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THE EQUIVARIANT K-THEORY OF COMMUTING 2-TUPLES IN SU(2)

by

Trefor Bazett

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2016 by Trefor Bazett

Abstract

The Equivariant K-Theory of Commuting 2-tuples in SU(2)

Trefor BazettDoctor of Philosophy

Graduate Department of MathematicsUniversity of Toronto

2016

In this thesis, we study the space of commuting n-tuples inG = SU(2),Hom(Zn, SU(2)).

We describe this space geometrically via providing an explicit G-CW complex structure,

an equivariant analog of familiar CW- complexes. For the n = 2 case, this geometric de-

scription allows us to compute various cohomology theories of this space, in particular

the G-equivariant K-Theory K∗G(Hom(Z2, SU(2))), both as an R(SU(2))-module and

as an R(SU(2))-algebra, where R(SU(2)) denotes the representation ring of SU(2).

This space is of particular interest in the study of quasi-Hamiltonian spaces M, which

come equipped with a moment map that takes values in the Lie group, φ : M → G. For

the specific example M = G × G with moment map given by the commutator map,

φ−1(e) = Hom(Z2, G). Finite dimensional quasi-Hamiltonian spaces have a bijective

correspondence with certain infinite dimensional Hamiltonian spaces, and we compute

relevant components of this larger picture in addition to φ−1(e) = Hom(Z2, SU(2)) for

this example.

ii

Acknowledgements

I wish to thank my joint supervisors Lisa Jeffrey and Paul Selick for their support,patience and advice. I cannot sufficiently express my gratitude. I also wish to thankmy committee members Dror Bar-Natan and Eckhard Meinrenken.

I wish to thank Alfonso Gracia-Saz and Sean Uppal for their passion and commit-ment to teaching mathematics; they have been invaluable at helping me develop as ateacher.

My family has been an inexhaustible well of support and encouragement for myentire life. In particular, I must thank my brother Mark for keeping me sane andgrounded. And last but could never be least, my amazing wife Amy, who has madeeverything in my life better for the last fifteen years.

iii

Contents

Contents iv

1 Introduction 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Classical Results and Context 4

3 Preliminaries 53.1 G-CW Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Equivariant K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 On SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 G-CW Complex Calculations 114.1 G-CW structures for various G-spaces . . . . . . . . . . . . . . . . . . . . 114.2 G-CW structure for Hom(Z2, G) . . . . . . . . . . . . . . . . . . . . . . . . 164.3 G-CW structure for Hom(Zn, G) . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Equivariant K-theory computations 225.1 K∗G(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Module structure of K∗G(Hom(Z2, G)) . . . . . . . . . . . . . . . . . . . . 24

5.2.1 The 1-skeleton of Hom(Z2, G) . . . . . . . . . . . . . . . . . . . . . 245.2.2 The 2-skeleton of Hom(Z2, G) . . . . . . . . . . . . . . . . . . . . . 26

5.3 Other cohomology theories of Hom(Z2, G) . . . . . . . . . . . . . . . . . 305.3.1 K∗T (Hom(Z2, G)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3.2 H∗(Hom(Z2, G);Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3.3 H∗T (Hom(Z2, G);Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3.4 H∗G(Hom(Z2, G);Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

iv

5.4 Algebra structure for K∗G(Hom(Z2, G)) . . . . . . . . . . . . . . . . . . . . 375.5 Module Structure of various other spaces . . . . . . . . . . . . . . . . . . 395.6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Quasi-Hamiltonian Systems 446.1 Quasi-Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Kirwan Surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.3 Relating to future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Bibliography 50

v

Chapter 1

Introduction

1.1 Overview

In this thesis, our principal objective is the study of the commuting n-tuples inSU(2), (g1, · · · , gn) ∈ (SU(2))n | gigj = gjgi ∀i, j. Since a homomorphism is de-termined by the mappings of the generators, this is the space Hom(Zn, SU(2)), as itwill be henceforth denoted in this thesis. This is an SU(2)-space where SU(2) acts onHom(Zn, SU(2)) by conjugation on each component.

A standard methodology to compute homology and cohomology groups associ-ated to topological spaces is to establish a CW-complex structure on the spaces. Theappropriate equivariant analog of the familiar CW-complexes is the notion of a G-CWcomplex developed independently by [Mat] and [Ill1]. From the skeleta filtration Xn ofa G-CW space X , we can consider the corresponding long exact sequences on variouscohomology theories corresponding to Xn → Xn+1 → Xn+1/Xn, much as we wouldfor CW-complexes.

A fundamental result in equivariant K-theory is Bott periodicity, which we de-scribe in more detail following [Seg] in section 3.2. Bott periodicity is the fact thatKqG(X) ∼= Kq+2

G (X) for a compact G-space X . Thus all even KqG(X) are equal to each

other, as are all odd KqG(X) equal to each other. One consequence of this is that on

equivariant K-theory the long exact sequence induced by Xn → Xn+1 → Xn+1/Xn

yields a six term cyclical exact sequence. As quotients Xn+1/Xn can be well described,these six term exact sequences provide a methodology to compute the unknown termsin the sequence. Thus by establishing an explicit G-CW complex for a space, this geo-

1

CHAPTER 1. INTRODUCTION 2

metric description gives us considerable power in computing equivariant cohomologytheories.

We apply this methodology to a range of G-spaces, but our principal computation isto establish a G-CW complex structure for G = SU(2) on Hom(Zn, G). From this, in thecase of n = 2, we go on to use the long exact sequences corresponding to the skeletalfiltration to compute X = Hom(Z2, SU(2)) on various cohomology theories such asK∗G(X), ordinary H∗(X,Z), K∗T (X), H∗G(X), and H∗T (X) where T is the maximal torusof diagonal elements in SU(2). The former two, as modules, have previously beencomputed by [AG] and [BJS], respectively, for X = Hom(Z2, G).

Beyond providing an alternate and pleasingly geometric computation that mo-tivates future work using this G-CW approach, our results extend the literature inthree ways. Firstly, our geometric G-CW decompositions allow us to also establish themultiplicative structure, thus computing K∗G(Hom(Z2, G)) as an R(G)-algebra, not justas an R(G)-module. Additionally, our methods allow us to compute the additive andmultiplicative structure on the other cohomology theories mentioned above. Finally,we equip the commuting n-tuples in SU(2) with a G-CW structure that generalizes the2-tuples case, from which the various cohomology theories mentioned above could becomputed analogously. However, we lack a closed form solution for the general n caseand note that computations quickly become cumbersome. As such, only the n = 2 casehas been explicitly written out in this thesis.

One motivation for this study ofHom(Z2, G) is the quasi-Hamiltonian systems intro-duced by [AMM]. Indeed, the commutator map from G×G→ G provides an exampleof such a quasi-Hamiltonian system with Hom(Z2, G)/G as the symplectic quotientwhere G acts regularly on Hom(Z2, G). The space Hom(Z2, G)/G is of importance insymplectic geometry. As described in [Jef], the moduli space M = Hom(π1(Σ), G)/G

can by identified via the holonomy map as gauge equivalence classes of flat connectionson a trivial principal G bundle over a closed, oriented 2-manifold Σ of genus g. In thegenus 1 case we get π1(Σ) = Z2. See [Jef2] for background. There are a range of otherspaces related to the larger pictures of quasi-Hamiltonian systems and their bijectivecorrespondence with particular Hamiltonian LG systems, and for this example withG = SU(2) we additionally compute K∗G(G) and K∗G(G×G) as modules. Finally, werelate them to larger questions in quasi-Hamiltonian systems.

CHAPTER 1. INTRODUCTION 3

1.2 Outline of the Thesis

This thesis is organized by chapter as follows:

• In chapter 2 we outline various results in the literature relevant to this thesis.In particularly, we discuss known computations on equivariant K-theory andcohomology for several relevant spaces and where our results match or extendthem.

• In chapter 3 we provide an introduction into the objects and tools that we willuse in this thesis. In particular, we define G-CW complexes in section 3.1 anddiscuss a few important facts. In section 3.2 we introduce Equivariant K-theory,following [Seg]; in particular, we cite various propositions that will be criticalfor our computations on equivariant K-theory. In section 3.3 we provide a briefoverview of equivariant cohomology. Finally, in section 3.4 we discuss variousfacts about the Lie group SU(2).

Henceforth, all of the G-spaces under consideration in this thesis are for G =

SU(2).

• In chapter 4 we establish G-CW complex structures on all the various spaces weare interested in computing in this thesis. In particular, we study G, G×G andHom(Zn, G). Of critical importance is describing the attaching maps in our G-CWstructure sufficiently clearly so that we can compute equivariant K-theory fromthem.

• In chapter 5, we compute the equivariantK-theory of our range of spaces asR(G)-modules. Note that while we have a G-CW structure on commuting n-tuples, weonly go on to compute equivariant K-theory in the n = 2 case. In section 5.3 wethen study other equivariant cohomology theories of Hom(Z2, G). Additionally,we make explicit the R(G)-algebra structure for K∗G(Hom(Z2, G)) in section 5.4.In section 5.5 the module structure of various other spaces is computed. In 5.6,we discuss some potential for future work.

• In chapter 6, we discuss the relationship between Hom(Z2, G) and the otherspaces we compute here and the quasi-Hamiltonian systems that originally mo-tivated this study. We conduct a review of the literature on analogs of Kirwansurjectivity, and conclude the thesis with suggestions for further work along thesethemes.

Chapter 2

Classical Results and Context

In this chapter we summarize prior computations of various cohomology theories inthe literature that overlap with our results and describe where we match the literature.

K∗G(G) was first computed in 2000 by [BZ] for simply connected Lie groups Gacting on themselves by conjugation as the algebra of Grothendieck differentials on therepresentation ring, K∗G(G) ∼=G Ω∗R(G)/Z. We will match, as expected, this result in thecase of G = SU(2) in section 5.1.

The space Hom(π,G) for π a finitely generated discrete group, and G a Lie group,was studied by [AC]. In particular, for π a free Abelian group of rank equal to n, coho-mology groups were explicitly computed in the n = 2 and n = 3 cases for G = SU(2).[BJS] further study Hom(Zn, SU(2)), computing the integral cohomology groups forall positive n, in agreement with [AC2], an erratum to the original [AC] in the n = 3

case. Our results in this thesis match the n = 2 case on integral cohomology as noted insection 5.4.

[AG] provide a computation for K∗SU(2)(Hom(Z2, G)) as an R(G)-module. This re-sult uses the Atiyah-Hirzebruch-Segal Spectral Sequence and Segal’s localization to re-duce the problem to computing to studying the fixed point set H∗(Hom(Z2, G)

T;R(G)).

We reproduce this result in section 5.2 through computing K∗SU(2)(Hom(Z2, G)) viaproviding a G-CW structure on Hom(Z2, G).

Finally, [HJS1] and [HJS2] are twin papers that compute K∗G(ΩG) for G = SU(2)

acting on itself by conjugation as a module and as an algebra.

4

Chapter 3

Preliminaries

This section contains a brief overview of G-CW complexes, equivariant K-theory and afew facts on SU(2) that will be used in our computations. It is by no means intended tobe exhaustive on these subjects.

3.1 G-CW Complexes

An equivariant analog of the familiar CW-Complexes, referred to as G-CW complexesin this thesis, was developed independently by [Mat] and [Ill1]. Let X be a G-space fora compact Lie group G; that is, X is a Hausdorff topological space with a left action ofG on X . Following [May],

Definition 1. For each k, let Hk denote a closed subgroup of G. A G-CW complex X is aunion of G-spaces Xn where X0 is a disjoint union of orbits G/Hk, referred to as 0-cells, andXn+1 is determined inductively by attaching (n+1)-cells (G/Hk)×Dn+1 to Xn via attachingG-maps σk : (G/Hk)× Sn → Xn

That is, Xn+1 is determined by the pushout of the following diagram:

tk(G/Hk)× Sntkσk //

Xn

tk(G/Hk)×Dn+1 // Xn+1.

(3.1)

A point of the form (eHk, t) ∈ (G/Hk) × Sn is fixed under the Hk action, and somaps via σk into the fixed point set of Hk, denoted (Xn)Hk . Thus, an attaching G-mapσk is determined by its restriction Sn → (Xn)Hk .

5

CHAPTER 3. PRELIMINARIES 6

The degree of a cell of the form (G/H)×Dn cell is n, and such cells are referred toas n-cells. The dimension of such a cell is (Dim(G/H) + n). The space Xn, formed byattaching cells of degree up to and including n, is referred to as the n-skeleton of X .One interesting characteristic is that lower degree cells can sometimes have higherdimensions than higher degree cells. A G-CW subcomplex Y of a G-CW complex X isa subcollection of cells from the G-CW complex for X that is itself a G-CW complex.

Given two G-CW complexes X and Y , one can form a (G × G)-CW complex onX × Y by taking all products of cells

(G/H ×Dn)× (G/K ×Dm) ∼= ((G×G)/(H ×K))×Dn+m.

If a G-CW complex for each (G × G)/(H ×K) can be determined, then every ((G ×G)/(H × K)) × Dn+m cell can be written as a G-cell and thus a G-CW complex forX × Y can be determined.

The following lets us interpret the quotients of skeleta from our G-CW complexas suspensions. For E a G-bundle, let T (E) denote the Thom Space. In particular,we write T ((G/Hk) × Dn) to refer to the Thom space of the product G-bundle withbase G/Hk with the usual action and trivial action on Dn. We will also let S(X)

denote the unreduced suspension, a G-space where g acts on an element in X × I viag(x, t) = (gx, t).

Proposition 1. The n-skeleton Xn of a G-CW complex X satisfies

Xn/Xn−1 ∼=G

∨k

T ((G/Hk)×Dn) ∼=G

∨k

(Sn ∧ ((G/Hk) t pt)) 'G∨k

(Sn((G/Hk) t pt))

with k indexing all n-cells of our G-CW complex.

Proof. The first identification follows from the observation that an attaching map fora cell of the form G/Hk × Dn glues G/Hk × Sn into the (n − 1)-skeleton resulting inthe quotient of the product disk G-bundle by the product sphere G-bundle, henceT ((G/Hk) × Dn). The second identification follows from the more general fact thatfor a vector bundle ξ, the fibrewise direct sum T (ξ ⊕ Rn) ∼=G S

n ∧ T (ξ) applied to thecase of ξ being the zero dimensional G-bundle over base G/Hk which has Thom spaceT (ξ) = G/Hk t pt.

We now quote Theorem J on page 368 of [Mat2] which shows that the inclusion


of a G-subskeleton is a G-cofibration. That is, that it satisfies the homotopy extensionproperty.

Proposition 2 (Homotopy Extension Property). Let f0 : X → Z be a G-map of X into anarbitrary G-space Z. Let gi : Y → Z be a G-homotopy of g0 = f0|Y where Y is a G-subcomplexof X . Then there is a G-homotopy ft : X → Z such that ft|Y = gt.

The following standard property of cofibrations is also valid in the equivariant case.Standard proofs of the nonequivariant case can be found in [May2] and [Sel]. The proofof the extension from the nonequivariant to the equivariant case for the dual theoremregarding G-fibrations is Theorem A.3 of [HS], and the proof for G-cofibrations followsanalogously.

Proposition 3. Let j : A→ X be a G-cofibration for a G-space A with A closed in the G-spaceX . Let f, g : A→ Y be G-homotopic. Then the pushouts of j with f and g are G-homotopic.

In particular, if A is G-contractible, we get that X 'G X/A.

3.2 Equivariant K-theory

We will consider the case of G being a compact Lie group and X a compact G-space.The group KG(X) is defined as the Grothendieck group of isomorphism classes of G-equivariant complex vector bundles over X . Throughout this section, we follow [Seg]’sapproach outlining the basic definitions and propositions.

Consider the map j : X → pt. We define the reduced equivariant K-theory on Xas KG(X) = coker(j∗ : KG(pt)→ KG(X)).

Equivariant K-theory generalizes from two trivial cases. Firstly, when X = pt,

KG(pt) = R(G) (3.2)

where R(G) is the representation ring of G. Indeed, a G-bundle over a point is arepresentation of G. Secondly, when G is trivial, KG(X) ∼= K(X), ordinary K-theory.The map X → pt induces a map KG(pt) ∼= R(G)→ KG(X) which makes equivariantK-theory an R(G)-algebra.

Further, there are two extreme cases of the action of G on X . Firstly, if G actstrivially on X , then any vector bundle can be thought of as an equivariant vector


bundle, giving us a map K(X) → KG(X). Together with the ring homomorphismKG(pt) ∼= R(G)→ KG(X) induced by the map X → pt , we get a ring homomorphismµ : R(G)⊗K(X)→ KG(X).

Proposition 4. If G acts trivially on X , the map µ described above is an isomorphism.

Secondly, for G acting freely:

Proposition 5. If G acts freely on X then the quotient map π : X → X/G induces anisomorphism

π∗ : K(X/G)→ KG(X) (3.3)

For a right G-space X , and a left G-space Y , let X ×G Y := (X × Y )/G where X × Yadmits the G-action defined by g · (x, y) = (xg−1, gy).

Proposition 6. For H a closed subgroup of G, and an H-space X , we can form the compactG-space (G ×H X)/H := (G ×H X). The map j : H → G induces an isomorphismj∗ : KG(G×H X)→ KH(X). In particular, we have KG(G/H) ∼= KH(pt) ∼= R(H).

Perhaps the most important theorem in equivariant K-theory is the Thom isomor-phism, presented here in the form of Theorem 3.1 of [Gre]. An alternate reference isTheorem 6.1.4 in [AS].

Proposition 7 (Thom Isomorphism). For E a G-bundle over a compact G-space X, there isa natural Thom isomophism

KG(X)∼=G−−→ KG(E)

For q ≥ 0, we can define

KqG(X) = KG(Sq ∧ (X t pt)). (3.4)

A consequence of the Thom isomorphism gives us Bott periodicity which is anisomorphism Kq

G(X) ∼= Kq+2G (X), as described in Theorem 3.2 of [Gre] or Theorem 2.7.1

of [AS]. We thus only need to consider K0G(X) and K1

G(X) as KqG(X) is the same for all

even q, and the same for all odd q. We will therefore refer to equivariant K-theory asbeing Z/2-graded.

Another consequence of Bott periodicity is that the typical long exact sequencebecomes a cyclical six term sequence that will be critical for many of our computations:


Proposition 8 (Exact Sequences). Let A → X be a G-cofibration with A closed. Then thefollowing six term sequence is exact.

K0G(A)

K0G(X)oo K0

G(X/A)oo

K1G(X/A) // K1

G(X) // K1G(A)

OO(3.5)

In particular, we will apply these exact sequences in the case where A is a G-subskeleton as proposition 2 of section 3.1 asserts that the inclusion of a G-subskeletonis a G-cofibration.

3.3 Equivariant cohomology

While we are largely interested in equivariant K-theory, we will also record various re-sults on equivariant cohomology as well. We thus recall some essential facts regardingequivarariant cohomology in this section.

Let X be a G-space for a topological group G. We can construct the homotopyquotient EG×GX := (EG×X)/G where EG→ BG := EG/G is a universal principalG-bundle. The spaceEG×X is homotopy equivalent toX , but has the advantage thatGacts freely on it. Further, the homotopy quotient is well defined as the classifying spaceBG is unique up to homotopy. We then define the equivariant integral cohomologyring as

H∗G(X;Z) := H∗(EG×G X;Z).

We will stop making the Z explicit from now on.

If G acts freely on X , then the fibres of EG ×G X → X/G are just EG which isG-contractible, so EG ×G X ∼=G X/G. This gives that H∗G(X) = H∗(X/G). For Xa point, we get that EG ×G pt = EG/G which was defined to be BG. Much likeequivariant K-theory, the map X → pt thus induces an H∗G(pt)-algebra structure onH∗G(X).

Now let G be a compact Lie group, and T ⊂ G be a maximal torus. Finally, let


W := N(T )/T be the corresponding Weyl group. Proposition 1 of [Bri] shows that

H∗G(pt) = H∗T (pt)W .

We will consider our specific case of G = SU(2) and T = S1 in section 5.4.

3.4 On SU(2)

Let G = SU(2) which consists of all 2× 2 unitary matrices with determinant 1. Topo-logically, SU(2) is homeomorphic to S3. All maximal tori in SU(2) are conjugate to

T =

(eiθ 0

0 e−iθ

): θ ∈ [0, 2π)

, which is homeomorphic to S1, and any two max-

imal tori intersect at ±e, where e denotes the identity 2 × 2 matrix. It is helpful tovisualize e and -e as the poles on S3. Two elements in SU(2) commute precisely if theylie on a common maximal torus. Finally, G/T is homeomorphic to S2.

Elements in G\±e can be diagonalized as g = hth−1 with

t ∈ T+ :=

(t =

eiθ 0

0 e−iθ

): θ ∈ (0, π)

, where h is uniquely determined up to

right multiplication by elements in T . Alternatively, such elements can be diagonalized

as g = hth−1 with t ∈ T− :=

(t =

eiθ 0

0 e−iθ

): θ ∈ (π, 2π)

, where h is uniquely

determined up to right multiplication by elements in T . G has a G-projection G→ G/T

and for G\±e, denoting G set minus the poles, a projection G\±e → T+. We willuse these facts extensively when describing G-CW complexes on SU(2) and spacesrelated to SU(2) in chapter 4.

Let w =

(0 1

−1 0

), then the Weyl group W := N(T )/T is given by W = T,wT

and has a W action on T that takes t 7→ t−1. The element w acts on G/T correspondingto the antipodal map on S2.

For G = SU(2) and T the maximal torus, R(T ) ∼= Z[b, b−1], the representation ringof T with b the canonical one dimensional representation of T . Also, R(G) ∼= Z[v], therepresentation ring of G with v the standard two dimensional representation of G. Thenontrivial action of the Weyl group is given by b 7→ b−1 and the restriction of v to T isgiven by b⊕ b−1. The ring R(T ), as a group, is R(G)⊕R(G).

Chapter 4

G-CW Complex Calculations

Henceforth, let G = SU(2), acting on itself by conjugation. In this chapter we identifyG-CW structures on several G-spaces, in particular for Hom(Zn, G). We begin with themore elementary examples to develop our intuition. Then, in sections 4.2 and 4.3, wediscuss commuting 2-tuples and n-tuples, respectively. These G-CW structures will beused to compute the equivariant K-theory of these spaces in chapter 5.

A list of theG-CW complexes computed in this section is tabulated below, excludingHom(Zn, G):

Table 4.1: G-CW ComplexesSpace CellsG t2(G/G)×D0 , (G/T )×D1

(G/T )× (G/T ) t2(G/T )×D0 , (G/∗)×D1

G×G t4(G/G)×D0 , t4(G/T )×D1 , t2(G/T )×D2, (G/∗)×D3

G ∨G t3(G/G)×D0 , t2(G/T )×D1

Hom(Z2, G) t4(G/G)×D0 , t4(G/T )×D1 , t2G/T ×D2

((G×G)\Hom(Z2, G))+ G/G×D0 , G/(Z/2)×D3

4.1 G-CW structures for various G-spaces

Example 1. G = SU(2)

The group SU(2) has two G-fixed points under the conjugation action, ±e. We thuslet the zero skeleton X0 consists of two 0-cells of the form (G/G)×D0 corresponding

11

CHAPTER 4. G-CW COMPLEX CALCULATIONS 12

to the two points ±e ∈ SU(2). Topologically, we view this as the poles of SU(2), whichis topologically S3.

Recall from page 9 of section 3.4, that elements in G\±e can be diagonalizedas g = hth−1 with t ∈ T+ := t ∈ T | 0 < θ < π where h is uniquely determinedup to right multiplication by elements in T. As described in section 3.4, G\±e thushas projections to G/T and to T+. Thus, we let X1 consist of a 1-cell of the form(G/T ) ×D1 attached to X0 via the attaching map that collapses each of the two con-nected components of G/T × S0 to the two 0-cells, respectively. Identifying D1 withT+, described earlier, we have a G-homeomorphism from (G/T )×D1 to G\±e givenby (gT, t) 7→ gtg−1 where G acts on (G/T )×D1 by left multiplication on the first factorand trivially on the second. Note that we had two choices for this diagonalization,depending on whether we assert that t ∈ T+ or T−. We chose the former.

Topologically, one can think of this as the construction of S3 by taking S2 × D1

and collapsing the ends S2 × ∂D1 to two points. Another way to think of this spaceis that SU(2) is the G-space S(G/T ) where the G action on the suspension satisfiesh · (gT, t) = (hgT, t).

Example 2. G×G (as a (G×G)-CW complex)

For G×G, we begin by considering the collection of cells formed by taking formalproducts of cells in G. This results in four 0-cells (G/G)× (G/G)×D0×D0 , four 1-cells(G/T )× (G/G)×D1 ×D0 and one 2-cell (G/T )× (G/T )×D1 ×D1.

Our larger goal, however, is to rewrite this as a G-CW complex, not a G×G-CWcomplex. The zero and one degree cells in the resulting (G×G)-CW complex can beimmediately rewritten as four 0-cells (G/G)×D0 and four 1-cells (G/T )×D1 to formthe 1-skeleton of a G-CW complex. The only nontrivial (G × G) cell to rewrite as aG-cell, to be done shortly, is the 2-cell.

The 1-skeleton for G×G, denoted (G×G)1, thus consists of a hollow square wherethe four vertices represent the four 0-cells consisting of pairs (±e,±e), and the foursegments represent the four 1-cells, the pairs with ±e in one factor and any elementsfrom G/T in the other.


In the following diagram, representing (G×G)1, a non vertex point on the squarerepresents a copy of G/T. The space represented then consists of four copies of thesuspension of G/T , topologically S3, labeled A,B,C,D, and glued together as per thediagram. Note that this can be visualized as S2 × S1 that has been ’pinched’ at the fourvertices.

Diagram 1. (G×G)1 (e,-e)(e,e)

(-e,-e)(-e,e)

B

D

CA

ss

ssAttaching G-maps for the 1-skeleton are determined as per the diagram, where for

a specific 1-cell (G/T )×D1, (G/T )× ∂D1 attaches to the two adjacent vertices inducedby the G-map from G/T → ∗. For instance, the 1-cell consisting of pairs (e, g),∀g ∈ Gis attached to the vertices (e, e) and (e,−e). Each of the four segments in Diagram 1can be thought of as a copy of G. Note that for each such segment, as in Example 1,we have two choices of diagonalization to use in the homeomorphism from G/T ×D1

to the G \ ±e. We choose to diagonalize g = hth−1 with t ∈ T+. This choice willbecome relevant when we aim to attach higher degree cells when discussing the G-CWcomplex for Hom(Z2, G) in section 4.2.

Finally, we have the top 2-cell (G/T × G/T ) × D2. Here the attaching map σ :

(G/T )× (G/T )×∂D2 → (G×G)1 has the ∂D2 = S1 factor identified with the boundarysquare of Diagram 1. At each of the four vertices, the left factors have the collapsingmap G/T ×G/T → ∗. Along the A and C segments, we have the projection map ontothe first factorG/T ×G/T → G/T . Along the B and D segments, we have the projectionmap onto the first second G/T ×G/T → G/T .

Thus far we have managed to rewrite the 0-skeleton and 1-skeleton as G-CWcomplexes. However, we still have a top 2-cell G/T ×G/T ×D2 that is not written as aG-CW complex. We thus turn to studying the space G/T ×G/T .

Example 3. G/T ×G/T

G/T × G/T is a G-space with the diagonal action, and we aim to describe it as aG-CW complex.


Let τ denote the tangent bundle of G/T , and let ε denote the trivial complex linebundle over G/T . Lemma 3.1 of [HJS2] provides a G-equivariant homeomorphismΘ : P(τ ⊕ ε)→ (G/T )× (G/T ). Here, the diagonal elements are given by the subspaceP(τ) ∼=G G/T .

We describe a G-CW complex on P(τ ⊕ ε) as follows, noting that the fibre of thisbundle is S2. There is a single 0-cell of the form G/T ×D0. These map to the pointsat infinity in P(τ ⊕ ε), which, under Θ, become the diagonal elements. Describing thenon-diagonal elements are a single 2-cell of the form G/T × D2 with attaching mapG/T × ∂D2 = G/T × S1 → G/T ×D0 induced by collapsing S1 to a point. On the righthand factors, this is the construction of the sphere S2 by attaching the boundary of D2,which is homeomorphic to the complex plane, to a point.

Example 4. G×G (now as a G-CW Complex)

The 0- and 1-skeletons of G × G, originally written as (G × G)-CW cells, wererewritten as G-CW cells in Example 2.

Now, however, we can use the result of Example 3 to rewrite the top 2-cell (G×G)×D2

as a G-CW complex. As such in addition to the four 0-cells and four 1-cells describedpreviously, we also have one (G/T × D0) × D2 ∼=G G/T × D2 2-cell and one 4-cell(G/T × D2) × D2 ∼=G G/T × D4. Here, the attaching map for the 2-cell (recall wethought of these as the diagonal elements) is to attach G/T × ∂D2. Here the ∂D2 = S1

wraps around the boundary square of Diagram 1, while the G/T maps by the identitymap, except for the corners where it collapses to a point.

For the 4-cell, ∂D4 = (∂D2 ×D2)⋃

(D2 × ∂D2). For the G/T × S1 ×D2 component,the middle factor corresponds to the same ∂D2 = S1 collapsing to a point to form asphere, as when we considered (G/T )× (G/T ). Thus this component is attached bycollapsing to G/T × pt × D2 and identifying this with the 2-cell G/T × D2 thoughtof as the diagonal elements. For the G/T × D2 × S1 component, we use the the G-homeomorphism of Example 3 to identify this with a point in G/T ×G/T × S1 wherethe S1 is thought of as the boundary square of Diagram 1. The attaching map ontothe four 1-cells of the form G/T ×D1 in the 1-skeleton is thus projection from the firstfactor along the A and C segments of S1 and projection from the second factor alongthe B and D segments. Finally, all of G/T ×G/T collapses to a point at the four verticesin S1.


Example 5. G ∨G

For G ∨G, with the wedge attaching the North pole of the first copy to the Southpole of the second copy, X0 consists of three 0-cells (G/G)×D0, labeled S1, N1 = S2, N2,and two 1-cells (G/T )×D1. The attaching maps for one of the two 1-cells, thought ofas the bottom copy of SU(2), collapse to a point the G/T factor while for the secondfactor the ∂D1 = S0 endpoints are attached to S1 and N1 = S2. For the second 1-cell,thought of as the top copy of SU(2), the G/T factor likewise collapses to a point whilethe endpoints ∂D1 = S0 are attached to N1 = S2 and N2.

Example 6. Non-commuting 2-tuples

In this example we reinterpret work following the presentation of page 25, Proposi-tion 6.11 of [AG] in our G-CW language, and in a way that makes for slightly easiercomputations. We will give a G-CW structure on the one point compactification of thenon-commuting 2-tuples.

Following [AG], set Y := (G×G)\Hom(Z2, G). For the commutator map φ : G×G→ G,let φ| denote the restriction of φ to the noncommuting 2-tuples, φ| : Y → G\1. As perProposition 4.7 of [AC], this is a locally trivial G-fibre bundle with fibre

F := φ−1(−1) = (x1, x2, ) ∈ G×G | [x1, x2] = −1,

and base su2, the Lie algebra of SU(2) with the adjoint representation. As SU2 isSU(2)-contractible we get a homotopy equivalence Y 'G F × su2 'G F × D3 byidentifying the Lie algebra of SU(2) with the Lie algebra R3. Following from [AG],F ∼=G G/Z(G) ∼=G G/(Z/2) where G acts on G/(Z/2) by left translation.

Now consider the one-point compactification Y + which identifies F × ∂D3 witha point. We can thus equip Y + with a G-CW structure via a single point, G/G × D0

as the 0-skeleton, and then a 3-cell G/(Z/2)×D3 identified by the trivial map on bothfactors G/(Z/2)× ∂D3 → G/G×D0.


4.2 G-CW structure for Hom(Z2, G)

We build our intuition first through an explicit description of commuting n-tuplesin the n = 2 case (which is the only case we will go on to compute the equivariantK-theory of) before describing the general n-tuple case in the next section. Considercommuting 2-tuples in G,

Hom(Z2, G) = (g1, g2) ∈ G×G | g1g2g−11 = g2.

Elements in G commute if they are on the same maximal torus; all maximal tori areconjugate to each other and intersect at the two poles, ±e. Let the 0-skeleton consist ofpairs (±e,±e), described by four 0-cells (G/G)×D0. Let the 1-skeleton consist of pairs(±e, g) or (g,±e) for g ∈ G, described by four 1-cells (G/G)×D1. The attaching mapsall work precisely as in Diagram 1 and thus (Hom(Z2, G))1 = (G×G)1.

For a pair (g1, g2) ∈ Hom(Z2, G)\(Hom(Z2, G))1, recall from section 3.4 that we hadtwo options (up to right multiplication by elements of T) for how to diagonalize g1.The action of the Weyl group alternates between these presentations by acting on G/Tby the antipodal map and on the maximal torus T by taking t → t−1. We choose todiagonalize as g1 = h1t1h

−11 for h1 ∈ G/T and t1 ∈ T+. Aiming to determine a cell

of the form (G/T ) × D2, now that a choice in G/T has been fixed by our choice fordiagonalizing g1, for g2 we must now use g2 = h1t2h

−11 where t2 may be in either T− or

T+. That is,

Hom(Z2, G)\(Hom(Z2, G))1 ∼= (G/T × (T\±e)2)/W ∼= ((G/T )× t4D2)/W.

The W action is needed because otherwise we would be double counting given ourtwo ways to diagonalize; the nontrivial element of the Weyl group alternates betweenthese presentations. Applying W,

Hom(Z2, G)\(Hom(Z2, G))1 ∼= (G/T )× t2D2.

We thus get two (G/T )×D2 2-cells where the first copy refers to pairs where t1, t2 areboth in T+ and the second copy refers to pairs where t1 ∈ T+ but t2 ∈ T−1. We canthink of the two copies as representing pairs that are on the same side of the maximaltorus, and on opposite sides, respectively.


To explicitly define our G-homeomorphism

σ : (G/T )× t2D2 → Hom(Z2, G)\(Hom(Z2, G))1

we first identify ((G/T )×t2D2) with ((G/T )× T+ × T+) t ((G/T )× T+ × T−) where

t ∈ T− if t−1 ∈ T+. Our maps should be compatible with our choice of writing ele-ments in the 1-skeleton in terms of T+. For the first connected component, elementsare already written this way so our G-map takes (gT, t1, t2) → (gt1g

−1, gt2g−1). For

the second connected component, the third factor is written in terms of T− and soour G-map becomes (gT, t1, t2) → (gt1g

−1, (w · g)t−12 (w · g)−1). As such, the attaching

map for the first connected component (G/T )× ∂D2 → (Hom(Z2, G))1 is induced bythe identity map on G/T and the identification of ∂D2 with the boundary square inDiagram 1. The attaching map for the second connected component is induced by thesame map along the boundary square but has the identity map on G/T along the Aand C segments of ∂D2 and the antipodal map on G/T along the segments B and D.

Therefore our G-CW complex for Hom(Z2, G) consists of the same four 0-cells ofthe form (G/G)×D0 and four 1-cells of the form (G/T )×D1 as (G×G)1, along withtwo 2-cells (G/T )×D2, attached as above.

4.3 G-CW structure for Hom(Zn, G)

Now consider the generalization to commuting n-tuples in G,

Hom(Zn, G) = (g1, · · · , gn) ∈ Gn | gigjg−1i = gj ∀i, j (4.1)

which for G = SU(2) can be described as

Hom(Zn, G) = (g1, · · · , gn) ∈ Gn | all gi lie in a common maximal torus. (4.2)

We define a filtration on Hom(Zn, G) via

(Hom(Zn, G))k = (g1, · · · , gn) ∈ Hom(Zn, G) | gi 6∈ ±e for at most k of the gi(4.3)

which satisfies(Hom(Zn, G))n = Hom(Zn, G). (4.4)


We equip each (Hom(Zn, G))k with a G-CW structure inductively as follows. Forconvenience, define

Fn(m) := the number of m-faces of an n-hypercube = 2n−m(n

m

). (4.5)

The 0-skeleton consists of Fn(0) = 2n points:

(Hom(Zn, G))0 = (g1, · · · , gn) ∈ Hom(Zn, G) | gi ∈ ±e ∀i (4.6)

and thus can be given a G-CW structure by considering 2n discrete cells (G/G)×D0.Now consider

(Hom(Zn, G))k\(Hom(Zn, G))k−1 =

(g1, · · · , gn) ∈ Hom(Zn, G) | gi 6∈ ±e for exactly k of the gi. (4.7)

In each n-tuple, we have n− k of the coordinates gi ∈ ±e and each such choice canbe thought of as defining a k-face where we are generalizing the notion of the k-facesof an n-hypercube. There are Fn(k) such k-faces, and taking interiors in our chosenmanner leaves each face disjoint. The points in each face then consist of k nontrivialcoordinates, and so for notational convenience we will shorten our n-tuple with k

nontrivial coordinates to a k-tuple. That is,

(Hom(Zn, G))k\(Hom(Zn, G))k−1 ∼= tFn(k)(g1, · · · , gk) ∈ Hom(Zk, G) | gi 6∈ ±e∀i.(4.8)

Recall that (g1, · · · , gk) ∈ Hom(Zk, G) precisely when all gi lie in a common maximaltorus. Given an element (g1, · · · , gk) ∈ Hom(Zk, G), our goal is to diagonalize this as(g1, · · · , gk) = (h1t1h

−11 , · · · , hktkh−1

k ). Recall that for each component gi we have twochoices on how to diagonalize and we choose that ti ∈ T+ for each i. With h1 deter-mined, then either hi = h1 or hi = w · h1 for all 2 ≤ i ≤ k. That is, either gi = h1tih

−11 or

gi = (w · h1)ti(w · h1)−1 for 2 ≤ i ≤ k.

Insisting that all gi 6∈ ±e, breaks (Hom(Zn, G))k\(Hom(Zn, G))k−1 into disjointsections, characterized by whether each of the 2nd through kth nontrivial coordinatesare located on the same side of the maximal torus as g1 or on the opposite side. Forinstance, in the commuting 2-tuples, there are two top cells corresponding to pairson the same side of the maximal torus, and pairs on the opposite sides of the max-


imal torus. Thus elements in a particular k-face of (Hom(Zn, G))k\(Hom(Zn, G))k−1

are uniquely determined by the following three pieces of data: an element of G/T , ak-tuple (t1, · · · , tk) ∈ (T+)k, and the knowledge of which of the latter k − 1 coordinateswe ought to apply the antipodal map to, the latter having 2k−1 possibilities giving uswhat we will term the 2k−1 leaves for each k-face. A particular (G/T )×Dk cell recordsthe first two pieces of data in the first and second factor, respectively, while the third isbrought by the attaching map to be discussed shortly.

Algebraically, we have

(Hom(Zn, G))k\(Hom(Zn, G))k−1 ∼= tFn(k)(G/T × (T\±e)k)/W∼= tFn(k)((G/T )× t2kI

k)/W (4.9)

where again the W action eliminates the double counting. Applying W ,

(Hom(Zn, G))k\(Hom(Zn, G))k−1 ∼= tFn(k)((G/T )× t2k−1Ik)

∼= tFn(k)2k−1((G/T )× Ik) ∼= t2n−1(nk)

((G/T )× Ik). (4.10)

This can be thought of as having 2k−1 k-cells of the form (G/T )×Dk correspondingto each of our Fn(k) k-faces.

Putting this together, we state the cells in the G-CW structure:

0− skeleton : 2n cells of the form (G/G)×D0 (4.11)

k − skeleton, 1 ≤ k ≤ n : 2n−1

(n

k

)cells of the form (G/T )×Dk (4.12)

While the cells have been listed, we still need to specify the attaching maps. The1-cells are attached analogously to the attaching map for the (single) 1-cell in our G-CWcomplex for G. That is, we have a G-map from (G/T ) × ∂D1 → ±e that identifiesS0 with the points ±e along with the map G/T → ∗. The difference now is weidentify the right hand factor in each (G/T ) × D1 with one of the Fn(1) edges in ann-hypercube and (G/T )× ∂D1 is attached to the corresponding vertices.

The analogy with the right hand factors and an n-hypercube continue, where wecontinue to attach the boundaries of the 2k−1 G/T ×Dk cells corresponding to eachof the various Fn(k) k-faces in the n-hypercube into the (k − 1)-skeleton surrounding


each such k-face where the map on the right hand factor sends ∂Dk to various Dk−1 inthe way expected for cubes. The trick, however, comes in the first factor and we needto be careful about the map from G/T → G/T even if the maps on the second factorfollow precisely as one expects from the skeleta of an n-hypercube.

To define our G-homeomorphism

σ : (G/T )×Dk → a k-face of (Hom(Zn, G))k\(Hom(Zn, G))k−1 (4.13)

we send (gT, t1, · · · , tk) to (g1, · · · , gk) where g1 = gt1g−1 and for 2 ≤ i ≤ k either

gk = gtkg−1 or (w · g)tk(w · g)−1), depending on which cell we are identifying. Each of

the 2k−1 cells for a particular k-face correspond with a particular choice of whether ornot to apply the antipode in coordinates 2 through k, and this data is now recorded inthe G-map.

We can make the data encapsulated by the G-homeomorphism precise by definingan index Λk = Zk2 with a 1 in the ith coordinate corresponding to applying the antipodeat that coordinate and a 0 means not doing this. Given our above preference, the firstcoordinate is thus always a 0. A cell corresponding to index (0, 1, 1), for instance, wouldconsist of those 3-tuples without any coordinates being ±e which had the first elementon one side of the maximal torus, and the other two on the other side. Then a cellconsists of a (G/T )×Dk together with a choice of λ ∈ Λk which determines a G-mapσλ : (G/T )×Dk → a k-face of (Hom(Zn, G))k\(Hom(Zn, G))k−1 where

σλ(gT, t1, · · · , tk) = ((wλ1 · g)t1(wλ1 · g)−1, · · · , (wλk · g)tk(wλk · g)−1). (4.14)

For the attaching G-maps from (G/T )× ∂Dk → (Hom(Zn, G))k−1 we begin with ak-cell (G/T )×Dk identified by choice of index λ ∈ Λk that has a 0 in the first coordinate.We identify Dk with a solid k-cube. Then ∂Dk can be written as a union of (k− 1)-facesto that k-cube. Now consider a particular (k − 1)-face of (G/T )× ∂Dk analogously toconsidering a particular (k− 1)-face of ∂Dk. Choosing a face is equivalent to ignoring aparticular coordinate in our index so the face can be thought of as being indexed by anelement λ ∈ Λk−1 whose first coordinate is not necessarily a 0. On the corresponding(k−1)-face in (Hom(Zn, G))k−1, we have 2k−2 (k−1)-cells (G/T )×Dk−1, each identifiedwith a particular choice of index δ ∈ Λk−1 that has a 0 in the first coordinate.


We thus have two possibilities for an attaching map:

(gT, t1, · · · , tk−1) 7→ (gT, t1, · · · , tk−1) or (gT, t1, · · · , tk−1) 7→ ((w · g)T, t1, · · · , tk−1).

These possibilities correspond to whether the induced index λ has a 0 in the first co-ordinate, in which case there is a cell from the (k − 1)-skeleton with a correspondingindex and we don’t need to apply w, or whether it has a 1, in which case we do applyw. Notice that there will always be two k-cells whose boundary attaches to particular(k − 1)-cell, given the two-fold choice in the first coordinate.

Chapter 5

Equivariant K-theory computations

In this chapter we compute the module structure of the various G-CW complexesdetermined in chapter 4, in particular, in section 5.2, K∗G(Hom(Z2, G)). In sections 5.3and 5.4 we additionally compute the module structure on other cohomology theoriesfor Hom(Z2, G). In section 5.5 we discuss the algebra structure on various cohomologytheories, in particular for K∗G(Hom(Z2, G)). In 5.6 we consider some of the other spacespreviously discussed. Finally, we discuss some possible generalizations in section 5.7.

Let Xn denote the n-skeleton of the various spaces described in the table at thebeginning of chapter 4. The module structure for the equivariant K-theory of thesespaces computed in this chapter are tabulated below.

Table 5.1: K∗G(X) computationsSpace K0

G(X) K1G(X)

G 0 R(G)(G/T )× (G/T ) R(G)⊕R(G)⊕R(G) 0

G×G R(G) R(G)⊕R(G)Hom(Z2, G) R(G)⊕ (R(G)⊕R(G))/〈(v,−2)〉 R(G)⊕R(G)

5.1 K∗G(G)

For this example, X1 = G. Recalling our G-CW complex for G, we can identity X0 andX1/X0 as follows

X0 ∼=G S0

X1/X0 ∼=G Thom(G/T ×D1) 'G S((G/T ) t pt)

22

CHAPTER 5. EQUIVARIANT K-THEORY COMPUTATIONS 23

where pt is used to denote the point at infinity in the Thom space.

We now compute, making use of Proposition 1 of section 3.1.

K0G(X0) = K0

G(2 pts) = K0G(pt) = R(G)

K1G(X0) = K1

G(2 pts) = K1G(pt)) = 0

K0G(X1/X0) = K0

G(Thom((G/T )×D1)) = K0G(S((G/T ) t pt))

= K1G((G/T ) t pt) = K1

G(G/T ) = 0

K1G(X1/X0) = K1

G(Thom((G/T )×D1)) = K1G(S((G/T ) t pt))

= K0G((G/T ) t pt) = K0

G(G/T ) = R(T )

As such, the six term exact sequence of Proposition 6 of section 3.2 correspondingto X0 → X1 → X1/X0 is the following.

R(G)

i∗

K0G(X)oo 0oo

R(T ) // K1G(X) // 0

OO (5.1)

We thus aim to study the kernel and cokernel of i∗.

The map of spaces i : X1/X0 → S(X0) is given by i : S(G/T t pt)→ S(ptt pt). Themap is induced by the map G/T → ∗.

To aid in visualizing this, the space X1/X0 consists of G (visualized as S3) with thenorth and south poles identified. Alternatively consider the G-space Y consisting of Gwith a line attaching the north and south poles where all points on this line are fixedunder the action of G. As the line is G-contractible, proposition 3 of section 3.1 givesthat X1/X0 'G Y . To include into S(X0) ∼=G S

1 we attach a cone over X1 ∼=G S(G/T ).This is thus equivalent to collapsing G/T to a point.

We thus need to study the map i∗ : K1G(pt) = R(G)→ K1

G(G/T ) = R(T ). We saw insection 3.4 that we could write R(G) = Z[v] and R(T ) = Z[b, b−1] where v = b+b−1. Thering R(T ) can be identified as an R(G)-module as R(G)⊕ R(G). Indeed, identifyingb−1 = v− b, we can give R(T ) the 1, b basis as an R(G)-module. Hence the collapsingmap G/T → ∗ induces the map R(G)→ R(G)⊕R(G) that takes v 7→ (v, 0). Finally,


K0G(X) = ker(i∗) = 0

K1G(X) = coker(i∗) = R(G)

Note that this corresponds with our expected answer from ordinary reduced co-homology with integer coefficients which, as G is topologically just S3, has a singlecopy of Z in degree three (which appears here in degree one due to the Z2 grading inequivariant K-theory). Indeed, one can write out the corresponding exact sequence forordinary integral cohomology and obtain the same result. This result also correspondsto [BZ]’s result for Kq

G(G) in the case of G = SU(2).

5.2 Module structure of K∗G(Hom(Z2, G))

From the G-CW structure described in section 4.2, we will compute the module struc-ture of its G-equivariant K-theory in this section.

5.2.1 The 1-skeleton of Hom(Z2, G)

Recall from theG-CW structured described in section 4.2, that the 0-skeleteon, denoted X0

consisted of four disjoint points (thought of as cells of the form G/G×D0) correspond-ing to the four fixed points (±e,±e). Our 1-skeleton had four 2-cells of the formG/T ×D1 attached according to Diagram 1.

Diagram 1. (Hom(Z2, G))1

(e,-e)(e,e)

(-e,-e)(-e,e)

B

D

CA

ss

ssBy Proposition 1 of section 3.1, we have X1/X0 'G

∨4(S((G/T )t∞)). We can now

compute the G-equivariant K-theory of X0 and X1/X0.

K0G(X0) = K0

G(t4pt) = ⊕4K0G(pt) = ⊕4R(G) (5.2)

K1G(X0) = K1

G(t4pt) = ⊕4K1G(pt) = 0 (5.3)


K0G(X1/X0) = K0

G(∨4

(S((G/T ) t ∞))) = K1G(∨4

((G/T ) t ∞))

= K1G((t4G/T ) t ∞)) = K1

G(t4G/T ) = ⊕4K1G(G/T ) = ⊕4K

1T (pt) = 0

(5.4)

K1G(X1/X0) = K1

G(∨4

(S((G/T ) t ∞))) = K0G(∨4

((G/T ) t ∞))

= K0G((t4(G/T )) t ∞)) = K0

G(t4(G/T )) = ⊕4K0G(G/T )

= ⊕4K0T (pt) = ⊕4R(T ) = ⊕8R(G)

(5.5)

In the above computations, we have made repeated use of two facts. First, we havean isomorphism K0

H(pt) = R(G) where in our case we’ve used H = T and H = G.Second, we have an isomorphism Kq

G(A tB) = KqG(A)⊕Kq

G(B).

Let us give explicit generators for the two above spaces with nontrivial equivariantK-theory. In particular, in equation 5.2 we observed that K0

G(X0) = K0G(t4pt) =

⊕4K0G(pt). Let xi ∈ K0

G(X0) denote the image under the disjoint union isomor-phism of a generator for K0

G(pt), which is isomorphic to R(G). Here, the i repre-sents the ith point in the disjoint union. Likewise, from equation 5.5, we have thatK1G(X1/X0) = K0

G(t4(G/T )) = ⊕4K0T (pt). Let yi ∈ K1

G(X1/X0) be the image under thedisjoint union isomorphism of a generator for K0

T (pt) = R(T ) where the i representsthe ith point in the disjoint union.

For the unreduced version of the six term exact sequence described in Proposition 6of section 3.2 for X0 → X1 → X1/X0 we get

⊕4R(G)

i∗

K0G(X1)oo 0oo

⊕4R(T ) // K1G(X1) // 0

OO . (5.6)

Our goal is thus to study i∗.

We can visualize the 1-skeleton as four copies of S(G/T ) glued together accordingto Diagram 1. The attaching maps work by having, for each of the four cells G/T ×D1

in the 1-skeleton, two versions of the collapsing map G/T → pt. This is analogous tohow, when studying SU(2) itself, the two connected components ofG/T×∂D1 collapsedown to the two endpoints. The difference now is that we are doing the same thingfour times and have to keep track of which two points the G/T × ∂D1 components


collapse down to.

We will write the map i∗ : ⊕4R(G) → ⊕R(T ) as a basis with the domain andcodomain written in the bases x1, · · · , x4 and y1, · · · , y4 described above, respectively.We will denote by j∗ the inclusion j∗ : R(G)→ R(T ) given by j∗(v) = (v, 0) ∈ R(G)⊕R(G) induced by the collapsing map j : G/T → pt. The kernel of this map is 0 andthe cokernel is R(G). Putting this all together, we have the map i∗ : ⊕4R(G)→ ⊕R(T )

described by the matrix j∗ j∗ 0 0

0 j∗ j∗ 0

0 0 j∗ j∗

j∗ 0 0 j∗

(5.7)

Note that this is completely analogous to how we could compute the homology ofthe edges of a square by giving it at a CW complex with four vertices and four linesegments.

Our goal is to compute the kernel and cokernel of this matrix. Because the matrixhas rank 3, the kernel is one copy of R(G). The copy of R(G) coming from only havingrank 3 and the four copies of R(G) coming from each j∗ map having cokernel R(G)

gives us that

K0G(X1) = ker(i∗) = R(G) (5.8)

K1G(X1) = coker(i∗) = ⊕5R(G) (5.9)

Note that our computation matches what we would expect on ordinary cohomology.Indeed, ignoring the G action, X1 is homotopy equivalent to (

∨4 S

3)∨S1 and thus has

five copies of the integers in odd degrees, and one copy in even degrees.

5.2.2 The 2-skeleton of Hom(Z2, G)

Recall that in addition to the 1-skeleton of the previous section, the G-CW complexfor Hom(Z2, G), as described in section 4.2, had two 2-cells of the form G/T × D2.Proposition 1 of section 3.1 thus gives us that.


X2/X1 ∼=G

∨2

(S2(G/T t pt)) (5.10)

Computing K∗G(X2/X1) gives us

K0G(X2/X1) = K0

G

(∨2

((S2(G/T ) t pt)

)= K0

G((G/T ) t (G/T ) t pt) =

= K0G((G/T ) t (G/T )) = K0

G(G/T )⊕K0G(G/T ) =

⊕2

R(T )

(5.11)

K1G(X2/X1) = K1

G

(∨2

((S2(G/T ) t pt)

)= K1

G((G/T ) t (G/T ) t pt) =

= K1G((G/T ) t (G/T )) = K1

G(G/T )⊕K1G(G/T ) = 0

(5.12)

The second to last isomorphism in equation 5.11 comes from the general isomor-phism that Kq

G(A t B) = KqG(A) ⊕ Kq

G(B) for G-spaces A and B. For i ∈ 1, 2, letyi ∈ K0

G(X2/X1) denote the image under the disjoint union isomorphism of a generatorfor K0

G(G/T ) which is in turn isomorphic to R(T ). Here the i represents the ith copy ofG/T in the disjoint union.

Combining with our prior computation of X1 which we saw had an isopmorphismK1G(X1) = ⊕5R(G) we get the six term exact sequence corresponding to X1 → X2 →

X2/X1:

0

K0G(Hom(Z2, G))oo ⊕4R(G)oo

0 // K1G(Hom(Z2, G)) // ⊕5R(G)

i∗

OO(5.13)

Consider the maps i : X2/X1 → S(X1) and j : S(X1) → S(X1/X0). The composi-tion is the cellular map d = j i : X2/X1 → S(X1/X0). We thus get

d∗ : K1G(X1/X0)→ K1

G(X1)→ K0(X2/X1) (5.14)

Substituting our prior computations for the quotient spaces we get

d∗ : K0G(t4(G/T ))→ K0

G(t4(G/T )). (5.15)

Note that the corresponding map being in even degree has all spaces zero.


Our goal is thus to study this map d∗. We first turn to the study of a different mapthat will be used in this consideration.

We denote by a the antipodal map a : G/T → G/T given by a(gT ) = (w · g)T wherew is the nontrivial element of the Weyl group. We denote by a∗ : K0

G(G/T )→ K0G(G/T )

the induced map in even degree. Via the isomorphism K0G(G/T ) = K0

T (pt) = R(T ),this gives a map that we will denote with the same symbol, a∗ : R(T )→ R(T ).

Now recall that our 1-skeleton was described by the following diagram.

Diagram 2. (Hom(Z2, G))1

(e,-e)(e,e)

(-e,-e)(-e,e)

B

D

CA

ss

ssTo this diagram we attach our two top cells. One of the two cells is induced by the

the identity map Id : G/T → G/T along all four segments. The other cell is induced bythe identity map Id : G/T × G/T along the A and C segments, but by the antipodalmap a : G/T → G/T on the B and D segments. Recalling from section 5.2.1 that wedefined x1, · · · , x4 as generators for each factor in K0

G(X1/X0) = ⊕4R(T ) thought of ascoming from each of the four (G/T ) factors in a disjoint union. Further, we had definedpreviously in this section generators y1 and y2 for K1

G(X2/X1) = ⊕2R(T ). The map d∗

in these bases is then given by the matrix(Id∗ Id∗ Id∗ Id∗

Id∗ a∗ Id∗ a∗

)(5.16)

which reduces to (Id∗ Id∗ Id∗ Id∗

0 a∗ − Id∗ 0 a∗ − Id∗

)(5.17)

To compute the kernel and cokernel of this matrix, we need to study the mapa∗ − Id∗ : R(T )→ R(T ).

Recall that R(T ) = Z[b, b−1]; that is, Laurent polynomials in the variable b. To


express this as an R(G)-module, we set v = b + b−1 and so we can express a genericelement as p+ qb where p, q ∈ Z[v]. For instance, in this basis b−1 = v(1) + (−1)b.

Recall that the antipodal map a : G/T → G/T was given by a(gT ) = (w · g)T . Asthe induced action of w on K-theory is to interchange b and b−1, so we get the mapa∗ : Z[b, b−1] → Z[b, b−1] given by (a∗)(p + qb) = p + qb−1. Hence, (a∗ − Id)(p + qb) =

qv − 2qb. Hence,

ker((a∗ − Id) : R(T )→ R(T )) = R(G) (5.18)

coker((a∗ − Id) : R(T )→ R(T )) = (R(G)⊕R(G))/〈(v,−2)〉 (5.19)

We now return to the study of d∗ in its entirety, where the reduced matrix gives us

ker(d∗) = ⊕5R(G) (5.20)

coker(d∗) = (R(G)⊕R(G))/〈(v, 2b)〉 (5.21)

Recall that d∗ = i∗ j∗ was a composition and our real goal is to study i∗ that appearin our six term exact sequence.

Firstly, note that j∗ is surjective. Indeed we had the following short exact sequencecoming from our computation of the 1-skeleton:

K1G(X1/X0)

j∗−→ K1G(X1)→ 0. (5.22)

Hence,coker(i∗) = coker(d∗) = (R(G)⊕R(G))/〈(v, 2b)〉 (5.23)

Finally, we note that d∗ has five copies ofR(G) in its kernel, while j∗ has three copiesof R(G) in its kernel, combined such that two copies of R(G) are left in the kernel of i∗.

Putting this all together, we get

K0G(Hom(Z2, G)) = coker(i∗) = (R(G)⊕R(G))/〈(v,−2)〉 (5.24)

K1G(Hom(Z2, G)) = ker(i∗) = R(G)⊕R(G). (5.25)

Note that this result matches that of [AG].


5.3 Other cohomology theories of Hom(Z2, G)

We repeat the computation, firstly of T -equivariant K-theory. Secondly we verifythat it works out as expected on ordinary integral cohomology, matching the resultsfrom [BJS] and [AC]. We then turn to G− and T−equivariant K-theory.

5.3.1 K∗T (Hom(Z2, G))

We can repeat the computation of section 5.2 tersely on T -equivariant K-theory. Speak-ing broadly, the interesting parts of our computation are encoded by the variousattaching maps between G/T and G/T and we aim to study these maps. In [HJS2] itwas shown that, as an R(T )-algebra, K∗G(G/T ) is computed to be

R(T )[L]/〈L2 − (b−1 − b)L+ 1〉,

for L a complex line bundle over G/T described in [HJS2] that, for our purposes, weneed merely know is Weyl invariant. Thus on reduced T -equivariant K-theory, asR(T)-modules, we get K0

T (G/T ) = R(T )⊕R(T ) and K1T (G/T ) = 0.

K0T (X2/X1) = K0

T

(∨2

((S2(G/T ) t pt))

)= K0

T ((G/T ) t (G/T ) t pt) =

= K0T ((G/T ) t (G/T )) =

⊕2

(R(T )⊕R(T )

(5.26)

K1T (X2/X1) = K1

T

(∨2

((S2(G/T ) t pt)

)= K1

T ((G/T ) t (G/T ) t pt) =

= K1T ((G/T ) t (G/T )) = 0

(5.27)

The computation for the the 1-skeleton X1 works exactly analogously, only withR(T ) everywhere instead of R(G). We won’t repeat the computation explicitly but wenonetheless get in an identical way

K0T (X1) = ker(i∗) = R(T ) (5.28)

K1T (X1) = coker(i∗) = ⊕5R(T ). (5.29)

We thus get the following six term exact sequence on equivariant K-theory corre-


sponding to the exact sequence of spaces X1 → X2 X2/X1

0

K0T (Hom(Z2, G))oo ⊕4R(T )oo

0 // K1T (Hom(Z2, G)) // ⊕5R(T )

i∗

OO(5.30)

To study i∗ we again study the map given by the composition

d∗ : K1G(X1/X0)

j∗−→ K1G(X1)

i∗−→ K0(X2/X1) (5.31)

where substituting our prior computations for the quotient spaces we get

d∗ : K0T (t4(G/T ))→ K0

T (t4(G/T )) (5.32)

The map d∗ can be described by the same matrix, only now we have to slow down fromsketching what has been, thus far, a completely analogous computation to carefullystudy the key map a∗ − Id∗ that appears upon reducing this matrix, much as we didwhen we worked G-equivariantly.

Recalling that we had described K0G(G/T ) as R(T )[L]/〈L2 − (b−1 − b)L+ 1〉, we can

write a generic element p1 + q1b + p2L + q2bL where p1, q1, p2, q2 are polynomials inv = b+ b−1. The antipodal map acts by by taking b to b−1 while taking L to itself. Hencea∗ : K0

G(G/T )→ K0G(G/T ) takes

p1 + q1b+ p2L+ q2bL 7→ p1 + q1b−1 + p2L+ q2b

−1L (5.33)

Now we can consider a∗ − Id∗ : K0G(G/T )→ K0

G(G/T ) which is given by

(a∗ − Id∗)(p1 + q1b+ p2L+ q2bL) = q1(b− b−1) + q2(b− b−1)L (5.34)

Now that we understand this key map, we can assemble the rest of the argument asin the G-equivariant case to get

K0T (Hom(Z2, G)) = coker(i∗) = R(T )⊕

(R(T )/〈b−1 − b〉

)(5.35)

K1T (Hom(Z2, G)) = ker(i∗) = R(T )⊕R(T ). (5.36)

Note that we thus get a torsion term in T -equivariant K-theory whose correspond-ing term on G-equivariant K-theory is not a torsion term. For an alternative way to


view this, we can consider (∑∞

i=0 aivi,∑∞

i=0 bivi) ∈ R(G)⊕ R(G) with at most a finite

number of nonzero ai, bi ∈ Z. Under the identification that (v,−2) = (0, 0), we can thusidentify (

∑∞i=0 aiv

i,∑∞

i=0 bivi) = (a0,

∑∞i=0(bi + 2ai+1)vi) so this nontrivial summand in

our cokernel can be identified with Z⊕ Z[v]. Hence this is not a free R(G)-module. Yetanother representation of a generic element is(

∞∑i=0

aivi,∞∑i=0

bivi)

)=

(a0 +

∞∑i=1

(1/2bi−1 + ai)vi, 0

).

Hence, if we invert the 2 ∈ R(G), this becomes a free module. This is analogous to the2-torsion in degree 4 that occurs in the integral cohomology calculation that disappearsupon inverting 2.

We note that in this caseK∗G(Hom(Z2, G)) = (K∗T (Hom(Z2, G)))W . Indeed,R(T )W =

R(G). Further, T -equivariantly we have a torsion term consisting of R(T )/〈b−1 − b〉which is a polynomial in just the variable b. Hence, upon taking Weyl invariance, onlythe constant polynomials survive, and so (R(T )/〈b−1 − b〉)W = Z . This is the samecopy of the integers that appears in the G-equivariant computation, hence the equalityafter taking Weyl invariants.

5.3.2 H∗(Hom(Z2, G);Z)

We begin on ordinary integral cohomology and verify that it works as expected. Recallthat the 1-skeleton consists of X1 = (

∨4 SG/T ) ∨ S1. Here we can exploit the fact that

nonequivariantly this is homotopy equivalent to (∨

4 S3) ∨ S1. The quotient X2/X1 =∨

2(S2(G/T ) ∨ S2) which is topologically∨

2(S4 ∨ S2) . The long exact sequence oncohomology for X1 → X2 → X2/X1 thus has all zeros in even degree for X1 terms andzeros in odd degree for X2/X1 terms. We are left with the following exact sequences ofthe form:

0← H2q(X2)← H2q

(∨2

(S4 ∨ S2)

)i∗←− H2q−1

((∨4

S3) ∨ S1

)← H2q−1(X2)← 0

which by desuspending can be rewritten for q ≥ 1


0← H2q(X2)← H2q−2

(∨2

(S2 ∨ S0)

)i∗←− H2q−2

((∨4

S2) ∨ S0

)← H2q−1(X2)← 0

and so we aim to compute the kernel and cokernel of i∗ as before.

For the rightmost S0 factors in the wedge, this induces on integral cohomology thediagonal map H0(pt) = Z→ H0(pt)⊕H0(pt) = Z⊕ Z. This results in a single factor ofZ coming from the cokernel of this map in degree 2.

For the left-hand factors, first let us consider the antipodal map a : G/T → G/T

thought of now topologically as a : S2 → S2. Using the isomorphism H2(G/T ) = Z,this induces the map a∗ : H2(S2;Z) = Z→ H2(S2;Z) = Z that is just multiplication by−1. The map a∗ − Id we saw previously is thus multiplication by −2. Analogouslyto section 5.2.2 we denote x1, · · · , x4 as generators for each factor in H2(

∨4G/T ) = Z4

and denote y1, y2 as generators for each factor in H2(∨

2 S2) = Z2. We thus get the map

from H2(∨

2 S2)→ H2(

∨4G/T ) given by the matrix in this basis:(

1 1 1 1

1 −1 1 −1

).

As this matrix has kernel Z2 and cokernel Z2, putting all of this together gives us

Hq(Hom(Z2, G)) =

Z q = 2

Z⊕ Z q = 3

Z/2 q = 4

0 else.

Note that this result matches that of [BJS] and [AC].

5.3.3 H∗T (Hom(Z2, G);Z)

Now let us considerH∗T (Hom(Z2, G);Z), following from the introduction to equivariantcohomology in section 3.3.

Following the presentation of [HJS2], let b denote the weight 1 one-dimensional


representation of T , as before. Let b = cT1 (b) ∈ H2T (pt;Z) = H2(BT ;Z) be the equivari-

ant Chern class of b. We thus get H∗T (pt;Z) = Z[b]. The nontrivial element of the Weylgroup W acts on Z[b] via w(b) = −b.

[HJS2] computed that

H∗T (G/T ;Z) = Z[b][L]/〈L2 − b2〉

where L is the first Chern class of the bundle L used in section 5.3.1; for our currentpurposes here it suffices to note that it is a complex line bundle that is Weyl invariant.

In particular, we observe that as a Z module we get that

H2q+1T (G/T ;Z) = 0

andH2qT (G/T ;Z) = Z⊕ Z

with the factors generated by bq and Lbq−1.

Now consider the antipodal map between G/T and G/T which is given by multi-plying by the nontrivial element of the Weyl group. This clearly induces the trivial mapon odd degrees. Further, a∗ : H4q

G (G/T ) = Z⊕ Z→ H4qG (G/T ) = Z⊕ Z is given by

a∗(pb2q + qLb2q−1) = pw · b2q + qw · (Lb2q−1) = pb2q − qLb2q−1

for p, q ∈ Z, as the Weyl group acts trivially on L and even powers of b, and bymultiplication by -1 on odd powers of b. Repeating this story in degree 4q + 2, we get

a∗(pb2q+1 + qLb2q) = pw · b2q+1 + qw · (Lb2q) = −pb2q+1 + qLb2q.

Repeating these computations for the map we will really care about, a∗ − Id, givesus

(a∗ − Id)(pb2q + qLb2q−1) = −2qLb2q−1

in degree 4q and

a∗(pb2q+1 + qLb2q) = −pb2q+1


in degree 4q-2. In either even degree case, we thus get a cokernel with a Z summandand a Z2 summand, only with the order of the factors differing between the two cases.

Now let us considerHom(Z2, G). Following the computation from section 5.2, recallthat our 1-skeleton consists of X1 = (

∨4 SG/T ) ∨ S1 and our quotient has the form

X2/X1 =∨

2(S2(G/T ) ∨ S2) . The long exact sequence on T -equivariant cohomologyfor X1 → X2 → X2/X1 thus has all zeros in even degree for X1 terms and zeros inodd degree for X2/X1 terms. We are left with

0← H2qT (X2)← H2q

T

(∨2

(S2(G/T ) ∨ S2))

)i∗←− H2q−1

T

(∨4

SG/T ) ∨ S1

)← H2q−1

T (X2)← 0

which by desuspending can be rewritten for q ≥ 1 as

0← H2qT (X2)← H2q−2

T

(∨2

((G/T ) ∨ S0)

)i∗←− H2q−2

T

((∨4

G/T ) ∨ S0

)← H2q−1

T (X2)← 0.

We thus need to compute the kernel and cokernel of i∗. For the right hand S0 factorsin the wedge, we get the diagonal map H2q−2

T (pt) = Z → H2q−2T (pt) ⊕ H2q−2

T (pt) =

(Z⊕ Z)⊕ (Z⊕ Z). This has trivial kernel, but picks up a factor of (Z⊕ Z) coming fromthe cokernel in all even degrees 2q for q ≥ 2.

We have previously analyzed the map a∗ : H4qG (G/T ) = Z⊕Z→ H4q

G (G/T ) = Z⊕Z.For the left factors in the wedge, for q ≥ 1, and for 1 ≤ i ≤ 4, denote by xi ∈H2q−2T (

∨4(G/T )) the image under the isomorphism for a wedge, H2q−2

T (∨

4(G/T )) =

⊕4H2q−2T (G/T ), of a generator for H2q−2

T (G/T ). Here i denotes the ith G/T factor.Likewise y1, y2 be generators coming from the two summands in H2q−2

T (G/T ). Theattaching maps work as before, giving us with respect to these bases the matrix(

Id Id Id Id

Id a∗ Id a∗

).

Recall that we had previously studied the map a∗− Id, which occurs upon reducingthe matrix, and observed that the cokernel of this map was Z⊕Z2 and the kernel of thismap was of two copies of H2q−2

T ((G/T );Z) = Z⊕ Z in degree 2q − 2 for q ≥ 2. Puttingall of this together gives us


H0T (Hom(Z2, G)) = 0

H1T (Hom(Z2, G)) = 0

H2T (Hom(Z2, G)) = Z

H2q−1T (Hom(Z2, G)) = ⊕4Z for q ≥ 2

H2qT (Hom(Z2, G)) = (Z2 ⊕ Z)⊕ (Z⊕ Z) for q ≥ 2.

5.3.4 H∗G(Hom(Z2, G);Z)

Finally, let us consider the case of H∗G(Hom(Z2, G);Z). Following [HJS2] again, wedenote t := b2 which lives in degree 4. Then

H∗G(pt;Z) = H∗(BG;Z) = (H∗(BT ;Z))W = Z[t].

The arguments of section 3 of [HJS2] while stated rationally all work integrally as well.In particular we have that

H∗G(G/T ;Z) = Z[t][L]/〈L2 − t〉

where L is as before.

In particular, we observe that as a Z module we get that

H2q+1G (G/T ;Z) = 0

andH2qG (G/T ;Z) = Z

with the factors generated by tq in degrees 4q and generated by Ltq−1 in degrees 4q-2 .

Since w(b) = −b, w(t) = t. As such, all of H∗G(G/T ;Z) is invariant under the actionof w. The core map that we will have to study is a∗− Id : H∗G(G/T ;Z)→ H∗G(G/T ;Z) isthus just the zero map. Repeating what we did for H∗T (Hom(Z2, G)), we get segmentsof the long exact sequence corresponding to X1 → X2 → X2/X1 that look like

0← H2qG (X2)← H2q

G (∨2

(S2(G/T ) ∨ S2))i∗←− H2q−1

G ((∨4

SG/T ) ∨ S1)← H2q−1G (X2)← 0


which by desuspending can be rewritten for q ≥ 1 as

0← H2qG (X2)← H2q−2

G (∨2

((G/T ) ∨ S0))i∗←− H2q−2

G ((∨4

G/T ) ∨ S0)← H2q−1G (X2)← 0.

As before, for the right hand S0 factors in the wedge, we get the diagonal mapH2q−2G (pt) = Z→ H2q−2

G (pt)⊕H2q−2T (pt) = Z⊕ Z. This has trivial kernel, but picks up a

factor of Z coming from the cokernel in all even degrees 2q for q ≥ 1.

Exactly as for the T -equivariant case, we get, for q ≥ 1, generators x1, · · ·x4 comingfrom

⊕4 H

2q−2G (G/T ) and generators y1, y2 coming from ⊕2H

2q−2G (G/T ). The attaching

maps work as before giving us the matrix with respect to these bases(Id Id Id Id

Id a∗ Id a∗

),

which reduces to (Id Id Id Id

0 0 0 0

).

Because of our earlier analysis that the antipodal map acts via the identity. Since as aZ-module each H2q

G (G/T ) is a single factor of Z in all even degrees, we get three copiesof the integers in the kernel, and 1 in cokernel in degree 2q for q ≥ 1 .

Putting this all together we get:

H0G(Hom(Z2, G)) = 0

H1G(Hom(Z2, G)) = 0

H2G(Hom(Z2, G)) = Z

H2q−1G (Hom(Z2, G)) = ⊕3Z for q ≥ 2

H2qG (Hom(Z2, G)) = ⊕2Z for q ≥ 2.

5.4 Algebra structure for K∗G(Hom(Z2, G))

Recall our six term exact sequence for the 1-skeleton:


K0G(t4pt)

i∗

K0G(X1)oo 0oo

K1G(X1/X0) // K1

G(X1) // 0

OO . (5.37)

Recall from proposition 1 of section 3.1 that the quotient X1/X0 is a suspension.Hence, both spaces on the left have trivial multiplicative structure and so we get agraded ring homomorphism onto K∗G(X1), which thus also has trivial multiplicativestructure.

Similarly, we have a six term exact sequence for the 2-skeleton

0

K0G(Hom(Z2, G))oo ⊕4R(G)oo

0 // K1G(Hom(Z2, G)) // ⊕5R(G)

i∗

OO(5.38)

where K∗G(Hom(Z2, G)) is determined by the kernel and cokernel of i∗. As computedin section 5.2, this works out, on unreduced, to be

K0G(Hom(Z2, G)) = coker(i∗) = R(G)⊕ (R(G)⊕R(G)) /〈(v,−2)〉 (5.39)

K1G(Hom(Z2, G)) = ker(i∗) = R(G)⊕R(G). (5.40)

Denoting K∗G(X) = K0G(X)⊕K1

G(X) we get

K∗G(Hom(Z2, G)) = (R(G)⊕ (R(G)⊕R(G)) /〈(v,−2)〉)⊕ (R(G)⊕R(G)) (5.41)

Let us denote by x1, · · · , x5 five generators for K∗G(Hom(Z2, G)) coming from eachof the five R(G) factors. Note that x1, x2, x3 come from even degree where x4, x5

come from odd degree. To determine the R(G)-algebra structure we need to knowhow to multiply these. Observe that, again by Proposition 1 of section 3.1, X2/X1

is a suspension. Hence, like X1, the space X2/X1 has trivial multiplicative struc-ture on equivariant K-theory. We thus have a surjective graded ring homomorphismK0G(X2/X1) ⊕ K1

G(X1) K∗G(Hom(Z2, G)). The domain has trivial multiplicativestructure and hence K∗G(Hom(Z2, G)) also has trivial multiplicative structure.


The only interesting further feature to record comes from the (R(G)⊕R(G)) /〈(v,−2)〉term which imposes the relation vx2 − 2x3 = 0 for this choice of generators. Puttingthis all together, the R(G)-algebra structure for K∗G(Hom(Z2, G)) is thus

K∗G(Hom(Z2, G)) = R(G)[x1, x2, x3, x4, x5]/〈xixj = 0 ∀i, j, vx2 − 2x3〉, (5.42)

where x1, x2, x3 are in even degree and x4, x5 are in odd degree.

Likewise, the same argument gives trivial multiplicative structure on the othercohomology theories under consideration.

5.5 Module Structure of various other spaces

Example 1. G/T ×G/T

Recall that our G-CW complex for (G/T )× (G/T ) yields the following spaces

X0 ∼=G G/T ×D0

X2/X0 ∼=G Thom(G/T ×D2) 'G S2(G/T t pt)

by Proposition 1 of section 3.1.

Computing on K∗G() gives us

KqG(X0) = Kq

G(G/T ) =

R(T ) if q even;

0 if q odd

and

KqG(X2/X0) = Kq

G(S2((G/T ) t pt) = Kq+2G ((G/T ) t pt)

= Kq+2G (G/T ) =

R(G)⊕R(G) if q even;

0 if q odd.(5.43)


We thus get the following six term exact sequence corresponding to X0 → X2 →X2/X0. Note that we do unreduced as G/T does not have a G fixed point.

R(G)⊕R(G)

K0G(X2)oo R(G)⊕R(G)oo

0 // K1G(X2) // 0

OO. (5.44)

The sequence splits, and thus we get

K0G(G/T ×G/T ) = ⊕4R(G) (5.45)

K1G(G/T ×G/T ) = 0 (5.46)

Note that computing the analogous exact sequence on ordinary reduced cohomol-ogy with integer coefficients for (G/T )× (G/T ), which is topologically S2 × S2, givestwo copies of Z in degree 2and one copy of Z in degree 4 as expected.

Example 2. G×G

Let us first recall Diagram 1 that was used in describing the G-CW complex.

Diagram 1. (G×G)1

(e,-e)(e,e)

(-e,-e)(-e,e)

B

D

CA

ss

ssDiagram 1 depicts our 1-skeleton X1 ∼=G (G × G)1 which, being the same as the

1-skeleton for Hom(Z2, G), we have already computed in section 5.2. Denoting G×Gby X , we note that as per Proposition 1 of Section 3.1

X/X1 ∼=G Thom((G/T )× (G/T )×D2) 'G S2((G/T )× (G/T )) ∨ S2.

We thus get

K0G(X1) = 0

K1G(X1) =

⊕5

R(G)


K0G(X/X1) = K0

G(S2(((G/T )×(G/T ))tpt)) = K0G((G/T )×(G/T ))

⊕K0G(S0) =

⊕4

R(G)

K1G(X/X1) = K1

G(S2(((G/T )× (G/T )) t pt)) = K1G((G/T )× (G/T ))

⊕K1G(S0) = 0

We thus get the following six term exact sequence on equivariant K-theory corre-sponding to the exact sequence of spaces X1 → X2 X2/X1

0

K0G(G×G)oo ⊕4R(G)oo

0 // K1G(G×G) // ⊕5R(G)

i∗

OO(5.47)

To determine i∗, corresponding to the identity map from S1 → S1 we get an identitybetween the right-most R(G) factors. Recall that the attaching map for the 2-cell ofthe form(G/T ) × (G/T ) × D2 is determined by maps (G/T ) × (G/T ) → G/T thatare projections onto the first factor along the A and C segments of Diagram 1, andprojections onto the second factor along the B and D segments of Diagram 1. InKqG((G/T ) × (G/T )), there were two copies of R(G), one for each of the two copies

of G/T , and a final copy of R(G) coming from the G/T × D2 cell representing thediagonal elements. If we denote x1, · · · , x5 as generators of the summands in ⊕5R(G)

and y1, · · · , y4 as generators of the summands in ⊕4R(G) then the map i∗ : ⊕5R(G)→⊕4R(G) is described in this basis is given by

Id 0 Id 0 0

0 Id 0 Id 0

Id Id Id Id 0

0 0 0 0 Id

Thus,

K0G(G×G) = coker(i∗) = R(G)

K1G(G×G) = ker(i∗) = R(G)⊕R(G)

Note that doing the analogous computation on reduced ordinary cohomology givesthe same result as that expected when considering the space topologically as S3 × S3;


that is, two copies of Z in degree 3 and one in degree 6.

Example 3. (G×G)\Hom(Z2, G))+

Note that Z/2 is the centre of SU(2). Using ourG-CW complex for ((G×G)\Hom(Z2, G))+,we compute:

KqG(((G×G)\Hom(Z2, G))+) = Kq

G(Thom(G/(Z/2)×D3))

= KqG(S3(G/(Z/2) t ∗)) = Kq+1

G (G/(Z/2) t ∗)

Recall that for normal subgroups H of G,

KqG(G/H) ∼= Kq

H(pt) ∼=

R(H) if q even;

0 if q odd.

Thus,

KqG((G×G)\Hom(Z2, G))+) ∼=

0 if q even;

R(Z/2) if q odd.

This computation lets us state a result regarding i∗ : KqG(G×G)→ Kq

G(Hom(Z2, G)),which is equivalent to Proposition 6.11 of [AG].

Theorem 1. Let i : Hom(Z2, G)→ G2 be the inclusion map. Then

i∗ : K1G(G×G)

∼=−→ K1G(Hom(Z2, G))

is an isomorphism and there is a short exact sequence of R(G)-modules

0→ K0G(G×G)→ K0

G(Hom(Z2, G))→ R(Z/2)→ 0

Proof. Consider the six term exact sequence corresponding to the pair (G×G,Hom(Z2, G)).The result follows from the above computation for the noncommuting 2-tuples togetherwith the isomorphism

KqG(G×G,Hom(Z2, G)) = Kq

G((G×G)\Hom(Z2, G))+)


5.6 Future Work

There are two immediate generalizations of our present computation of the commuting2-tuples.

• Commuting n-tuples in SU(2). The G-CW complex is written down in chapter 3,from which KG should follow. Heuristically, the important information encodedby our G-CW complex are various maps between G/T and G/T ; either theidentity map or the antipodal map. Hence, analogously to the n = 2 case, one willget matrices whose components are either the identity map or the antipodal map(or the zero map). A closed form solution may not be possible in the general case,but at least it should be able to be programmed into a computer for arbitrary n.The relevant information is encoded in the cellular maps Kq+1

G (Xk/Xk−1) →KqG(Xk+1/Xk) which is given by an 2n

(nk

)× 2n

(nk+1

)matrix (by the results of

section 4.3 that works by analogy with the k-faces of an n-cube). The entriesof this matrix consist of either the identity map or the antipodal map. As theattaching maps are done analogously to how one attaches the canonical CWstructure for an n-cube, a convenient description of the entries of these matricesshould be able to be determined. The n = 2 case we did explicitly is also aided bythe fact that we have only one nontrivial step X2/X1 → S(X1) to write out anexact sequence for; for higher n there will need to be multiple steps computingthe skeleta in turn, and the earlier skeleta will not be free.

• Commuting m-tuples (or just 2-tuples) in SU(n). For SU(n) the maximal torus isa torus of dimension n− 1 opposed to only being dimension 1 as it was in ourSU(2) case. This adds some complexity, but nonetheless constructing a G-CWcomplex on Hom(Zm, SU(n)) from which one can compute various cohomologytheories seems promising.

Chapter 6

Quasi-Hamiltonian Systems

In this chapter we outline the key ideas regarding quasi-Hamiltonian systems. Insection 6.1, we will introduce the concept of quasi-Hamiltonian systems and note thatthe spaces we have been computing throughout this thesis are motivated, in part, bythe spaces in a particular example of a quasi-Hamiltonian system. In section 6.2, weoutline the results in the literature about Kirwan surjectivity. In section 6.3, we discusssome potential areas for future work.

6.1 Quasi-Hamiltonian Systems

Alekseev, Malkin, and Meinrenken [AMM] introduced quasi-Hamiltonian G-spaceswhich are certain finite-dimensional G-spaces M equipped with a G-valued quasi-Hamiltonian moment map Φ : M → G. Following their presentation, for a compact Liegroup G, we make a choice of an invariant positive definite inner product 〈·, ·〉 on theLie algebra G. Let θ, θ ∈ Ω1(G,G) denote the left- and right-invariant Maurer-Cartanforms, respectively. Let χ = 1

12〈θ, [θ, θ]〉 denote the canonical bi-invariant 3-form on G.

Finally, let νξ denote the generating vector field on M for an element ξ ∈ G. Then aquasi-Hamiltonian G-space is a G-manifold M with an invariant 2-form ω ∈ Ω(M)G

and a G-map φ ∈ C∞(M,G)G such that

1. dω = −φ∗χ

2. i(νξ)ω = 12φ∗〈θ + θ, ξ〉

3. for each x ∈M , kerωx = νξ, ξ ∈ ker(Adφ(x) + 1)

These finite dimensional quasi-Hamiltonian G-spaces are in bijective correspon-dence with certain infinite dimensional Hamiltonian LsG-spaces M that fit into the

44

CHAPTER 6. QUASI-HAMILTONIAN SYSTEMS 45

following diagram where M is a pullback of the bottom right square and the columnsare fibrations with fibre the space ΩG. The space Φ−1(e)/G is referred to as the sym-plectic quotient M//G.

ΩG

= // ΩG

Φ−1(0)

//

=

M

π

Φ // LG∗

Hol

Φ−1(e)

//MΦ // G

(6.1)

where Hol is the holonomy map to the path space PG composed with the end-pointevaluation map.

Consider the following specific example of a quasi-Hamiltonian G-space. LetG = SU(2) which acts on itself by conjugation. The space M = (G × G)h with thediagonal action and equipped with the G-map Φ : (G × G)h → G given by takingthe commutator map Φ(a1, . . . , ah, b1, . . . , bh) =

∏hi=1[ai, bi] provides an example of a

quasi-Hamiltonian G-space. The corresponding symplectic quotient M//LsG is themoduli space of flat connections on the trivial principal G-bundle over a compact,connected 2-manifold Σ of genus h with boundary ∂Σ = S1 (See [AMM]). In particular,we will consider the h = 1 case which has kernel Φ−1(e) = Hom(Z2, G). For this specificcase we relabel parts of the diagram:

ΩG

= // ΩG

Hom(Z2, G)

//

=

M

π

Φ // LG∗

Hol

Hom(Z2, G)

// G×G Φ // G

(6.2)

While this chapter is near the end of this thesis, a key motivation for our work is tounderstand this diagram on equivariant K-theory. In so doing, while the principalcomputation is the equivariant K-theory of the symplectic quotient Hom(Z2, G), wehave also matched [BZ]’s result for G, and we remind the reader of the computationin [AG] of G×G and the work on ΩG done by [HJS1].


6.2 Kirwan Surjectivity

We collect in this section an overview of the original Kirwan surjectivity and its variousanalogs.

Kirwan [Kir] showed that for a compact Lie group G and a compact HamiltonianG-space M with proper moment map µ : M → G∗, the function ‖µ‖2 : M → R is aG-equivariantly perfect Morse function on M. It follows that the inclusion µ−1(0) →M

induces a surjection on rational G-equivariant cohomology

H∗G(M ;Q) H∗G(µ−1(0);Q). (6.3)

When 0 is a regular value of µ, µ−1(0)/G is a symplectic orbifold called the sym-plectic quotient M//G. G then acts locally freely on µ−1(0) and so we have a surjectioncalled the Kirwan map

κ : H∗G(M ;Q) H∗(M//G;Q). (6.4)

Harada and Landweber [HL] extended this result to G-equivariant K-theory

K∗G(M) K∗G(µ−1(0)) (6.5)

If the G action is free, the second space is isomorphic to K∗(M//G).

There is an infinite-dimensional analogue of the above situation which results inanalogous surjectivity theorems. The compact group G is replaced with the BanachLie group LsG := γ : S1 → G | γ is of Sobolev class s for s > 3/2. A HamiltonianLsG-space consists of an infinite-dimensional symplectic Banach manifold M togetherwith an LsG-action and moment map Φ : M→ LsG

∗.

Bott, Tolman, and Weitsman [BTW] proved that the inclusion Φ−1(0) →M, for 0 aregular value, induces a surjection on rational G-equivariant cohomology :

H∗G(M;Q) H∗G(Φ−1(0);Q) ∼= H∗(M//LsG;Q). (6.6)

Harada and Selick [HS] proved the G-equivariant K-theory analog

K∗G(M) K∗G(Φ−1(0)) ∼= K∗(M//LsG), (6.7)

where the latter isomorphism holds when G acts freely.


The third setting to be considered is that of the quasi-Hamiltonian G-spaces in-troduced by Alekseev, Malkin, and Meinrenken [AMM] which are certain finite-dimensional G-spaces equipped with a G-valued quasi-Hamiltonian moment mapΦ : M → G. These finite dimensional quasi-Hamiltonian G-spaces are in bijectivecorrespondence with certain infinite dimensional Hamiltonian LsG-spaces M that fitinto the following diagram where M is a pullback of the bottom right square and thecolumns are fibrations with fibre ΩG.

ΩG

= // ΩG

Φ−1(0)

//

=

M

π

Φ // LG∗

Hol

Φ−1(e)

//MΦ // G

(6.8)

Indeed, this correspondence is instrumental in [BTW] and [HS]’s proofs of surjectiv-ity in the Hamiltonian LsG-space situation for rational G-equivariant cohomology andG-equivariant K-theory, respectively.

In the quasi-Hamiltonian case, the inclusion Φ−1(e) →M fails to necessarily inducesurjections on either rational G-equivariant cohomology or G-equivariant K-theory.Nonetheless, [BTW] proved a similar result in a way that depended on the surjectivityfor Hamiltonian LsG-spaces. Loosely, they proved that H∗G(Φ−1(e)) is generated as aring by the image of H∗G(M)→ H∗G(Φ−1(e)) together with classes that originate in thefibre.

6.3 Relating to future work

For G = SU(2), one of our motivations for studying Hom(Z2, G) was to understandthis example in the larger context of quasi-Hamiltonian G-spaces. We have computedthe equivariant K-theory of the symplectic quotient, the base G×G, and the fibre ΩG

for our example, but we have not computed this for the infinite dimensional space M.We note a few facts regarding M. The space is defined as a pullback. If we restrict thebase to just commuting pairs in G×G then we get the map (G×G)|Hom(Z2,G) → G via


the commutator map, and the pullback for this restriction is a product as all such pairsmap to e. However, M itself is not a product topologically which is unfortunate as itprevents us from easily writing down a G-CW complex from our knowledge of thebase and the fibre. Note, however, that it is indistinguishable on rational cohomologyfrom a product because the commutator map factors through the smash product whichis topologically S3 ∧ S3 ∼= S6 → S3 and π6(S3) = Z12.

As such, an immediate future goal is to attempt to resolve this difficulty and beable to describe M in our example more fully, or perhaps, at least, the low dimensionalskeleta of it. [BTW] and [HS] provide a filtration of this space where it is proven that thesurjectivity of the inclusion φ−1(0) →M induces a surjection on rational cohomologyand equivariant K-theory, respectively. This surjection is true for the inclusion intoeach space in the filtration, and thus even understanding the first of these filtrationcomponents would be an advantage.

One immediate generalization from our present situation is to study the exampleof a quasi-Hamiltonian system where M = SU(2)2n with the moment map being aproduct of commutators as before. This has the symplectic quotient the moduli spaceof flat connections on a closed 2-manifold Σ of genus n. We have done the n = 1 case.Note that this does not have commuting n-tuples in SU(2) as the symplectic quotient.The G-CW complex for M is simple enough and of course ΩG remains as the fibre, butwork is needed to establish the G-CW structure on the symplectic quotient in this case.

While the above has all been specific to G = SU(2) we now consider G a compactLie group. An important long term goal is to extend this body of Kirwan surjectivityresults to the quasi-Hamiltonian setting on G-equivariant K-theory. For our specificexample, φ−1(e) →M does not induce a surjection on equivariant K-theory. However,we conjecture that a K-theoretic analog of [BTW]’s result in rational equivariant coho-mology for the quasi-Hamiltonian case holds true. Specifically, from [HS] the inclusionΦ−1(0) → M induces a surjection on G-equivariant K-theory. As Φ−1(0) = Φ−1(e),some set of generators of K∗G(M) are needed to generate K∗G(Φ−1(e)). On cohomology,the result of [BTW] uses cochains and the Leray-Hirsch theorem, unavailable to us onequivariant K-theory. Nonetheless, it is conjectured that an analog can be stated onG-equivariant K-theory. Here, one source is the image π∗(K∗G(M)) ⊂ K∗G(M). Unlikethe Hamiltonian G-space and Hamiltonian LsG-space situations, this source alone isinsufficient to ensure surjectivity. Additionally, it is conjectured that one needs genera-


tors originating in an appropriate sense from the fibre ΩG.

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