + a2 b1 ). This provides a bilinear multiplication R8 × R8 −→ R8 free of zero divisors. One can use the same formula to define multiplication of “bioctonions”, by regarding a1 , a2 , b1 , b2 themselves as octonions. This gives a bilinear “Cayley-Dickson” multiplication µ : R16 × R16 −→ R16 which is long known to have zero divisors. However [7] some restrictions of µ to “partial multiplications” will be zero-divisior free. Two such

Geometric Dimension

restrictions are the Âľr and the Âľc below: Âľr :

R9 Ă&#x2014; R16

Âľc : R10 Ă&#x2014; R10

â&#x2C6;&#x2019;â&#x2020;&#x2019; R16 â&#x2C6;&#x2019;â&#x2020;&#x2019; R16

where for Âľr we take (a1 , a2 ) only with a1 â&#x2C6;&#x2C6; R, and for Âľc we take (a1 , a2 ), (b1 , b2 ) with a1 â&#x2C6;&#x2C6; C, b1 â&#x2C6;&#x2C6; C. These two zero-divisor free multiplications are â&#x20AC;&#x153;compatibleâ&#x20AC;? in that they coincide over the mutual subdomain R9 Ă&#x2014; R10 . Both multiplications are in fact â&#x20AC;&#x153;orthogonalâ&#x20AC;?, in the sense that the norm of the product equals the product of the norms of the factors. For a line Îť through the origin in Rn write [Îť] for the corresponding point of the real projective space RP nâ&#x2C6;&#x2019;1 . Let Îˇ0 be the 6-dimensional [Îť] vector bundle over RP 9 obtained by setting up over [Îť] the fiber Îˇ0 , given by the short exact sequence idÎť â&#x160;&#x2014;Ë&#x2020;Âľc [Îť] 0 â&#x2C6;&#x2019;â&#x2020;&#x2019; Îť â&#x160;&#x2014; Îť â&#x160;&#x2014; R10 â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; Îť â&#x160;&#x2014; R16 â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; Îˇ0 â&#x2C6;&#x2019;â&#x2020;&#x2019; 0

â&#x2020;&#x2019; where Âľ Ë&#x2020;c sends Îťâ&#x160;&#x2014; v â&#x160;&#x201A; Îťâ&#x160;&#x2014;R10 to Âľ c (ÎťĂ&#x2014; â&#x2C6;&#x2019; v ). To see the geometric mean10 is canonically isomorphic to ing of this sequence, note that Îťâ&#x160;&#x2014; Îť â&#x160;&#x2014; R R10 . Exactness means that, over RP 9 , the Whitney sum of 16 copies of the Hopf line bundle has 10 independent sections, with Îˇ0 as direct sum RP 9 = Z/32Z, complement. Note that Îˇ0 isnâ&#x20AC;&#x2122;t stably trivial, as KO with Îť â&#x2C6;&#x2019; as generator. When [Îť] happens to be in RP 8 , this exact sequence fits into a commutative diagram Îť â&#x160;&#x2014; Îť â&#x160;&#x2014; R10 Îť â&#x160;&#x2014; Îť â&#x160;&#x2014; R16

idÎť â&#x160;&#x2014;Ë&#x2020; Âľc â&#x2030;&#x2C6;

Îť â&#x160;&#x2014; R16

[Îť]

Îˇ0

idÎť â&#x160;&#x2014;Âľr

This is to say that over RP 8 , Îˇ0 is isomorphic to the 6-dimensional 10 trivial bundle â&#x20AC;&#x153;constantâ&#x20AC;? fiber is the normal bundle of Îťâ&#x160;&#x2014;(Îťâ&#x160;&#x2014;R ) whose 16 inside Îť â&#x160;&#x2014; Îť â&#x160;&#x2014; R , which is framed by the standard normal frame of R10 inside R16 . By pinching RP 8 to a point and identifying all [Îť] Îˇ0 8 [Îť]â&#x2C6;&#x2C6;RP

into one vector space according to such framing, one obtains a 6-dimensional vector bundle over S 9 , denoted again by Îˇ0 , which represents the

K. Y. Lam and D. Randall

9 ). Indeed it is not hard to show that this Îˇ0 further generator of KO(S extends to a vector bundle Îˇ 0 over S 9 â&#x2C6;Ş8 e10 , but weâ&#x20AC;&#x2122;ll omit the details. Such Îˇ 0 can provide a concrete alternative to the vector bundle Îˇ in Â§2.

Spheres of dimension 4k, k 5

4k ) = Z has Theorem 4.1. For k 5 any nonzero element x in KO(S gd(x) = 2k + 1. This is the first part of Theorem (1.1) of Davis-Mahowald [5]. It was actually established much earlier in [3]. Very briefly, the complexiâ&#x160;&#x2014;C 4k ) â&#x2C6;&#x2019;â&#x2C6;&#x2019; 4k ) is a monomorphism. If x is reprefication map KO(S â&#x2C6;&#x2019;â&#x2020;&#x2019; K(S sented by a vector bundle Îś of R-dimension m then x = 0 â&#x2021;&#x2019; x â&#x160;&#x2014; C = 0. Thus x â&#x160;&#x2014; C has nontrivial Chern character, and so Îś â&#x160;&#x2014; C must have a nonzero 2k th Chern class. This already forces m 2k. If m = 2k, then Îś â&#x160;&#x2014; C â&#x2030;&#x2C6; Îś â&#x160;&#x2022; Îś as real vector bundles and in terms of Euler classes Ď&#x2021; 0 = c2k (Îś â&#x160;&#x2014; C) = Ď&#x2021;4k (Îś â&#x160;&#x2022; Îś) = Ď&#x2021;2k (Îś) Ď&#x2021;2k (Îś) = 0 â&#x2C6;&#x2014; (S 4k ). This contradiction because cup product is taken in the ring H forces m 2k + 1.

Finally, the fact that there exists indeed a (2k + 1)-dimensional vec 4k ) has been estor bundle which represents the generator x of KO(S tablished quite early in [3]. This is again established in [5], through a comparison of the Postnikov tower of BSO with its connective covers [9]. We are not aware of any other existence proofs in the literature. In any event, with the above theorems, the geometric dimension of nonzero theory of an arbitrary sphere of dimensions differelements in the KO ent from 1,2,4,8,12 and 16 is now completely determined. There is little difficulty in handling these exceptional dimensions case by case. Indeed the answer to all nontrivial cases are already tabulated at the end of [5]. Postscript: The content of this article was presented by the first author at the 2008 meeting of the Sociedad Matematica Mexicana in Guanajuato, in a special session on algebraic topology dedicated to Professor Sam Gitler. At that time the authors were unaware of the paper by M. Cadek and M. Crabb entitled â&#x20AC;?Gâ&#x2C6;&#x2019;structures on spheresâ&#x20AC;?, published in Proc. London Math. Soc. (3)93(2006), 791-816. In an appendix to their paper, Cadek and Crabb included a self-contained proof of the

Geometric Dimension

Davis-Mahowald Theorem 1.1 in [5]. Their proof and ours, while independently arrived at, overlap to some extent. In our treatment more emphasis was placed on explicit constructions, such as the algebraic description of the 6−dimensional vector bundle η over S 9 . Also, as a conclusive step, we provided an original proof that any 5−dimensional vector bundle over a sphere of dimension at least 9 must be stably trivial. Our approach is elementary throughout, using standard K−theory. Such an approach, we believe, is much in line with Sam Gitler’s mathematical style. Acknowledgement The authors are grateful to the editors of Morfismos for an invitation to publish this write-up of the 2008 talk. They wish to record their sincere appreciation to Sam Gitler’s long term friendship and encouragement to both of them. The second author is partially supported by a Distinguished Professorship at Loyola University. Kee Yuen Lam Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada, lam@math.ubc.ca

Duane Randall Department of Mathematics, Loyola University, New Orleans, LA 70118, USA, randall@loyno.edu

References [1] Adams J. F.; Geometric dimension of bundles over RP n , International 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, New York (1975).

K. Y. Lam and D. Randall

[7] Lam K.Y.; Construction of non-singular bilinear maps, Topology 6 (1967), 423-426. [8] Lam K.Y.; A formula for the tangent bundle of flag manifolds and related 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.

Morfismos, Vol. 18, No. 2, 2014, pp. 51–65 Morfismos, Vol. 18, No. 2, 2014, pp. 51–65

The equivariant cohomology rings of regular nilpotent Hessenbergcohomology varieties inrings Lie type A: The equivariant of regular Research Announcement nilpotent Hessenberg varieties in Lie type A: Announcement Hiraku Abe,Research Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda Hiraku Abe, Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda Dedicated to the memory of Samuel Gitler (1933-2014).

Abstract Dedicated to the memory of Samuel Gitler (1933-2014). 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 to Abstract give a systematic method for producing an explicit presentation Let n be a fixed positive integer and h : {1, 2, ..., n} → {1, 2, ..., n} by generators and relations of the equivariant and ordinary cohoa Hessenberg function. The main result of this manuscript is to mology rings (with Q coefficients) of any regular nilpotent Hesgive a systematic method for producing an explicit presentation senberg variety Hess(h) in type A. Specifically, we give an explicit by generators and relations of the equivariant and ordinary cohoalgorithm, depending only on the Hessenberg function h, which mology rings (with Q coefficients) of any regular nilpotent Hesproduces the n defining relations {fh(j),j }nj=1 in the equivariant senberg variety Hess(h) in type A. Specifically, we give an explicit cohomology ring. Our result generalizes known results: for the algorithm, depending only on the Hessenberg function h, which case h = (2, 3, 4, . . . , n, n), which corresponds ton the Peterson vain the equivariant produces the n defining relations {fh(j),j } etn ) given previriety P etn , we recover the presentation of HS∗ (Pj=1 cohomology ring. Our result generalizes known results: for the ously by Fukukawa, Harada, and Masuda. Moreover, in the case case h = (2, 3, 4, . . . , n, n), which corresponds to the Peterson vah = (n, n, . . . , n), for which the corresponding regular nilpotent riety P etn , we recover the presentation of HS∗ (P n etn ) given previHessenberg variety is the full flag variety F ags(C ), we can exously by Fukukawa, Harada, and Masuda. Moreover, in the case plicitly relate the generators of our ideal with those in the usual h = (n, n, . . . , n), for which the corresponding regular nilpotent Borel presentation of the cohomology ring of F ags(Cnn ). The Hessenberg variety is the full flag variety F ags(C ), we can exproof of our main theorem includes an argument that the restricplicitly relate the generators of our ideal with those in the usual tion homomorphism HT∗ (F ags(Cn )) → HS∗ (Hess(h)) is surjective. Borel presentation of the cohomology ring of F ags(Cn ). The In this research announcement, we briefly recount the context and proof of our main theorem includes an argument that the restricstate our results; we also give a sketch of our proofs and conclude tion homomorphism HT∗ (F ags(Cn )) → HS∗ (Hess(h)) is surjective. with a brief discussion of open questions. A manuscript containing In this research announcement, we briefly recount the context and more details and full proofs is forthcoming. state our results; we also give a sketch of our proofs and conclude with a brief discussion of open questions. A manuscript 2010 Mathematics Subject Classification: 55N91, 14N15. containing more details and full proofs is forthcoming.

Keywords and phrases: Equivariant cohomology, Hessenberg varieties, flag 2010 varieties. Mathematics Subject Classification: 55N91, 14N15. Keywords and phrases: Equivariant cohomology, Hessenberg varieties, flag varieties. 51 51

Abe, Harada, Horiguchi, and Masuda

Introduction

This paper is a research announcement and is a contribution to the volume dedicated to the illustrious career of Samuel Gitler. A manuscript containing full details is in preparation [1]. Hessenberg varieties (in type A) are subvarieties of the full flag variety F ags(Cn ) of nested sequences of subspaces in Cn . Their geometry and (equivariant) topology have been studied extensively since the late 1980s [6, 8, 7]. This subject lies at the intersection of, and makes connections between, many research areas such as: geometric representation theory [26, 14], combinatorics [12, 23], and algebraic geometry and topology [5, 20]. Hessenberg varieties also arise in the study of the quantum cohomology of the flag variety [22, 25]. The (equivariant) cohomology rings of Hessenberg varieties has been actively studied in recent years. For instance, Brion and Carrell showed an isomorphism between the equivariant cohomology ring of a regular nilpotent Hessenberg variety with the affine coordinate ring of a certain affine curve [5]. In the special case of Peterson varieties P etn (in type A), the second author and Tymoczko provided an explicit set of generators for HS∗ (P etn ) and also proved a Schubert-calculus-type “Monk formula”, thus giving a presentation of HS∗ (P etn ) via generators and relations [16]. Using this Monk formula, Bayegan and the second author derived a “Giambelli formula” [3] for HS∗ (P etn ) which then yields a simplification of the original presentation given in [16]. Drellich has generalized the results in [16] and [3] to Peterson varieties in all Lie types [10]. In another direction, descriptions of the equivariant cohomology rings of Springer varieties and regular nilpotent Hessenberg varieties in type A have been studied by Dewitt and the second author [9], the third author [18], the first and third authors [2], and Bayegan and the second author [4]. However, it has been an open question to give a general and systematic description of the equivariant cohomology rings of all regular nilpotent Hessenberg varieties [19, Introduction, page 2], to which our results 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 be well-known by experts but for which we did not find an explicit proof in the literature: namely, that the natural restriction homomorphism HT∗ (F ags(Cn )) → HS∗ (Hess(h)) is surjective when Hess(h) is a regular nilpotent Hessenberg variety (of type A).

Cohomology rings of Hessenberg varieties

Background on Hessenberg varieties

In this section we briefly recall the terminology required to understand the statements of our main results; in particular we recall the definition of a regular nilpotent Hessenberg variety, denoted Hess(h), along with a natural S 1 -action on it. In this manuscript we only discuss the Lie type A case (i.e. the GL(n, C) case). We also record some observations regarding the S 1 -fixed points of Hess(h), which will be important in later sections. By the flag variety we mean the homogeneous space GL(n, C)/B which 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 values in sequence, h = (h(1), h(2), . . . , h(n) = n). Let N : Cn → Cn be a linear 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 ) | N Vi ⊆ Vh(i) for all i = 1, . . . , n} ⊆ F ags(Cn ). If N is nilpotent, we say Hess(N, h) is a nilpotent Hessenberg variety, and if N is a principal nilpotent operator then Hess(N, h) is called a regular nilpotent Hessenberg variety. In this manuscript we restrict 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 principal nilpotent operator, i.e. N has one Jordan block with eigenvalue 0. Next recall that the following standard torus    g   1        g 2   ∗ ∈ C (i = 1, 2, . . . n) | g (2) T =   i ..     .       gn

acts on the flag variety F lags(Cn ) by left multiplication. However, this T -action does not preserve the subvariety Hess(h) in general. This problem can be rectified by considering instead the action of the following

Abe, Harada, Horiguchi, and Masuda

circle subgroup S of T , which does preserve Hess(h) ([17, Lemma 5.1]): ďŁź ďŁąďŁŤ ďŁś g ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁˇ ďŁ˝ ďŁ˛ďŁŹ g 2 ďŁˇ ďŁŹ â&#x2C6;&#x2014; | g â&#x2C6;&#x2C6; C . (3) S := ďŁŹ ďŁˇ .. ďŁ´ ďŁ´ ďŁ¸ ďŁ . ďŁ´ ďŁ´ ďŁ´ ďŁ´ ďŁž ďŁł gn

(Indeed it can be checked that S â&#x2C6;&#x2019;1 N S = gN which implies that S preserves Hess(h).) Recall that the T -fixed points F lags(Cn )T of the flag variety F lags(Cn ) can be identified with the permutation group Sn on n letters. More concretely, it is straightforward to see that the T -fixed points are the set {( ew(1) â&#x160;&#x201A; ew(1) , ew(2) â&#x160;&#x201A; Âˇ Âˇ Âˇ â&#x160;&#x201A; ew(1) , ew(2) , ..., ew(n) = Cn ) | w â&#x2C6;&#x2C6; 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) â&#x2C6;Š (F lags(Cn ))T so we may view Hess(h)S as a subset of Sn .

Statement of the main theorem

In this section we state the main result of this paper. We first recall some notation and terminology. Let Ei denote the subbundle of the trivial vector bundle F lags(Cn ) Ă&#x2014; Cn over F lags(Cn ) whose fiber at a flag Vâ&#x20AC;˘ is just Vi . We denote the T -equivariant first Chern class of the line bundle Ei /Eiâ&#x2C6;&#x2019;1 by Ď&#x201E;Ë&#x153;i â&#x2C6;&#x2C6; HT2 (F lags(Cn )). Let Ci denote the one dimensional representation of T through the map T â&#x2020;&#x2019; Câ&#x2C6;&#x2014; given by diag(g1 , . . . , gn ) â&#x2020;&#x2019; gi . In addition we denote the first Chern class of the line bundle ET Ă&#x2014;T Ci over BT by ti â&#x2C6;&#x2C6; H 2 (BT ). It is well-known that the t1 , . . . , tn generate H â&#x2C6;&#x2014; (BT ) as a ring and are algebraically independent, so we may identify H â&#x2C6;&#x2014; (BT ) with the polynomial ring Q[t1 , . . . , tn ] as rings. Furthermore, it is known that HTâ&#x2C6;&#x2014; (F lags(Cn )) is generated as a ring by the elements Ď&#x201E;Ë&#x153;1 , . . . , Ď&#x201E;Ë&#x153;n , t1 , . . . , tn . Indeed, by sending xi to Ď&#x201E;Ë&#x153;i and the ti to ti we obtain that HTâ&#x2C6;&#x2014; (F lags(Cn )) is isomorphic to the quotient Q[x1 , . . . , xn , t1 , . . . , tn ]/ (ei (x1 , . . . , xn ) â&#x2C6;&#x2019; ei (t1 , . . . , tn ) | 1 â&#x2030;¤ i â&#x2030;¤ n).

Cohomology rings of Hessenberg varieties

Here the ei denote the degree-i elementary symmetric polynomials in the relevant variables. In particular, since the odd cohomology of the flag variety F lags(Cn ) vanishes, we additionally obtain the following: (4)

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

As mentioned in Section 2, in this manuscript we focus on a particular circle 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 by t ∈ H 2 (BS), where by C we mean the standard one-dimensional representation of S through the map S → C∗ given by diag(g, g 2 , . . . , g n ) → g. Analogous to the identification H ∗ (BT ) ∼ = Q[t1 , . . . , tn ], we may also ∗ identify H (BS) with Q[t] as rings. Consider the restricion homomorphism (5)

HT∗ (F ags(Cn )) → HS∗ (Hess(h)).

Let τi denote the image of τ˜i under (5). We next analyze some algebraic relations satisfied by the τi . For this purpose, we now introduce some polynomials 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 of natural numbers satisfying n ≥ i ≥ j ≥ 1. These polynomials should be visualized 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 entries in the lower-triangular part of the matrix, i.e. the part at or below the diagonal. The definition of the fi,j is inductive, beginning with the case when i = j, i.e. the two indices are equal. In this case we make the following 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 .

Abe, Harada, Horiguchi, and Masuda

Again for convenience we define fâ&#x2C6;&#x2014;,0 := 0 for any â&#x2C6;&#x2014;. Informally, we may visualize each fi,j as being associated to the lower-triangular (i, j)-th entry in an n Ă&#x2014; n matrix, as follows: ďŁś ďŁŤ 0 ÂˇÂˇÂˇ ÂˇÂˇÂˇ 0 f1,1 ďŁˇ ďŁŹ f2,1 f2,2 0 ÂˇÂˇÂˇ ďŁˇ ďŁŹ ďŁˇ ďŁŹ ďŁˇ ďŁŹ f3,1 f3,2 f3,3 . . . (9) ďŁˇ ďŁŹ ďŁˇ ďŁŹ .. ďŁ¸ ďŁ . fn,1 fn,2 Âˇ Âˇ Âˇ fn,n To make the discussion more concrete, we present an explicit example.

Example 1. Suppose n = 4. Then the fi,j have the following form. fi,i = pi (1 â&#x2030;¤ i â&#x2030;¤ 4) f2,1 = (x1 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; t)p1 f3,2 = (x1 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; t)p1 + (x2 â&#x2C6;&#x2019; x3 â&#x2C6;&#x2019; t)p2 f4,3 = (x1 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; t)p1 + (x2 â&#x2C6;&#x2019; x3 â&#x2C6;&#x2019; t)p2 + (x3 â&#x2C6;&#x2019; x4 â&#x2C6;&#x2019; t)p3 f3,1 = (x1 â&#x2C6;&#x2019; x3 â&#x2C6;&#x2019; t)(x1 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; t)p1 f4,2 = (x1 â&#x2C6;&#x2019; x3 â&#x2C6;&#x2019; t)(x1 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; t)p1 + (x2 â&#x2C6;&#x2019; x4 â&#x2C6;&#x2019; t){(x1 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; t)p1 + (x2 â&#x2C6;&#x2019; x3 â&#x2C6;&#x2019; t)p2 } f4,1 = (x1 â&#x2C6;&#x2019; x4 â&#x2C6;&#x2019; t)(x1 â&#x2C6;&#x2019; x3 â&#x2C6;&#x2019; t)(x1 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; t)p1 For general n, the polynomials fi,j for each (i, j)-th entry in the matrix (9) above can also be expressed in a closed formula in terms of certain polynomials â&#x2C6;&#x2020;i,j for i â&#x2030;Ľ j which are determined inductively, starting on the main diagonal. As for the fi,j , we think of â&#x2C6;&#x2020;i,j for i â&#x2030;Ľ j as being associated to the (i, j)-th box in an n Ă&#x2014; n matrix. In what follows, for 0 < k â&#x2030;¤ n â&#x2C6;&#x2019; 1, we refer to the lower-triangular matrix entries in the (i, j)-th spots where iâ&#x2C6;&#x2019;j = k as the k-th lower diagonal. (Equivalently, the k-th lower diagonal is the â&#x20AC;&#x153;usualâ&#x20AC;? diagonal of the lower-left (n â&#x2C6;&#x2019; k) Ă&#x2014; (n â&#x2C6;&#x2019; k) submatrix.) The usual diagonal is the 0-th lower diagonal in this terminology. We now define the â&#x2C6;&#x2020;i,j as follows. 1. First place the linear polynomial xi â&#x2C6;&#x2019; it in the i-th entry along the 0-th lower (i.e. main) diagonal, so â&#x2C6;&#x2020;i,i := xi â&#x2C6;&#x2019; it. 2. Suppose that â&#x2C6;&#x2020;i,j for the (k â&#x2C6;&#x2019; 1)-st lower diagonal have already been defined. Let (i, j) be on the k-th lower diagonal, so i â&#x2C6;&#x2019; j = k. Define j â&#x2C6;&#x2020;iâ&#x2C6;&#x2019;j+ â&#x2C6;&#x2019;1, (xj â&#x2C6;&#x2019; xi â&#x2C6;&#x2019; t). â&#x2C6;&#x2020;i,j := =1

Cohomology rings of Hessenberg varieties

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 boxes which are in the “diagonal immediately above the (i, j) box” (i.e. the boxes which are in the (k − 1)-st lower diagonal), but we omit any boxes to 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 =

∆i−j+k,k .

k=1

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 ) denote the corresponding regular nilpotent Hessenberg variety equipped with the circle S-action described above. Then the restriction map HT∗ (F ags(Cn )) → HS∗ (Hess(h)) is surjective. Moreover, there is an isomorphism of Q[t]-algebras HS∗ (Hess(h)) ∼ = Q[x1 , . . . , xn , t]/I(h) sending xi to τi and t to t and we identify H ∗ (BS) = Q[t]. Here the ideal 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. Any Hessenberg function h : {1, 2, . . . , n} → {1, 2, . . . , n} determines a subspace of the vector space M (n × n, C) of matrices as follows: an (i, j)-th entry is required to be 0 if i > h(j). If we represent a Hessenberg function h by listing its values (h(1), h(2), · · · , h(n)), then the Hessenberg subspace can be described in words as follows: the first column (starting from the left) is allowed h(1) non-zero entries (starting from the top), the second column is allowed h(2) non-zero entries, et cetera. For

Abe, Harada, Horiguchi, and Masuda

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)-th entry of the matrix (9), the ideal I(h) can be described as being “generated by the fi,j in the boxes at the bottom of each column in the Hessenberg space”. For instance, in the h = (3, 3, 4, 5, 7, 7, 7) example above, 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 case the corresponding regular nilpotent Hessenberg variety has been wellstudied and it is called a Peterson variety P etn (of type A). Our result above is a generalization of the result in [11] which gives a presentation of HS∗ (P etn ). 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 HS∗ (P etn ) ∼ = 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 of the ordinary cohomology ring. Indeed, since the odd degree cohomology groups of Hess(h) vanish [29], by setting t = 0 we obtain the ordinary

Cohomology rings of Hessenberg varieties

cohomology. Let fË&#x2021;i,j := fi,j (x, t = 0) denote the polynomials in the variables xi obtained by setting t = 0. A computation then shows that fË&#x2021;i,j =

j k=1

=j+1

(xk â&#x2C6;&#x2019; x ).

(For the case i = j we take by convention have the following.

=j+1 (xk

â&#x2C6;&#x2019; x ) = 1.) We

Corollary 3.2. Let the notation be as above. There is a ring isomorphism Ë&#x2021; H â&#x2C6;&#x2014; (Hess(h)) â&#x2C6;ź = Q[x1 , . . . , xn ]/I(h) Ë&#x2021; where I(h) := fË&#x2021;h(j),j | 1 â&#x2030;¤ j â&#x2030;¤ n .

Remark 2. Consider the special case h = (n, n, . . . , n). In this case the condition in (1) is vacuous and the associated regular nilpotent Hessenberg variety is the full flag variety F ags(Cn ). In this case we can Ë&#x2021; Ë&#x2021; n, . . . , n) = I(n, explicitly relate the generators fË&#x2021;h(j)=n,j of our ideal I(h) n r with the power sums pr (x) = pr (x1 , . . . , xn ) := k=1 xk , thus relating our presentation with the usual Borel presentation as in (4), see e.g. [13]. More explicitly, for r be an integer, 1 â&#x2030;¤ r â&#x2030;¤ n, define qr (x) = qr (x1 , . . . , xn ) :=

n+1â&#x2C6;&#x2019;r k=1

=n+2â&#x2C6;&#x2019;r

(xk â&#x2C6;&#x2019; x ).

Note that by definition qr (x) = fË&#x2021;n,n+1â&#x2C6;&#x2019;r so these are the generators of Ë&#x2021; n, . . . , n). The polynomials qr (x) and the power sums pr (x) can I(n, then be shown to satisfy the relations (12)

qr (x) =

râ&#x2C6;&#x2019;1

(â&#x2C6;&#x2019;1)i ei (xn+2â&#x2C6;&#x2019;r , . . . , xn )prâ&#x2C6;&#x2019;i (x).

i=0

Remark 3. In the usual Borel presentation of H â&#x2C6;&#x2014; (F ags(Cn )), the ideal I of relations is taken to be generated by the elementary symmetric polynomials. The power sums pr generate this ideal I when we consider the cohomology with Q coefficients, but this is not true with Z coefficients. Thus our main Theorem 3.1 does not hold with Z coefficients in the case when h = (n, n, . . . , n), suggesting that there is some subtlety in the relationship between the choice of coefficients and the choice of generators of the ideal I(h).

Abe, Harada, Horiguchi, and Masuda

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 relations fh(j),j = fh(j),j (τ1 , . . . , τn , t) = 0. The main technique of this part of the proof is (equivariant) localization, i.e. the injection (13)

HS∗ (Hess(h)) → HS∗ (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) this then implies that fh(j),j = 0 as desired. This part of the argument is rather long and requires a technical inductive argument based on a particular choice of total ordering on Hess(h)S which refines a certain natural partial order on Hessenberg functions. Once we show fh(j),j = 0 for all j, we obtain a well-defined ring homomorphism which sends xi to τi and t to t: (14)

ϕh : Q[x1 , . . . , xn , t]/(fh(j),j | 1 ≤ j ≤ n) → HS∗ (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 techniques used in e.g. [11]. The next key step in our proof of Theorem 3.1 relies on the following two key ideas: firstly, we use our knowledge of the special case where the Hessenberg function h is h = (n, n, . . . , n), for which the associated regular nilpotent Hessenberg variety is the full flag variety F ags(Cn ), and secondly, we consider localizations of the rings in question with respect to R := Q[t]\{0}. For the following, for h = (n, n, . . . , n) we let H := Hess(h = (n, n, . . . , n)) = F ags(Cn ) denote the full flag variety and let I denote the associated ideal I(n, n, . . . , n). In this case we know that the map ϕ := ϕ(n,n,...,n) is surjective since the Chern classes τi are known to generate the cohomology ring of F ags(Cn ). Since the Hilbert series of both sides are identical, we then know that ϕ is an isomorphism. The following commutative diagram is crucial for the remainder of the argument. R−1 Q[x1 , . . . , xn , t]/I  surj

R−1 ϕ

− −−∼ −− → =

R−1 HS∗ (H)  

− −−∼ −− → =

R−1 HS∗ (HS )  surj

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

Cohomology rings of Hessenberg varieties

The horizontal arrows in the right-hand square are isomorphisms by the localization theorem. Since ϕ is an isomorphism, so is R−1 ϕ. The rightmost and leftmost vertical arrows are easily seen to be surjective, implying that R−1 ϕh is also surjective. A comparison of Hilbert series shows that R−1 ϕh is an isomorphism. Finally, to complete the proof we consider the commutative diagram Q[x1 , . . . , xn , t]/I(h)  inj

−−−h−→ R−1 ϕ

HS∗ (Hess(h))  inj

h R−1 Q[x1 , . . . , xn , t]/I(h) −−− −→ R−1 HS∗ (Hess(h)) ∼

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

Open questions

We outline a sample of possible directions for future work. • In [24], Mbirika and Tymoczko suggest a possible presentation of the cohomology rings of regular nilpotent Hessenberg varieties. Using our presentation, we can show that the Mbirika-Tymoczko ring is not isomorphic to H ∗ (Hess(h)) in the special case of Peterson varieties for n − 1 ≥ 2, i.e. when h(i) = i + 1, 1 ≤ i < n and n ≥ 3. (However, they do have the same Betti numbers.) In the case n = 4, we have also checked explicitly for the Hessenberg 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 interest to understand the relationship between the two rings in some generality. • In [15], the last three authors give a presentation of the (equivariant) cohomology rings of Peterson varieties for general Lie type in a pleasant uniform way, using entries in the Cartan matrix. It would be interesting to give a similar uniform description of the cohomology rings of regular nilpotent Hessenberg varieties for all Lie types.

Abe, Harada, Horiguchi, and Masuda

• In the case of the Peterson variety (in type A), a basis for the S-equivariant cohomology ring was found by the second author and Tymoczko in [16]. In the general regular nilpotent case, and following ideas of the second author and Tymoczko [17], it would be of interest to construct similar additive bases for HS∗ (Hess(h)). Additive bases with suitable geometric or combinatorial properties could lead to an interesting ‘Schubert calculus’ on regular nilpotent Hessenberg varieties. • Fix a Hessenberg function h and let S : Cn → Cn be a regular semisimple linear operator, i.e. a diagonalizable operator with distinct eigenvalues. There is a natural Weyl group action on the cohomology ring H ∗ (Hess(S, h)) of the regular semisimple Hessenberg variety corresponding to h (cf. for instance [30, p. 381] and also [28]). Let H ∗ (Hess(S, h))W denote the ring of W -invariants where W denotes the Weyl group. It turns out that there exists a surjective ring homomorphism H ∗ (Hess(N, h)) → H ∗ (Hess(S, h))W which is an isomorphism in the special case of the Peterson variety. (Historically this line of thought goes back to Klyachko’s 1985 paper [21].) In an ongoing project, we are investigating properties of this ring homomorphism for general Hessenberg functions h. Hiraku Abe Advanced Mathematical Institute, Osaka City University, Sumiyoshi-ku, Japan, Osaka 558-8585, hirakuabe@globe.ocn.ne.jp

Megumi Harada Department of Mathematics and Statistics, McMaster University 1280 Main Street West, Hamilton, Ontario L8S4K1, Canada Megumi.Harada@math.mcmaster.ca

Tatsuya Horiguchi Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan d13saR0z06@st.osaka-cu.ac.jp

Mikiya Masuda Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan masuda@sci.osaka-cu.ac.jp

References [1] Abe H., Harada M., Horiguchi T., Masuda M.; The equivariant cohomology rings of regular Hessenberg varieties in Lie type A, in preparation.

Cohomology rings of Hessenberg varieties

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Abe, Harada, Horiguchi, and Masuda

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