by Trefor Bazett - University of...

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 in 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 descrip-

tion 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. This space is of particular interest as φ−1(e) in a quasi-Hamiltonian

system Mφ−→ G consisting of the G-space SU(2) × SU(2), together with a moment

map φ given by the commutator map. Finite dimensional quasi-Hamiltonian spaces

have a bijective correspondence with certain infinite dimensional Hamiltonian spaces,

and we additionally compute relevant components of this larger picture in addition to

φ−1(e) = Hom(Z2, SU(2)) for this example.

ii

Acknowledgements

I with to thank my joint supervisors Lisa Jeffrey and Paul Selick for their support,patience and advice. I can not sufficienlty 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 committ-ment to teaching mathematics; they have been invaluable at helping me develop as ateacher.

My family has been an inexhaustable 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. If there are any grammatical errorsleft in this thesis, please blame her.

iii

Contents

Contents iv

1 Introduction 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Historical Results and Context 4

3 Preliminaries 53.1 G-CW Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Equivariant K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 On SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 G-CW Complex Calculations 104.1 G-CW structures for various G-spaces . . . . . . . . . . . . . . . . . . . . 104.2 G-CW structure for Hom(Z2, G) . . . . . . . . . . . . . . . . . . . . . . . . 144.3 G-CW structure for Hom(Zn, G) . . . . . . . . . . . . . . . . . . . . . . . . 154.4 G-CW structure for ΩG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Equivariant K-theory computations 215.1 K∗G(G), K∗G(G×G), etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Module structure of K∗G(Hom(Z2, G)) . . . . . . . . . . . . . . . . . . . . 265.3 Module Structure of K∗T (Hom(Z2, G)) . . . . . . . . . . . . . . . . . . . . 295.4 Hom(Z2, G) on other cohomology theories . . . . . . . . . . . . . . . . . 30

5.4.1 H∗(Hom(Z2, G);Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.4.2 H∗T (Hom(Z2, G);Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4.3 H∗G(Hom(Z2, G);Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

iv

5.5 Algebra structure for K∗G(Hom(Z2, G)) . . . . . . . . . . . . . . . . . . . . 365.6 Module Structure of K∗G(ΩG) . . . . . . . . . . . . . . . . . . . . . . . . . 375.7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Quasi-Hamiltonian Systems 396.1 Quasi-Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Kirwan Surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3 Relating to future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Bibliography 44

v

Chapter 1

Introduction

1.1 Overview

In this thesis, our principle objective is the study of the commuting n-tuples in SU(2),(g1, · · · , g2) ∈ (SU(2))n | gigj = gjgi ∀i, j. Since a homomorphism is determined bythe mappings of the generators, this is the space Hom(Zn, SU(2)). This is an SU(2)-space where SU(2) acts on Hom(Zn, SU(2)) by conjugation on each component. Alter-natively, this space is homomorphisms from the fundamental group of (S1)n into Gwhich can be identified with the space of flat G-connections on (S1)n modulo basedgauge transformations. Our main goal is to understand this space on various cohomol-ogy theories, in particular SU(2)-equivariant K-theory.

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. In the specific case of equivariant K-theory, which is a Z2 gradedcohomology theory due to Bott Periodicity, this yields six term cyclical exact sequences.As quotients Xn+1/Xn can be well descibed, this provides a methodology to computethe unknown terms in the sequence. Thus by establishing an explicit G-CW complexfor a space, this geometric description gives us considerable power in computingequivariant cohomology theories.

1

CHAPTER 1. INTRODUCTION 2

We apply this methodology to a range of G-spaces, but our principal computationis to establish a G-CW complex structure for G=SU(2) on Hom(Zn, G). From this, inthe case 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). The former two, as modules,have previously been computed by [AG] and [BJS] respectively for X = Hom(Z2, G).

Beyond providing an alternate and pleasingly geometric computation that moti-vates future work using this G-CW approach, our results extend the literature in threeways. Firstly, our geometric G-CW decompositions allows us to also establish the multi-plicative structure, thus computing K∗G(Hom(Z2, G)) as an R(G)-algebra, not just as anR(G)-module. Additionally, our methods allow us to write this on other cohomologytheories in particular HG(Hom(Z2, G)). Finally, we equip the commuting n-tuples inSU(2) with a G-CW structure that generalizes the 2-tuples case, from which the variouscohomologies theories mentioned above could be computed analogously. However, welack a closed form solution for the general n case and note that computations quicklybecome cumbersome. As such, only the n=2 case has been explicitly written out in thisthesis.

One motivation for this study of Hom(Z2, G) is the quasi-Hamiltonian systemsintroduced by [AMM]. Indeed, the commutator map from G × G → G provides anexample of such a quasi-Hamiltonian system with Hom(Z2, G)/G as the symplecticquotient where G acts regularly on Hom(Z2, G). Hom(Z2, G)/G has orbifold singulari-ties and we can identify it with based gauge equivalence classes of the moduli space offlat connections on a closed 2-manifold Σ of genus 1 [Jef]. 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 weadditionally compute KG(ΩG), KG(G) and KG(G×G), and finally relate them to largerquestions in quasi-Hamiltonian systems.

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 dicuss known computations on equivariant K-theory and

CHAPTER 1. INTRODUCTION 3

cohomology 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). All of the G-spaces under consideration in thisthesis are for G= SU(2) and details about how one diagonalizes elements, forinstance, in this space are critical.

• In Chapter 4 we establish G-CW complexes on all the various spaces we are inter-ested in computing in this thesis. In particular, we study G, G×G, Hom(Zn, G),and ΩG. 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 equivariant K-theory of our range of spaces as R(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 and 5.4we then study Hom(Z2, G) on various other cohomology theories. Additionally,we make explicit the R(G)-algebra structure for K∗G(Hom(Z2, G)) in Section 5.5.We study ΩG in Section 5.6 and finally, in 5.7, we discuss some potential for futurework.

• 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 futher work along thesethemes.

1.3 Conventions

Henceforth in this thesis, unless otherwise specified when wishing to make a moregeneral statement, we will be considering G=SU(2), acting on itself by conjugation. Twill denote the maximal torus, as will be described in Section 3.4.

Chapter 2

Historical Results and Context

in this chapter we summarize prior computations on various cohomology theoriesin the literature that overlap with our result 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. We will match, as expected, this result in the case of G=SU(2) inSection 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 abelsian 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 for all pos-itive n, in agreement with [AC2], an errata to the original [AC] in the n=3 case. Ourresults in this thesis match the n=2 case on integral cohomology as noted in Section 5.4.

[AG] provide a computation for K∗SU(2)(Hom(Z2, SU(2))) as an R(G)-module. Thisresult uses the Atiyah-Hirzebruch-Segal Spectral Sequence and Segal’s localizationto reduce the problem to computing Hp

W (T 2,RT ), which is computed using a W-CWdecomposition of T 2. We reproduce this result in Section 5.2 through computingK∗SU(2)(Hom(Z2, SU(2)) via providing a G-CW structure on (Hom(Z2, SU(2)).

[HJS1] and [HJS2] are twin papers that compute K∗G(ΩG) for G = SU(2) acting onitself by conjugation as a module and as an algebra respectively. We will reinterpretthe last few steps in the module computation by making explicit a G-CW structure onΩSU(2) in Section 5.6.

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 normal CW-Complexes, referred to as G-CW complexes inthis thesis, was developed independently by [Mat] and [Ill1]. Let X by a G-space for acompact Lie group G; that is, X is a Hausdorff topological space with a left action of Gon X. Let Hk denote a closed subgroup of G for each k. Following [May],

Definition 1. A G-CW complex X is a union of G-spaces Xn where X0 is a disjoint union oforbits G/Hk, referred to as 0-cells, and Xn+1 is determined inductively by attaching (n+1)-cells(G/Hk)×Dn+1 to Xn via attaching G-maps σk : (G/Hk)× Sn → Xn

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

∐k(G/Hk)× Sn

∐k σk //

Xn

∐k(G/Hk)×Dn+1 // Xn+1.

(3.1)

A point of the form (eHk, t) ∈ (G/Hk) × Dn+1 is fixed under the Hk action, andso maps under σk into the fixed point set of Hk, denoted (Xn)Hk . Thus, an attachingG-map σk is determined by its restriction Sn → (Xn)Hk .

5

CHAPTER 3. PRELIMINARIES 6

By convention, a (G/H) ×Dn cell will be referred to as an n-cell, be said to havedegree n, and have dimension (Dim(G/H) + n). The space Xn, formed by attachingcells of degree up to and including n, is referred to as the n-skeleton of X. A G-CWsubcomplex Y of a G-CW complex X is a subcollection of cells from the G-CW complexfor 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×Yby 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 (in a noncanonicalway) then a G-CW complex for X × Y can be determined.

The following let’s us interpret the quotients of skeleta from our G-CW complex asa suspension. We will henceforth write T (G/Hk)×Dn) to refer to the Thom space ofthe product G-bundle with base G/T with the usual action and trivial action on Dn.

Proposition 1. For a G-CW skeleton Xn on a G-space X,

Xn/Xn−1 ∼=∨k

T ((G/Hk)×Dn) ∼=∨k

(Sn((G/Hk)∐∗)) ∼=

∨k

(Sn(G/Hk) ∨ Sn)

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

The first identification follows from the observation that an attaching map for aG/Hk × Dn cell glues G/Hk × Sn into the (n-1)-skeleton resulting in the quotient ofthe trivial disk bundle by the trivial sphere bundle, hence T (G/Hk ×Dn). The secondidentification follows from the more general fact that for a vector bundle ξ, the fibrewisedirect sum T (ξ ⊕ Rn) ' Sn ∧ T (ξ) applied to the case of ξ being the 0 dimensionalbundle over base G/Hk which has Thom space T (ξ) = G/Hk

∐pt.

3.2 Equivariant K-theory

Let G be a compact Lie group and X a compact G-space. Then [Seg] defines a Z/2-graded algebra K∗G(X) generated by equivalence classes of complex G-vector bundlesover X. We follow Segal’s approach outlining the basic facts.

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


KqG(pt) ∼=

R(G) if q even;

0 if q odd..

where R(G) is the representation ring of G. Secondly, when G is trivial,K∗G(X) = K∗(X),ordinary K-theory. The map X → pt induces a map K1

G(pt) = R(G)→ K1G(X) which

makes equivariant K-theory an R(G)-algebra.

Further, there are two extreme cases on the action of G on X. Firstly, if G actstrivially on X, then any vector bundle can be thought of as an equivariant vectorbundle, giving us a map K∗(X) → K∗G(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 2. If G acts trivially on X, the map µ described above is an isomorphism.

Secondly, for G acting freely:

Proposition 3. If G acts freely on X then the quotient map π : X → X/G induces anisomorphsim

π∗ : K(X/G)→ KG(X)

Proposition 4. For H a closed subgroup of G, and an H-space X, the map i : H → G inducesan isomorphism i∗ : K∗G(G×H X)→ K∗H(X)

In particular, we have

KqG(G/H) ∼= Kq

H(pt) ∼=

R(H) if q even;

0 if q odd..

The aforementioned Z/2 grading is a consequence of the critical property of BottPeriodicity:

Proposition 5. The Thom homomorphism φ∗ : KG(X)→ KG(T (E)) is an isomorphism forany G-vector bundle E on a compact G-space X.

A consequence of Bott Periodicity is that the typical long exact sequence becomes acyclical six-term sequence that is critical for many of our computations:


Proposition 6 (Exact Sequences). Let A be a closed G-subspace of a compact G-space X, thenthe following six-term sequence is exact.

K0G(A)

K0G(X)oo K0

G(X/A)oo

K1G(X/A) // K1

G(X) // K1G(A)

OO(3.2)

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×G X = (EG×X)/G where EG → BG := EG/G is a universal principalG-bundle. The homotopy quotient is thus homotopy equivalent (as EG is contractible)to X, but the action of G on the homotopy quotient is free. Further, our definition willbe well defined as the classifying space BG is unique up to homotopy. We then definethe equivariant integral cohomology ring 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, we just have H∗G(X) = H∗(X/G) as EG×G X is thus a bundleover X/G with contractible fibre. For X a point, we get that EG×G pt = BG. Muchlike equivariant K-theory, the map X → pt thus induces an H∗G(pt)-module structureon H∗G(X).

Now let G be a compact Lie group, and T a maximal torus in G. Finally, let W be thecorresponding Weyl group. Then, the splitting principle gives us 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)

G = SU(2) consists of all 2x2 unitary matrices with determinant 1. As a manifold, SU(2)

is homeomorphic to S3. All maximal tori are conjugate to T =

(eiθ 0

0 e−iθ

)| θ ∈

[0, 2π) which topologically is homeomorphic to S1 and any two maximal tori intersectat±I . Two elements in SU(2) commute precisely if they lie on a common maximal torus.

Elements in G\±I can be uniquely diagonalized up to right multiplication byelements in T as g = hth−1 with t ∈ T+ := t ∈ T | 0 < θ < π. G has G-projections G→ G/T , which is topologically homeomorphic to S2, and to G→ T+. Let

w =

(0 1

−1 0

), then the Weyl group is given by W = T,wT and has a W action on

T that takes t 7→ t−1 and on G/T corresponding to the antipodal action on S2.

For G=SU(2) and T the maximal torus, R(T ) ∼= Z[b, b−1], the representation ringof T with b the weight 1, one dimensional representation of T and 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. R(T ), as a group, is R(G)⊕R(G)

Chapter 4

G-CW Complex Calculations

In this chapter we identify G-CW structures on several G-spaces, in particular forHom(Zn, G). We begin with the more elementary examples to develop our intuition.Then in 4.2 and 4.3 we discuss commuting 2-tuples and n-tuples respectively beforeconcluding with ΩG

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

Table 4.1: G-CW ComplexesSpace Cells

G∐

2(G/G)×D0 , (G/T )×D1

(G/T )× (G/T )∐

2(G/T )×D0 , (G/∗)×D1

G×G∐

4(G/G)×D0 ,∐

4(G/T )×D1 ,∐

2(G/T )×D2, (G/∗)×D3

G ∨G∐

3(G/G)×D0 ,∐

2(G/T )×D1

Hom(Z2, G)∐

4(G/G)×D0 ,∐

4(G/T )×D1 ,∐

2G/T ×D2

ΩG (G/G)×D0 , (G/T )×D2 , (G/T )×D2(2r−1)

((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)

X0 consists of two (G/G)×D0 0-cells thought of as the North and South poles ofSU(2), which is topologically S3, and are fixed points under the conjugation action.

10

CHAPTER 4. G-CW COMPLEX CALCULATIONS 11

X = X1 consists of a (G/T ) × D1 cell attached to X0 via the attaching map that col-lapses each of the two connected components ofG/T×S0 to the two 0-cells respectively.Topologically, one can think of this as the construction of S3 by taking S2 × D1 andcollapsing the ends S2 × ∂D1 to two points. Identifying D1 with T+, described earlier,we have a G-homeomorphism from (G/T ) ×D1 to G\±e given by (gT, t) 7→ gtg−1

where G acts on (G/T )×D1 by left multiplication on the first factor and trivially on thesecond. Note that we have two choices for this diagonalization, depending on whetherwe assert that t ∈ T+ or t ∈ T−. We choose the former.

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

For G × G, we begin by considering the collection of cells formed by taking for-mal products of cells in G. This results in four (G/G) × (G/G) × D0 × D0 0-cells,four (G/T )× (G/G)×D1 ×D0 1-cells and one (G/T )× (G/T )×D1 ×D1 2-cell. Thezero and one degree cells in the resulting (G × G)-CW complex can be immediatelyrewritten as four (G/G) × D0 cells and four (G/T ) × D1 cells to form a G-CW com-plex. The only nontrivialG×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, 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 elements fromG/T ×D1 in the other.

In the following diagram, a non vertex point on the square represents a copy of G/T.The space then consists of four copies of the suspension of G/T, topologically S3, labeledA,B,C,D, and wedged together as per the diagram. Note that this can be visualized asS2 × S1 that has been ’pinched’ at the four vertices.

Diagram 1. (G×G)1

(e,-e)(e,e)

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

B

D

CA

ss

ss


Attaching G-maps for the one skeleton are determined as per the diagram where fora specific (G/T )×D1 1-cell, (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). Note that much like for G we make thechoice of diagonalization with elements t ∈ T+ at all points in Diagram 1.

Note that X1 is G-homotopy equivalent to (∨

4 S(G/T )∨S1 with trivial action on

the last factor.

Example 3. G/T ×G/T

We construct the following G-CW structure. It has a 0-cell G/T ×D0 and a 2-cellG/T ×D2 with an attaching map that glues ∂D2 = S1 to the point D0 on the secondfactors and identity on the first. We have thus constructed the space G/T × S2. Topo-logically this is the same as G/T × G/T which is topologically just S2 × S2 but theyhave different G-actions on the second factor.

Consider a generic point (gT, hT ) ∈ G/T×G/T which we can rewrite as (gT, g(g−1h)T ).We construct a G-homeomorphism fromG/T ×G/T → G/T ×D2 via (gT, g(g−1h)T ) 7→(gT,x) where x ∈ S2 is the antipode of the coset g−1hT ∈ G/T where we forget theG-action and thus have a point in S2. As such the D0 in G/T ×D0 is thought of as oneof the poles of S2 and points in G/T ×D0 map diagonally into G/T ×G/T . Stereopro-graphic projection from D2 into the sphere minus a pole determines the nondiagonalpoints.

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

The 0 and 1 skeletons of G×G, as a G-CW complex where established in Example 2.Thought of as a (G×G)-CW complex, we have this top (G/T ×G/T )×D2 2-cell wherethe attaching map σ : (G/T ) × (G/T ) × ∂D2 → X1 consists of collapsing to a pointat each of the four vertices, the projection map π1 : (G/T ) × (G/T ) → G(T ) onto thefirst factor along the A and C segments, and the projection π2 : (G/T )× (G/T )→ G(T )

onto the second factor along the B and D segments.

Now, however, we can rewrite this top cells as a G-CW complex, following Example3, and as such in addition to the four 0-cells and four 1-cells described previously, wealso have one(G/T ×D0)×D2 = G/T ×D2 2-cell and one (G/T ×D2)×D2 = G/T ×D4

4-cell. Here, the attaching map for the 2-cell (recall we thought of these as the diagonal


elements) is to attach G/T × ∂D2 where the ∂D2 = S1 wraps around the boundarysquare of Diagram 1 while the G/T maps by the identity map, except for the cornerswhere it collapses to a point. For the 4-cell, ∂D4 = (∂D2×D2)

⋃(D2×∂D2). As such the

G/T × S1 ×D2 component is attached by collapsing to G/T × pt×D2 and identifyingthis with the G/T × D2 cell thought of as the diagonal elements. The nondiagonalelements represented by points G/T ×D2 × S1 are identified along the boundary circleas described above, namely collapsing to a point at the four vertices, but projecting tothe first factor along the A and C segments and projecting to the second factor alongthe B and D segments.

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 (G/G)×D0 0-cells, labelled S1,N1 = S2, N2,and two (G/T ) × D1 1-cells attached as in the previous case with one of the 1-cellsattached to S1 and N1 = S2 and the other to N1 = S2 and N2.

Example 6. Non-commuting 2-tuples

In this example we reinterpret work in Proposition 6.11 of [AG] in our G-CW lan-gauge, and in a way that makes for slightly easier computations.

Following [AG], Y := (G × G)\Hom(Z2, G)). Consider the restriction of the com-mutator map φ : G× G → G to the noncommuting 2 tuples, φ| : Y → G\1. As per[AC], this is a locally trivial G-fibre bundle with fibre

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

which has a G-equivariant homotopy equivalence to

Y 'G F × SU2∼=G F ×D3

Following from [ACG], F ∼= G/Z(G) ∼= G/(Z/2) where G acts on G/(Z/2) by lefttranslation.

Now consider Y + which identifies F×∂D3 with a point. We can thus equip Y + witha G-CW structure via a single point, G/G×D0 as the zero skeleton, and then a 3-cellG/(Z/2)×D3 identified by the trivial map on both factors 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-tuples inthe n=2 case (which is the only case we will go on to compute the equivariant K-theoryof) before describing the general n-tuple case in the next section. Consider commuting2-tuples in G,

Hom(Z2, G) = (gi, gj) ∈ G×G | gigjg−1i = gj

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. Let the zero skeleton consist ofpairs (±e,±e), and the one skeleton consist of pairs (±e, g) or (g,±e) for g ∈ G. ThusHom(Z2, G)1 = (G×G)1 and is as in Diagram 1.

For a pair (g1, g2) ∈ Hom(Z2, G)\(Hom(Z2, G))1, we have two options for how todiagonalize g1 where the action of the Weyl group alternates between these presenta-tions by acting on G/T via the antipodal map and on the maximal torus T via takingt → t−1. We can either write g1 = h1t1h

−11 for h1 ∈ G/T and t1 ∈ T+ or we can write

g1 = (w ·h1)t−11 (w ·h1)−1 for w ·h1 ∈ G/T being the antipode of h1 and w · t1 = t−1

1 ∈ T−.Aiming to determine a (G/T )×Dn cell, the former of these two choices for g1 fixes ourchoice for an element in G/T as h1 and so 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 )×∐

4

D2)/W

Applying W,

Hom(Z2, G)\(Hom(Z2, G))1 ∼= (G/T )×∐

2

D2

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 )×∐

2

D2 → Hom(Z2, G)\(Hom(Z2, G))1

we first identify ((G/T )×∐

2D2) with ((G/T )× T+× T+)

∐((G/T )× T+× T−) where

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

−1, gt2g−1). For the sec-

ond connected component, the third factor is written in terms of T− and so our G-mapbecomes (gT, t1, t2)→ (gt1g

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

first connected component (G/T )× ∂D2 → (Hom(Z2, G))1 is induced by the identitymap on G/T and the identification of ∂D2 with the boundary square in Diagram 1, andfor the second connected component is induced by the same map along the boundarysquare but has the identity map on G/T along the A and C segments of ∂D2 and theantipodal map on G/T along the segments B and D.

Therefore our G-CW complex for Hom(Z2, G) consists of the same four (G/G)×D0

0-cells and four (G/T ) × D1 1-cells as (G × G)1, along with two (G/T ) × D2 2-cells,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 zero 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 G/G × D0 cells.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-skeletons of an n-hypercube. There are Fn(k) such k-faces, and taking interiors aswe are 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 knontrivial coordinates to a k-tuple. That is,

(Hom(Zn, G))k\(Hom(Zn, G))k−1 ∼=∐Fn(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. Recall further that for an element (g1, · · · , gk) ∈ Hom(Zk, G), we have a choiceon how to diagonalize g1 where the action of the Weyl group alternates between thesepresentations by acting on G/T via the antipodal map and on the maximal torus Tvia taking t → t−1. We can either write g1 = h1t1h

−11 for h1 ∈ G/T and t1 ∈ T+ or

we can write g1 = (w · h1)t−11 (w · h1)−1 for w · h1 ∈ G/T being the antipode of h1 and

w · t1 = t−11 ∈ T−. Preferring to diagonalize all coordinates with ti ∈ T+, the h1 ∈ G/T

specified by the diagonalization for g1 either is used, or its antipode is used, for all sub-sequence coordinates. That is, either gi = h1tih

−11 or gi = (w ·h1)ti(w ·h1)−1 for 2 ≤ i ≤ k.

Insisting that all gi 6∈ ±e, this breaks (Hom(Zn, G))k\(Hom(Zn, G))k−1 into dis-


joint sections, characterized by whether each of the 2nd through kth nontrivial coor-dinates are located on the same side of the maximal torus as g1 or on the oppositeside. For instance, in the commuting 2-tuples, there are two top cells corresponding tonontrivial pairs on the same side of the maximal torus, and pairs on the opposite sidesof the maximal torus. Thus elements in Hom(Zk, G) are uniquely determined by thefollowing three pieces of data: an element of G/T , an n-tuple (t1, · · · , tn) ∈ T+n , andthe knowledge of which of the latter k-1 coordinates we ought to apply the antipodalmap to, the latter having 2k−1 possibilities and corresponding to the 2k−1 cells for eachk-face. A particular (G/T ) × Dk cell records the first two pieces of data in the firstand second factor respectively, while the third is brought by the attaching map to bediscussed shortly.

Algebraically, we have


(G/T×(T\±e)k)/W ∼=∐Fn(k)

((G/T )×∐2k

Ik)/W.

(4.9)

Applying W,


((G/T )×∐2k−1

Ik) ∼=

∼=∐

Fn(k)2k−1

((G/T )× Ik) ∼=∐

2n−1(nk)

((G/T )× Ik). (4.10)

This can be thought of as having 2k−1 (G/T )×Dk k-cells corresponding to each ofour Fn(k) k-faces. For example, in Hom(Z2, G), we have two such 2-cells where thefirst copy refers to pairs where t1, t2 are both in T+ and the second copy refers to pairswhere t1 ∈ T+ but t2 ∈ T−. We can think of the two copies as representing pairs thatare on the same side of the maximal torus, and on opposite sides, respectively. The Waction identifies these two cells with two corresponding cells having the first coordinatebeing instead given by t1 ∈ T−.

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 a eachof the various Fn(k) k-faces in the n-hypercube into the k-1 skeleton surrounding eachsuch k-face where the map on the right hand factor sends ∂Dk to various Dk−1 in theway expected for cubes. The trick, however, comes in the first factor and we need to becareful about the map from G/T → G/T even if the maps on the second factor followprecisely as one expects from the skeleta of an n-hypercube.

Explicitly defining our G-homeomorphism

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

we send (gT, t1, · · · , tk) 7→ (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 corresponded with a particular choice of whetheror not to apply the antipode in coordinates 2 through k, and this data is now recordedin the G-map.

We can make the data encapsulated by the G-homeomorphism precise by definingan index Λn = Zn2 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 trivial coordinates who had the first element on oneside of the maximal torus, and the other two on the other side. Then a cell consists of a(G/T )×Dk together with a G-map σλ 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 a(G/T )×Dk k-cell identified by choice of index λ ∈ Λk that has a 0 in the first coordinate.We identify a particular (k-1)-face of (G/T )× ∂Dk analogous to the (k-1)-face of ∂Dk.Choosing a face is equivalent to ignoring a particular coordinate in our index so theface can be thought of as being indexed by an element λ ∈ Λk−1 whose first coordinateis not necessarily a 0. On the corresponding (k-1)-face in (Hom(Zn, G)k−1 we have 2k−2

(G/T )×Dk−1 cells, each identified with a particular choice of index δ ∈ Λk−1 that has a0 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 whetherthe induced index λ has a 0 in the first coordinate, in which case there is a cell fromthe k-1 skeleton with a corresponding index and we don’t need to apply w, or whetherit has a 1, in which case we do apply w. Notice that there will always be two k-cellswhose boundary attaches to particular (k-1)-cell, given the two fold choice in the firstcoordinate.

4.4 G-CW structure for ΩG

The G-space ΩG has a filtration Ωpoly,rG that is studied in detail in [HJS1]’s computationof the module and product structure of the G-equivariant K-theory of ΩG. For our pur-poses, a key fact is that this filtration, henceforth labeled F2r for consistency with [HJS1],has a convenient geometric description as quotients F2r/F2r−2 are G-homeomorphic toThom(τ 2r−1) where the G-bundle τ is G-isomorphic to the tangent bundle on P1.

As G/T ∼= P1, we get the following G-CW complex where the G action on thesecond components are the trivial SU(2) action on (D2)r ∼= Cr.

F0∼= ∗ ∼= (G/G)×D0

F2/F0∼= Thom(τ) ∼= G/T ×D2

F4/F2∼= Thom(τ 3) ∼= (G/T )×D6

...

F2r/F2r−2∼= Thom(τ 2r−1) ∼= (G/T )×D2(2r−1)


The attaching map (G/T ) × ∂D2 → (G/G) × D0 is induced via the map from(G/T ) → ∗ and the usual construction of S2 by gluing the boundary of D2 to thepoint F0. For the attaching G-map (G/T ) × ∂D6 → (G/G) × D2, consider D6 asD2×D2×D2 ∈ C3. A G-map fromD2×D2×D2 → D2 is given by (z1, z2, z3) 7→ z1+z2+z3

where the left hand disks are of radius 1 and the right hand disk is of radius 3. Theattaching G-map is this map on the second component and the identity on G/T on thefirst. For the general term, (z1, z2, ..., z2r−1) 7→ (z1, z2, ..., z2r−4, z2r−3 + z2r−2 + z2r−1).

Chapter 5

Equivariant K-theory computations

This chapter computes K∗G(X) of the various G-CW complexes determined in Chapter4. In Sections 5.3 and 5.4 we additionally compute Hom(Z2, G) on other cohomologytheories. In Section 5.5 we discuss the R(G) algebra structure for K∗G(Hom(Z2, G)). Fi-nally, we discuss some possible generalizations in section 5.7. The module structure forthe equivariant K-theory of the various spaces computed in this Chapter are tabulatedbelow.

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)

ΩG∏∞

k=1 R(G) 0

5.1 K∗G(G), K∗G(G×G), etc.

Example 1. KqG(G)

The spaces involved are as follows:

X0 ∼= S0

S(X0) ∼= S1 ∼= S(∗) ∨ S1

X1/X0 ∼= Thom(G/T ×D1) ∼= S(G/T ) ∨ S1

21

CHAPTER 5. EQUIVARIANT K-THEORY COMPUTATIONS 22

The map of spaces i : X1/X0 → S(X0) is induced by the map G/T → ∗ in the firstfactor and identity in the second.

We now compute K∗G:

K0G(X0) = K0

G(2pts) = R(G) = K1G(S(X0))

K1G(X0) = K1

G(2pts) = 0 = K0G(S(X0))

K0G(X1/X0) = K0

G(S(G/T ))⊕ K0G(S1) = K1

G(G/T )⊕ K1G(S0) = 0

K1G(X1/X0) = K1

G(S(G/T ))⊕ K1G(S1) = K0

G(G/T )⊕ K0G(S0) = R(G)⊕R(G)

We thus get the following six term exact sequence

R(G)

i∗

K0G(X)oo 0oo

R(G)⊕R(G) // K1G(X) // 0

OO (5.1)

i∗ : R(G)→ R(G)⊕R(G), induced by the map of spaces i, takes v 7→ (0, v).Therefore:

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 cohomology and obtain the same result. This result also corresponds to [BZ]’sresult for Kq

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

Example 2. KqG(G/T ×G/T )

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

X0 ∼= G/T ×D0

X2/X0 ∼= Thom(G/T ×D2) ∼= S2(G/T ) ∨ S2


by Prop 1 of Section 3.1.

Computing K∗G gives us

KqG(X0) ∼= Kq

G(G/T ) ∼=

R(G) if q even;

0 if q odd

and

KqG(X2/X0) ∼= Kq

G(S2(G/T )∨S2) ∼= Kq+2G (G/T )⊕Kq+2(S0) ∼=

R(G)⊕R(G) if q even;

0 if q odd.

We thus get the following six term exact sequence corresponding to X0 → X2 →X2/X0

R(G)

K0G(X2)oo R(G)⊕R(G)oo

0 // K1G(X2) // 0

OO(5.2)

The sequence splits, and thus we get

K0G(G/T ×G/T ) = R(G)⊕R(G)⊕R(G) (5.3)

K1G(G/T ×G/T ) = 0 (5.4)

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 2 and one copy of Z in degree 4 as expected.

Example 3. KqG(G×G)

Recall that our one skeleton X1 ∼= (G × G)1 is as in Diagram 1, and is thus G-homotopic to (

∨4 S(G/T ))

∨S1 where G acts trivially on the S1. Denoting X = G×G,

we not that as per Theorem 2,

X/X1 ∼= Thom((G/T )× (G/T )×D2) ∼= S2((G/T )× (G/T )) ∨ S2

We now compute KqG(X1), Kq

G(X/X1),


K0G(X1) = (

⊕4

K0G(S(G/T )))

⊕K0G(S1) = (

⊕4

K1G(G/T ))

⊕K1G(S0) = 0

K1G(X1) = (

⊕4

K1G(S(G/T )))

⊕K1G(S1) = (

⊕4

K0G(G/T ))

⊕K0G(S0) =

⊕5

R(G)

K0G(X/X1) = K0

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

⊕K0G(S0) =

⊕4

R(G)

K1G(X/X1) = K1

G(S2((G/T )× (G/T )) ∨ S2) = 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(5.5)

To determine i∗, note that corresponding to to the identity map from S1 → S1

we get an identity between the right most R(G) factors. Recall that the attachingmap for the (G/T )× (G/T )×D2 cell is determined by maps (G/T )× (G/T )→ G/T

that are projections onto the first factor along the A and C segments of Diagram 1,and projections 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 these bases 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 4. KqG(((G×G)\Hom(Z2, G))+)

Proof. Using our G-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))+)

5.2 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 for K∗G(Hom(Z2, G)) and K∗T (Hom(Z2, G)) and verify it works out as expectedfor ordinary cohomology H∗(Hom(Z2, G)). We begin the computation on reducedG-equivariant K-theory.

First note that the that K∗T (G/T ) was computed in [HJS1] to be

K∗T (G/T ) = R(T )[L]/ < L2 − (b+ b−1)L+ 1 > (5.6)

and so for the module structure we will use K0T (G/T ) = R(T ).

Recall that our one skeleton X1 ∼= (Hom(Z2, G))1 is the same as (G×G)1 and hencewe restate our previous computation.

K0G(X1) = 0 (5.7)

K1G(X1) =

⊕5

R(G) (5.8)

Further, we have (G/T )×D2 2-cells, and so Proposition 1 of 3.1 gives

X2/X1 ∼=∨2

Thom((G/T )×D2) ∼=∨2

((S2(G/T )) ∨ S2) (5.9)

and so


K0G(X2/X1) = K0

G(∨2

((S2(G/T )) ∨ S2)) =⊕

2

(K0G(G/T )

⊕K0G(S0)) =

⊕4

R(G)

(5.10)

K1G(X2/X1) = K1

G(∨2

((S2(G/T )) ∨ S2)) =⊕

2

(K1G(G/T )

⊕K1G(S0)) = 0. (5.11)


0

K0G(Hom(Z2, G))oo ⊕4R(G)oo

0 // K1G(Hom(Z2, G)) // ⊕5R(G)

i∗

OO(5.12)

And henceK0G(Hom(Z2, G)) = coker(i∗) (5.13)

K1G(Hom(Z2, G)) = ker(i∗) (5.14)

To determine i∗, first recall the diagram

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

(e,-e)(e,e)

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

B

D

CA

ss

ssFor each of our two 2-cells, ∂D2 ∼= S1 is identified with the boundary square in this

diagram, and hence we get the diagonal map from R(G)→ R(G)⊕R(G) on the rightmost factors. Recall that the attaching map for the first of the two (G/T )×D2 cells isinduced by the identity map on G/T along all four segments while the attaching mapfor the second of two (G/T )×D2 cells is induced by the identity on G/T for the A andC segments, but by the antipodal map a on G/T for the B and D segments. If we denotex1, · · · , x5 as generators of the summands in ⊕5R(G) and y1, · · · , y4 as generators ofthe summands in ⊕4R(G) then the map i∗ : ⊕5R(G) → ⊕4R(G) is described in these


bases is given by Id Id Id Id 0

Id a∗ Id a∗ 0

0 0 0 0 Id

0 0 0 0 Id

We aim to compute the kernel and cokernel of this matrix. After reducing this ma-

trix, the only nonobvious component to consider is the image of (a∗−Id) : K0G(G/T )→

K0G(G/T ), which we will study first on unreduced (a∗ − Id) : K0

G(G/T )→ K0G(G/T ).

Recall that K0G(G/T ) = K0

T (pt) which, as an R(T )-module, is just R(T ) = Z[b, b−1];that is, Laurent polynomials in the variable b. Identifying b−1 = v − b, we can express ageneric element as p+ qb where p, q ∈ Z[v].

Consider the antipodal map a : G/T → G/T given by a(gT ) = (w · g)T where w isthe nontrivial element of the Weyl group. As the induced action of w, the nontrivialelement of the Weyl group, on K-theory is to interchange b and b−1, a induces the mapa∗ : Z[b, b−1] → Z[b, b−1] given by (a∗)(p + qb) = p + qb−1. Hence, (a∗ − Id)(p + qb) =

q(b−1 − b).

The cokernel of this map is as an R(T)-module is thus:

coker((a∗ − Id) : K0G(G/T )→ K0

G(G/T )) = R(T )/ < b−1 − b > . (5.15)

However, we wish to write this as an R(G)-module as we are doing G-equivariantK-Theory. Recall that we were describing a generic element in R(T ) as p + qb wherep, q ∈ Z[v]. We can thus describe R(T) as an R(G)-module as R(G) ⊕ R(G) using the1, b basis. In particular, we can denote b−1 as 1v + (−1)b in this basis. Since as an R(T)module, a∗(p+qb) = p+qb−1 = p+vq−qb, writing it in this way as an R(G)-module weget that a∗ : R(G)⊕R(G)→ R(G)⊕R(G) thus takes a∗(p, q) = (p+vq,−q). Hence, as anR(G)-module, (a∗ − Id)(p, q) = (vq,−2q). The cokernel of this map as an R(G)-moduleis thus:

coker((a∗ − Id) : K0G(G/T )→ K0

G(G/T )) = (R(G)⊕R(G))/ < (v,−2) > . (5.16)

Finally, combining with the other components of the matrix thus gives


K0G(Hom(Z2, G)) = coker(i∗) = R(G)⊕ (R(G)⊕R(G))/ < (v,−2) > (5.17)

K1G(Hom(Z2, G)) = ker(i∗) = R(G)⊕R(G). (5.18)

Note that this result matches that of [AG].

5.3 Module Structure of K∗T (Hom(Z2, G))

We can repeat the computation of Section 5.2 tersely on T-equivariant K-theory knowingthat, in [HJS2], the R(T)-algebraK∗G(G/T ) is computed to beR(T )[L]/ < L2−(b−1−b)L+

1 >. Thus on reduced as R(T)-modules we get K0T (G/T ) = R(T ) and K1

T (G/T ) = 0.

K0T (X2/X1) = K0

T (∨2

((S2(G/T )) ∨ S2)) =⊕

2

(K0T (G/T )

⊕K0G(S0)) =

⊕4

R(T )

(5.19)

K1T (X2/X1) = K1

T (∨2

((S2(G/T )) ∨ S2)) =⊕

2

(K1T (G/T )

⊕K1G(S0)) = 0. (5.20)


0

K0T (Hom(Z2, G))oo ⊕4R(T )oo

0 // K1T (Hom(Z2, G)) // ⊕5R(T )

i∗

OO(5.21)

with the same matrix for i∗ only now written in terms of basis elements for ⊕5R(T ) and⊕4R(T ) as opposed to⊕5R(G) and⊕4R(G). The same key map, (a∗−Id) appears uponreducing and since as R(T)-modules K0

T (G/T ) = KG(G/T ) we have already computedthe cokernel above. Combining everything, we get

K0T (Hom(Z2, G)) = coker(i∗) = R(T )⊕R(T )⊕R(T )/ < b−1 − b > (5.22)


K1T (Hom(Z2, G)) = ker(i∗) = R(T )⊕R(T ). (5.23)

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 toview this, we can consider (

∑∞i=0 aiv

i,∑∞

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 case K∗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 polynomials 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 Invariance.

5.4 Hom(Z2, G) on other cohomology theories

Let’s repeat the computation on various other cohomology theories. In the case ofH∗(Hom(Z2, G);Z) we verify that the results match the expected results from [BJS] and[AC]. Secondly we do the case of H∗T (Hom(Z2, G);Z) and finally H∗G(Hom(Z2, G);Z)

neither of which has not been recorded in the literature.

5.4.1 H∗(Hom(Z2, G);Z)

We begin on ordinary integral cohomology and verify that it works as expected. Fol-lowing the computation from Section 5.2, recall that our 1 skeleton for Hom(Z2, G)


consists of X1 = (∨

4 SG/T ) ∨ S1 which is topologically (∨

4(S3) ∨ S1 and our quotienthas the form X2/X1 =

∨2(S2G/T ) ∨ S2) which is topologically

∨2(S4 ∨ S2) . The long

exact sequence on for cohomology for X1 → X2 → X2/X1 thus has all zeros in evendegree for X1 terms and zeros in odd degree for X2/X1 terms. We are left with thefollowing exact sequences of the 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 factors S0 vectors in the wedge, this induces on integral cohomol-ogy the diagonal map H0(pt) = Z→ H0(pt)⊕H0(pt) = Z⊕ Z. This results in a singlefactor of Z 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. This now induces the map a∗ :

H2(S2;Z) = Z → H2(S2;Z) = Z that is just multiplication by -1 on integral coho-mology. The a∗ − Id map we saw previously is thus multiplication by -2. As such,denoting x1, · · · , x4 as generators for each factor in H2(

∨4G/T ) = Z4 and denoting

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].

5.4.2 H∗T (Hom(Z2, G);Z)

Now let us consider H∗T (Hom(Z2, G)) , following from the introduction to equivariantcohomology in Section 3.3.

Following the presentation of [HJS2], let b denote the weight 1 one-dimensionalrepresentation of T, as before. Let b = cT1 (b) ∈ H2

T (pt;Z) = H2(BT ;Z) be the equivariantChern 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] compute that

H∗T (G/T ;Z) = Z[b][L]/〈L2 − b2〉.

where L is the first chern class of the bundle L described in [HJS2]; for our purposeshere it sufficies 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 )→ H4qG (G/T ) 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 whereId represents the identity map, gives us

(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(S2G/T )∨S2) . The long exact sequence on T-equivariant cohomology forX1 → X2 → X2/X1 thus has all zeros in even degree for X1 terms and zeros in odddegree for X2/X1 terms. We are left with

0← H2qT (X2)← H2q

T (∨2

(S2G/T ) ∨ S2))i∗←− H2q−1

T ((∨4

SG/T ) ∨ S1)← H2q−1T (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−1T (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.


For the left factors in the wedge, for q ≥ 1 let x1, · · ·x4 be generators for each ofthe summands in

⊕4 H

2q−2T (G/T ) and let y1, y2 be generators for the summands in

H2q−2T (G/T ). The attaching maps work as before giving us in this basis the matrix(

Id Id Id Id

Id a∗ Id a∗

)Recall that we had previously studied the map a∗ − Id which occurs upon reducing

the matrix and observed that the cokernel of this was Z⊕Z2 and a kernel of two copiesof H2q−2

T (G/T );Z) = Z⊕ Z in degree 2q − 2 for q ≥ 2. Putting all of this together givesus

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.

5.4.3 H∗G(Hom(Z2, G);Z)

Finally, let us consider the case of H∗G(Hom(Z2, G)). Following [HJS2] again, we denotet := 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

(S2G/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.

For the left factors in the wedge, for q ≥ 1 let x1, · · ·x4 be generators for each ofthe summands in

⊕4 H

2q−2G (G/T ) and let y1, y2 be generators for the summands in

H2q−2G (G/T ). The attaching maps work as before giving us in this basis the matrix(

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.

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

Recall that we had described both the quotient of skeleta, X2/X1, and the one skeletonX1, as wedges of suspensions:

X2/X1 ∼=∨2

((S2(G/T )) ∨ S2) (5.24)

X1 ∼= (∨4

S(G/T )) ∨ S1 (5.25)

As such, the multiplicative structure on any of our cohomology theories for eitherof these spaces is automatically trivial. Thus, if we consider the aforementioned sixterm exact sequence

0

K0T (Hom(Z2, G))oo ⊕4R(T )oo

0 // K1T (Hom(Z2, G)) // ⊕5R(T )

i∗

OO(5.26)

these maps are all ring homomorphisms and thus the trivial multiplicative structurecarries forward onto K∗T (Hom(Z2, G)). Denoting a generator for each of the foursummand in the module structure by xi, we thus get

K∗T (Hom(Z2, G)) = R(T )[x1, x2, x3, x4, x5]/〈xixj = 0 ∀i, j, (b−1 − b)x2〉. (5.27)

Repeating the computation G-equivariantly, we get

K∗G(Hom(Z2, G)) = R(T )[x1, x2, x3, x4, x5]/〈xixj = 0 ∀i, j, vx2 − 2x3〉. (5.28)


5.6 Module Structure of K∗G(ΩG)

Recall that following the work of [HJS1], we had established a G-CW complex forΩpolyG consisting of a point in degree zero and a (G/T ) × D2(2r−1) cell in each evendegree. Computing Kq

G,

KqG(F0) ∼= Kq

G(pt) ∼= 0 ∀q

KqG(F2r/F2(r−1)) ∼= Kq

G(Thom((G/T )×D2(2r−1))) ∼=

∼= KqG((G/T )

∐∗) ∼=

R(G)⊕R(G) if q even;

0 if q odd.

Thus our exact sequence

K0G(F2(r−1))

K0G(F2r)oo K0

G(F2r/F2(2r−1))oo

K1G(F2r/F2(2r−1)) // K1

G(F2r) // K1G(F2(r−1))

OO

becomes

K0G(F2(r−1))

K0G(F2r)oo R(G)⊕R(G)oo

0 // K1G(F2r) // K1

G(F2(r−1))

OO

where, by induction with K∗G(F0) as the basis case, K∗G(F2(r−1)) is 0 in odd degree and afree R(G)-module in even degree and hence the sequence in the top row splits,

KqG(F2r) ∼=

KqG(F2(r−1))⊕R(G)⊕R(G) if q even;

0 if q odd.

By induction we get

KqG(F2r) ∼=

∏2r

k=1 R(G) if q even;

0 if q odd.

Finally,


KqG(ΩG) ∼= Kq

G(ΩpolyG) ∼= lim←− KqG(Ωpoly,rG) ∼=

∏∞

k=1 R(G) if q even;

0 if q odd.

5.7 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 chould follow. Heuristically, the important information encodedby our G-CW complex is a 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, and even n=3 iscumbersome, but at least this should be able to be programmed for a computer toproduce the computation for a given value of n.

• Commuting m-tuples (or just 2-tuples) in SU(n). For SU(n) the maximal torus is atorus of dimension n-1 opposed to only being dimension 1 as it was in our SU(2)case. This adds some complexity, but nonetheless constructing a G-CW complexon Hom(Zm, SU(n)) from which one can compute various cohomology theoriesseems 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 and note that the spaces we have beencomputing throughout this thesis are motivated, in part, by the spaces in a particularexample of a quasi-Hamiltonian system. In Section 6.2 we outline the literature resultson Kirwan surjectivity. In Section 6.3 we discussed some potential for future work.

6.1 Quasi-Hamiltonian Systems

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

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

let νξ denote the genderating vector field on M for an element ξ ∈ G. Then a quasi-Hamiltonian G-space is a G-manifold M with an invariant 2-form ω ∈ Ω(M)G and aG-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

39

CHAPTER 6. QUASI-HAMILTONIAN SYSTEMS 40

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

ΩG

= // ΩG

Φ−1(0)

//

=

M

π

Φ // LG∗

Hol

Φ−1(e)

//M Φ // G

(6.1)

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 principle com-putation is the equivariant K-theory of the symplectic quotient Hom(Z2, G), we havealso reinterpreted the work of [HJS1] to compute ΩG and the unsurprisingG andG×G.

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 finite Hamiltonian


G-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] extend this result to G-equivariant K-theory

K∗G(M) K∗G(µ−1(0)) ∼= K∗(M//G) (6.5)

where the latter isomorphism holds when G acts freely.

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 inducesa surjection on rational G-equivariant cohomology for 0 a regular value:

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 analogue

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 bijective


correspondence 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)

//M Φ // 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] prove a similar result in a way that depends on the surjectivity forHamiltonian LsG-spaces. Loosely, they prove that H∗G(Φ−1(e)) is generated as a ring bythe image of H∗G(M)→ H∗G(Φ−1(e)) together with classes that originate in the fibre.

6.3 Relating to future work

One of our motivations was to understand this example in the larger context of quasi-Hamiltonian G-spaces. We have computed the equivariant K-theory of the symplecticquotient, the base G × G, and the fibre ΩG for our example, but we have not com-puted this for the infinite dimensional space M. We note a few facts regarding thisspace. Firstly, note that the the space is defined as a pullback. If we restrict the baseto just commuting pair in G × G then we get the map G × G|Hom(Z2,G) → G via thecommutator map, and the pull back for this restriction really is a product as all suchpairs map to e. However, M itself is not a product topologically which is unfortunateas it prevents us from easily writing down a G-CW complex from our knowledge ofeach factor. Note however that it is indistinguishable on rational cohomology froma product because the commutator map factors through the smash product which istopologically 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 student the exampleof a quasi-Hamiltonian system where M = SU(2)2n with moment map a productof commutators as before. This has the symplectic quotient the moduli space of flatconnections on a closed 2-manifold Σ of genus n. We have done the n=1 case. Note thatthis does not have commuting n-tuples in SU(2) as the symplectic quotient. The G-CWcomplex for M is simple enough and of course ΩG remains as the fibre, but work isneeded to establish the G-CW structure on the symplectic quotient in this case.

An important long term goal is to extend this body of Kirwan surjectivity resultsinto the quasi-Hamiltonian setting on G-equivariant K-theory. For our specific ex-ample, φ−1(e) → M does not induce a surjection on equivariant K-theory. However,we conjecture that a K-theoretic analogue of [BTW]’s result in rational equivariantcohomology for the quasi-Hamiltonian case holds true. Specifically, from [HS] theinclusion Φ−1(0) →M induces a surjection on K∗G(). As Φ−1(0) = Φ−1(e), some set ofgenerators of K∗G(M) are needed to generate K∗G(Φ−1(e)). On cohomology, the resultof [BTW] uses cochains and the Leray-Hirsch theorem, unavailable to us on equivariantK-theory. Nonetheless, it is conjectured that an anolog can be stated on equivariant Ktheory. Here, one source is the image π∗(K∗G(M)) ⊂ K∗G(M). Unlike the HamiltonianG-space and Hamiltonian LsG-space situations, this source alone is insufficient toensure surjectivity. Additionally, it is conjectured that one needs generators originatingin an appropriate sense from the fibre ΩG.

Bibliography

[AC] A. Adem, F. Cohen. Commuting elements and spaces of homomorphisms. Math.Ann. 338 (2007), no. 3, 587626

[AC2] A. Adem, F. Cohen. Commuting elements and spaces of homomorphisms:Erratum

[AG] A. Adem, J.M. Gomez. Equivariant K-theory of compact Lie group actionswith maximal rank isotropy. pp1-26, in Configuration Spaces: Geometry, Topologyand Combinatorics, Publ. Scuola Normale Superiore, Vol 14 (CRM Series), 2012(Brickhauser).

[AMM] A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued moment maps. . J.Differential Geometry 48:445–495, 1998.

[BJS] T. Baird, L.C. Jeffrey and P. Selick. The space of commuting n-tuples in SU(2).Illinois Journal of Mathematics, Vol 55, Number 3 (2011), 805-813.

[BTW] R. Bott, S. Tolman, and J. Weitsman. Surjectivity for Hamiltonian loop groupspaces. Invent. Math., 155(2):225–251, 2004, math.DG/0210036.

[BZ] J.-L. Brylinski and B. Zhang. Equivariant K-theory of compact connected liegroups. J. of K-theory, 20(1):23–36, 2000.

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[HJS2] M. Harada, L. Jeffrey, P. Selick, The product structure on the equivariant K-theory of the based loop group of SU(2) Quarterly Journal of Mathematics, Vol. 65,Issue 2.

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[HL] M. Harada and G. D. Landweber. Surjectivity for Hamiltonian G-spaces inK-theory. Trans. Amer. Math. Soc., 359:6001–6025, 2007.

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[Ill1] S. Illman Equivariant singular homology and cohomology for actions of compact Liegroups Proceedings of the Second Conference of Compact Transformation Groups(Univ. of Massachusetts, Amherst 1971) Lecture Notes in Mathematics, Vol 298, pp.403-415. Berlin, Heidelberg, New York: Spring 1972

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