COHOMOLOGY AND K-THEORY OF COMPACT LIE...

COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS

CHI-KWONG FOK

Abstract. In this expository article, we review the computation of the (de Rham) co-

homology of compact connected Lie groups and the K-theory of compact, connected and

simply-connected Lie groups.

Contents

1. Introduction 2

2. Cohomology of compact Lie groups 3

3. The cohomology ring structure 11

3.1. Hopf algebras and their classification 12

3.2. The map p∗ 15

4. Elements of K-theory 24

5. K-theory of compact Lie groups 25

References 28

Date: July 19, 2010.

1

2 CHI-KWONG FOK

1. Introduction

In this expository article we give an account of the computation of the (de Rham)

cohomology and K-theory of compact Lie groups based on the classical work of [CE] and

[A1], as well as the article [R]. These become standard results in the algebraic topology of

compact Lie groups.

For the computation of the cohomology groups of compact Lie groups, we demonstrate

the use of the averaging trick to show that it suffices to compute the cohomology using

left-invariant differential forms, which in turn have a natural correspondence with skew-

symmetric multilinear forms on the Lie algebra of the Lie group. In this way the whole

situation is reduced to computing the Lie algebra cohomology. One may further restrict to

the bi-invariant differential forms, the advantage of which is that these forms are automat-

ically closed. We shall then derive some interesting results about the topology of compact

Lie groups using this elegant technique.

As to the cohomology ring structure, we first review the basics of Hopf algebras, which

include the cohomology of compact connected Lie groups as motivating examples. Hopf’s

determination of the general ring structure of cohomology of compact connected Lie groups

by means of a result on the classification of Hopf algebras will then be presented. To obtain

more information about the cohomology ring we shall appeal to the well-known map

p : G/T × T → G(1)

(g, t) 7→ gtg−1

where T is a maximal torus of G. Using p, we can deduce information about H∗(G,R) by

computing H∗(G/T,R) and H∗(T,R). More precisely, it will be shown that

p∗ : H∗(G,R)→ (H∗(G/T,R)⊗H∗(T,R))W(2)

is a ring isomorphism, where W is the Weyl group of G. We will describe the W -module

structure of H∗(G/T,R) using Morse theory. Making use of invariant theory, in particular

the famous theorem by Borel that H∗(G/T,R) is isomorphic to the space harmonic polyno-

mials on t and Solomon’s result on W -invariants of differential forms on t with polynomial

coefficients(c.f. [So]), Reeder interpreted the right-hand side of (2) as W -invariant subspace

of differential forms with harmonic polynomial coefficients and gave a detailed description

of it(c.f. [R]). It will be shown in the end that we have

Theorem 1.1. If G is compact and connected, then H∗(G,R) is an exterior algebra gen-

erated by elements from odd cohomology groups.

COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 3

For K-theory, we will be only concerned about simply-connected compact Lie groups.

The structure of the K-theory is immediate once we know that K∗(G) is torsion-free and

apply the fact that rational cohomology ring and rational K-theory of a finite CW -complex

are isomorphic through the Chern character(c.f. [AH]). In fact

Theorem 1.2. If G is a compact, simply-connected Lie group, then K∗(G) is an exterior

algebra generated by elements in K−1(G) induced by fundamental representations of G.

The proof from [A1] will be reproduced, with the technical details coming from K-theory

carefully explained.

2. Cohomology of compact Lie groups

The treatment of this section is mainly based on [CE]. Let G be a compact connected

Lie group. We denote the space of differential n-forms of G by Ωn(G), and use Lg to mean

the map of left multiplication by g. A differential form ω ∈ Ω∗(G) is left-invariant if

L∗gω = ω for all g ∈ G

The space of left-invariant n-forms is a subspace of Ωn(G), which will be denoted by ΩnL(G).

Lemma 2.1. If ω ∈ ΩnL(G), then dω ∈ Ωn

L(G).

Proof. Note that for all g ∈ G,

L∗gdω = dL∗gω = dω

Lemma 2.2. If ω1 ∈ ΩnL(G), ω2 ∈ Ωm

L (G), then ω1 ∧ ω2 ∈ Ωm+nL (G).

Proof. Simply note that L∗g(ω1 ∧ ω2) = L∗gω1 ∧ L∗gω2.

By Lemma 2.1, Ω∗L(G) is a complex with differential d. Let H∗L(G) be the cohomology

of Ω∗L(G). Lemma 2.2 implies that H∗L(G) has a ring structure induced by wedge product.

Since Ω∗L(G) is a subspace of Ω∗(G), there is an inclusion map ι : Ω∗L(G) → Ω∗(G),

which induces

ι∗ : H∗L(G)→ H∗(G,R)

It is plain that ι∗ is a ring homomorphism. In fact we have

Theorem 2.3. ι∗ is a ring isomorphism.

4 CHI-KWONG FOK

Before giving a proof of theorem we shall introduce the ‘averaging trick’. Let ω ∈ Ωn(G).

Define

Iω =

∫GL∗gωdg

where dg denotes the normalized Haar measure of G. By the translation invariance of dg,

Iω ∈ ΩnL(G). One can easily show that the averaging operator I is linear, and commutes

with d and pullbacks.

Proof of Theorem 2.3. First we shall show that ι∗ is injective. Suppose that ω ∈ ΩnL(G)

and ω = dµ for some µ ∈ Ωn−1(G). Then

ω = Iω = Idµ = dIµ

So ω ∈ d(Ωn−1L (G)) and thus [ω] = 0 in H∗L(G).

To show that ι∗ is surjective, we shall, for any ω ∈ Ωn(G) with dω = 0, find µ ∈ Ωn−1(G)

such that ω + dµ ∈ ΩnL(G).

Claim 2.4. There exists µ ∈ Ωn−1(G) such that ω + dµ = Iω.

Proof. Recall the de Rham theorem, which asserts, for compact manifold M , the nonde-

generate pairing between the cohomology class [ω] ∈ Hn(M,R) and the homology class

[c] ∈ Hn(M,R) represented by a linear combination of n-chains c =∑

i riZi, where Zi is

closed n-dimensional submanifolds. The pairing is given by

〈[ω], [c]〉 =∑i

ri

∫Zi

ω

If we can show that ∫Zω − Iω = 0(3)

for any n-dimensional submanifold, then [ω − Iω] = 0 and hence ω − Iω is exact.

Note that, since G is connected, there exists a continuous path s : I → G between e and

g. Thus Lg is homotopic to IdG. It follows that the induced map

Lg∗ : Hn(G,R)→ Hn(G,R)

is identity, and that [gZ] = [Z] for any n-dimensional submanifold Z. With this in mind,

Equation (3) can be proved as follows.


∫Z

(ω − Iω) =

∫Zω −

∫Z

∫GL∗gωdg

=

∫Zω −

∫G

∫ZL∗gωdg (Fubini’s theorem)

=

∫Zω −

∫G

∫gZωdg

=

∫Zω −

∫G

∫Zωdg

= 0

Theorem 2.3 implies that it is sufficient to use only the left-invariant forms on G to

compute its cohomology. By way of left invariance, we can further restrict our attention

to skew-symmetric multilinear forms on g, the Lie algebra of G, as one can easily observe

that

Lemma 2.5. The map

ϕ : Ω∗L(G)→ (

∗∧g)∗

ω 7→ ω := ω|∧n TeG

is a ring isomorphism.

Proof. Note that ϕ is linear. Surjectivity is easy: given α ∈ (∧n g)∗, define ω ∈ Ωn(G)

such that

ωg((X1)g, · · · , (Xn)g) = α(Lg−1∗(X1)g, · · · , Lg−1∗(Xn)g)

for any vector fields Xi, 1 ≤ i ≤ n, on G. ω ∈ ΩnL(G) because

(L∗gω)h((X1)h, · · · , (Xn)h) = ωgh(Lg∗(X1)h, · · · , (Lg)∗(Xn)h)

= α(Lh−1∗(X1)h, · · · , Lh−1∗(Xn)h)

= ωh((X1)h, · · · , (Xn)h)

6 CHI-KWONG FOK

Let ω ∈ ΩnL(G) such that ω = 0. Then for any (X1)g, · · · , (Xn)g ∈ TgG,

ωg((X1)g, · · · , (Xn)g) = (L∗gω)e(Lg−1∗(X1)g, · · · , Lg−1∗(Xn)g)

= ωe(Lg−1∗(X1)g, · · · , Lg−1∗(Xn)g)

= 0

Note that ϕ is also a ring homomorphism. Together with the above we have shown that

indeed ϕ is a ring isomorphism.

Recall that the differential d on Ωn(G) can be explicitly defined by

dω(X1, · · · , Xn+1) =n+1∑i=1

(−1)iXiω(X1, · · · , Xi, · · · , Xn+1) +(4) ∑i<j

(−1)i+jω([Xi, Xj ], X1, · · · , Xi, · · · , Xj , · · · , Xn+1)

where X1, · · · , Xn+1 are vector fields on G.

Lemma 2.6. The map δ : (∧n g)∗ → (

∧n+1 g)∗ which makes the following diagram com-

mutes

ΩnL(G)

d //

ϕ

Ωn+1L (G)

ϕ

(∧n g)∗

δ// (∧n+1 g)∗

is given by

δα(ξ1, · · · , ξn+1) =∑i<j

(−1)i+jα([ξi, ξj ], ξ1, · · · , ξi, · · · , ξj , · · · , ξn+1)

Proof. Let (Xi)g = Lg∗(ξi) be the left-invariant vector field which restricts to ξi ∈ TeG ∼= g

at the identity e. Then

(ϕdω)(ξ1, · · · , ξn+1) = dω(X1, · · · , Xn+1)((Xi)g = Lg∗(ξi))

=∑i<j

(−1)i+jω([Xi, Xj ], X1, · · · , Xi, · · · , Xj , · · · , Xn)

(The first term of (4) vanishes as both ω and Xi are left invariant)

=∑i<j

(−1)i+jω([(Xi)e, (Xj)e], · · · , (Xi)e, · · · , (Xj)e, · · · , (Xn)e)

= (δϕω)(ξ1, · · · , ξn+1)


Theorem 2.7. H∗L(G) ∼= H∗((∧∗ g)∗, δ)

Proof. By Lemma 2.5 and Lemma 2.6, the two complexes (Ω∗L(G), d) and ((∧∗ g)∗, δ) are

isomorphic.

Definition 2.8. The cohomology ring H∗((∧∗ g)∗, δ) is called the Lie algebra cohomology

of g, denoted by H∗(g).

Corollary 2.9. H∗(G,R) ∼= H∗L(G) ∼= H∗(g), if G is a compact connected Lie group.

Remark 2.10. Corollary 2.9 says that the cohomology of a compact connected Lie group

is completely determined by local information(i.e. the Lie algebra cohomology of g). This

is made possible by virtue of the symmetry of G, as encapsulated in the trick of averaging

left multiplication and the left invariance of forms in Ω∗L(G).

Corollary 2.11. If g1 and g2 are isomorphic Lie algebras, then H∗(G1,R) ∼= H∗(G2,R)

where Gi is a compact connected Lie group such that TeGi = gi, i = 1, 2.

Example 2.12. Since su(2) ∼= so(3), H∗(SU(2),R) ∼= H∗(SO(3),R). Indeed, both are

isomorphic to R[x]/(x2).

Remark 2.13. The corollary becomes false if H∗(Gi,R) is replaced by H∗(Gi,Z), i = 1, 2.

A counterexample is provided by SU(2) and SO(3)(H2(SU(2),Z) = 0 whileH2(SO(3),Z) ∼=Z2).

Example 2.14. Let G = (S1)n, the n-dimensional torus. Then g ∼= Rn with trivial Lie

bracket. So δ ≡ 0 and

H∗((S1)n,R) ∼=∗∧Rn

One can take a step further by considering the complex of bi-invariant differential forms,

i.e. forms that are both left-invariant and right-invariant. Let

Ω∗B(G) = Ω∗L(G) ∩ Ω∗R(G)

denote the space of bi-invariant forms of G. Using the averaging trick as in the proof of

Theorem 2.3, we obtain

Proposition 2.15. The inclusion ι : Ω∗B(G) → Ω∗(G) induces a ring isomorphism ι∗ :

H∗B(G)→ H∗(G,R).

8 CHI-KWONG FOK

Definition 2.16. Let (∧∗ g)

∗Gdenote the subspace invariant under the adjoint action.

Let

ψ : Ω∗B(G)→ (∗∧g)∗G

map ω to its restriction ωe at the identity.

Proposition 2.17. (∗∧g)∗G = α ∈ (

∗∧g)∗|

n∑i=1

α(ξ1, · · · , [ξ, ξi], · · · , ξn) = 0 for any ξ, ξi ∈

g, 1 ≤ i ≤ n

Proof. Suppose∑n

i=1 α(ξ1, · · · , [ξ, ξi], · · · , ξn) = 0 for any ξ, ξi ∈ g for 1 ≤ i ≤ n. For

any given g ∈ G, there exists ξ ∈ g such that exp(tξ), 0 ≤ t ≤ 1, joins e and g, as G is

connected. So

d

dt

∣∣∣∣t=0

α(Adexp(tξ)ξ1, · · · ,Adexp(tξ)ξn)

=

n∑i=1

α(ξ1, · · · , [ξ, ξi], · · · , ξn) = 0

It follows that

α(Adgξ1, · · · ,Adgξn) = α(ξ1, · · · , ξn)

and so α ∈ (∧∗ g)∗G. The converse is easy.

Proposition 2.18. ψ is a ring isomorphism.

Proof. Since ψ is a ring homomorphism, it remains to show that it is both injective and

surjective. Injectivity can be shown in the same way as in the proof of Lemma 2.5. Given

α ∈ (∧∗ g)∗G, let ω ∈ Ω∗(G) be such that

ωg((X1)g, · · · (Xn)g) = α(Lg−1∗(X1)g, · · · , Lg−1∗(Xn)g)

Obviously ω ∈ ΩL(G). Moreover,

ωgh(Rh∗(X1)g, · · · , Rh∗(Xn)g)

=α(Lh−1g−1∗Rh∗(X1)g, · · · , Lh−1g−1∗Rh∗(Xn)g)

=α(Adh−1Lg−1∗(X1)g, · · · ,Adh−1Lg−1∗(Xn)g)

=α(Lg−1∗(X1)g, · · · , Lg−1∗(Xn)g)

=ωg((X1)g, · · · , (Xn)g)

Hence ω ∈ ΩR(G).


Similar to Lemma 2.6, we have

Proposition 2.19. The following diagram

ΩnB(G)

d //

ψ

Ωn+1B (G)

ψ

(∧n g)∗G

δ// (∧n+1 g)∗G

commutes

Corollary 2.20. The two complexes (Ω∗B(G), d) and ((∧∗ g)∗G, δ) are isomorphic, and

hence their cohomology rings H∗B(G) and H∗((∧∗ g)∗G, δ) are isomorphic as well.

Proposition 2.21. For any α ∈ (∧∗ g)∗G, δα = 0.

Proof. Fix k. Then∑j<k

(−1)j+kα([Xj , Xk], X1, · · · , Xj , · · · , Xk, · · · , Xn+1)

+∑j>k

(−1)j+kα([Xk, Xj ], · · · , Xk, · · · , Xj , · · · , Xn+1)

=∑j<k

(−1)k+1α(X1, · · · , Xj−1, [Xk, Xj ], Xj+1, · · · , Xk, · · · , Xn+1)

+∑j>k

(−1)k+1α(X1, · · · , Xk, · · · , [Xk, Xj ], · · · , Xn+1)

=0 by Proposition 2.17

It follows that

0 =

n+1∑k=1

∑j<k

(−1)j+kα([Xj , Xk], X1, · · · , Xj , · · · , Xk, · · · , Xn+1)

+∑j>k

(−1)j+kα([Xk, Xj ], · · · , Xk, · · · , Xj , · · · , Xn+1)

= 2

∑j<k

(−1)j+kα([Xj , Xk], X1, · · · , Xj , · · · , Xk, · · · , Xn+1)

= 2(δα)(X1, · · · , Xn+1)

10 CHI-KWONG FOK

Corollary 2.22. H∗((∧∗ g)∗G, δ) ∼= (

∧∗ g)∗G. Hence H∗(G,R) ∼= (∧∗ g)∗G.

Actually, Corollary 2.22 does not come in handy for explicit computation of H∗(G,R)

in general. We will give a fuller description of H∗(G,R) in the next Section. Nonetheless,

we can still deduce lower dimensional cohomology groups of G easily from Corollary 2.22.

Theorem 2.23. Let G be a compact connected semi-simple Lie group. Then H1(G,R) =

H2(G,R) = 0, H3(G,R) 6= 0.

Proof. If α ∈ g∗G, then α([X1, X2]) = 0 for all X1, X2 ∈ g. Since g is semi-simple, [g, g] = g.

Hence α must be 0 and g∗G = H1(G,R) = 0.

Let β ∈ (∧2 g)∗G. Then

β([X1, X2], X3) + β(X2, [X1, X3]) = 0

On the other hand, Proposition 2.21 implies that

(δβ)(X1, X2, X3) = −β([X1, X2], X3) + β([X1, X3], X2)− β([X2, X3], X1) = 0

The above two equations implies that β([X2, X3], X1) = 0 for all Xi ∈ g, 1 ≤ i ≤ 3. So

(∧2 g)∗G = H2(G,R) = 0.

Finally, consider the 3-form

γ(X1, X2, X3) = B([X1, X2], X3) ∈ (

3∧g)∗G

where B is the Killing form of g. Since B is non-degenerate when g is semi-simple, γ is

not zero.

Theorem 2.24. A compact connected Lie group G is semi-simple iff π1(G) is finite.

Proof. We shall first claim that H1(G,R) = 0 iff π1(G) is finite. Note that H1(G,R) = 0

iff H1(G,Z) is a torsion abelian group. Since G is compact, H1(G,Z) must be finitely

generated. Thus H1(G,Z) is a finite abelian group iff H1(G,R) = 0. By the universal

coefficient theorem,

H1(G,Z) ∼= Hom(H1(G,Z),Z)⊕ Ext(H0(G,Z),Z) ∼= Hom(H1(G,Z),Z)

So H1(G,Z) is a finite abelian group iff H1(G,Z) is also a finite abelian group. But

H1(G,Z) ∼= π1(G)ab = π1(G)(the last equality holds because π1(G) is abelian(c.f. [BtD],

Ch. V, Thm. 7.1)). This establishes the claim. If G is compact, connected and semi-

simple, then by Theorem 2.23, H1(G,R) = 0. So π1(G) is finite by the claim. If π1(G) is


finite, then H1(G,R) = 0, which in turn implies that [g, g] = g. Note that g is a reductive

Lie algebra and can be written as

g = z(g)⊕ h

where h is semi-simple. The condition [g, g] = g forces z(g) = 0. This completes the

proof.

Remark 2.25. From the proof of Theorem 2.24, we can see that if G is compact, connected

and semi-simple, then H1(G,Z) = 0. To compute H2(G,Z), we may again make use of

the universal coefficient theorem to get

H2(G,Z) ∼= Hom(H2(G,Z)⊕ Ext(H1(G,Z),Z)

Since H2(G,Z) must be a finite abelian group, and Hom(·,Z) is either a free abelian group

or 0, Hom(H2(G,Z),Z) = 0 and H2(G,Z) ∼= Ext(H1(G,Z),Z). But H1(G,Z) ∼= π1(G), so

H2(G,Z) = π1(G).

We are now in a position to answer the classical problem that asks which spheres afford

a Lie group structure. It turns out that

Theorem 2.26. S0, S1 and S3 are the only spheres that can be given Lie group structures.

Proof. We know that S0 ∼= Z2 and S1 ∼= U(1). If n ≥ 2 and Sn affords a Lie group structure,

then it must be semi-simple because π1(Sn) is trivial. By Theorem 2.23, H3(S3,R) 6= 0,

so n must be 3. As it turns out, S3 ∼= SU(2).

3. The cohomology ring structure

Determining the ring structure ofH∗(G,R) is a more subtle issue. Nevertheless, a general

description of the algebraic structure of H∗(G,R) has been known for a long time. In this

section, we will first review the basics of Hopf algebras, of which the cohomology ring of any

H-space is a motivating example, and a result, due to Hopf, on the classification of Hopf

algebras over a field of characteristic 0, from which one can easily deduce that H∗(G,R) is

an exterior algebra on l generators, where l is the rank of G. Next we implement a careful

study of the map (1) and H∗(G/T,R) to obtain more information about the l generators,

e.g. the degrees of the generators.

12 CHI-KWONG FOK

3.1. Hopf algebras and their classification. Most of the material in this section is

based on the part of Chapter 3 of [H1] on Hopf algebras. Let X be a connected H-space

equipped with a multiplication map µ : X × X → X. For convenience of exposition,

suppose further that R is a commutative ring with identity and that H∗(X,R) is a free

R-module. By Kunneth formula, H∗(X ×X,R) is isomorphic to H∗(X,R) ⊗R H∗(X,R)

as R-algebras. Composing the isomorphism with

µ∗ : H∗(X,R)→ H∗(X ×X,R)

we have a graded R-algebra homomorphism

∆ : H∗(X,R)→ H∗(X,R)⊗R H∗(X,R)

It is not hard to show that, if α ∈ Hn(X,R), then

∆(α) = α⊗ 1 + 1⊗ α⊕k∑i=1

α′i ⊗ α′′i

where the degrees of α′i and α′′i are less than n.

The above properties of H∗(X,R) led Hopf to define an algebra named after him.

Definition 3.1. A Hopf algebra A over R is a graded commutative R-algebra which

satisfies

(1) A0 = R.

(2) There exists a graded algebra homomorphism ∆, called comultiplication,

∆ : A → A⊗R A

such that if α ∈ An, then

∆(α) = α⊗ 1 + 1⊗ α+k∑i=1

α′i ⊗ α′′i

where the degrees of both α′i and α′′i are less than n.

If ∆(α) = α⊗ 1 + 1⊗ α, then α is called a primitive element.

Example 3.2. Let A = R[α], the polynomial algebra over R generated by an element of

even degree. We shall show that it is a Hopf algebra over R. Let us assume that there

exists a comultiplication ∆. Then ∆(α) = α⊗ 1⊕ 1⊗ α since there is no element in R[α]

with positive degree less than that of α. It follows that

∆(αn) = (α⊗ 1 + 1⊗ α)n =

n∑i=1

(n

i

)αi ⊗ αn−i


∆ thus defined is indeed a comultiplication.

Example 3.3. Let A =∧R(β), the exterior algebra over R generated by one odd degree

element α. Assume that there exists a comultiplication map ∆. Note that

0 = ∆(β2)

= ∆(β)2

= (β ⊗ 1 + 1⊗ β)2

= β2 ⊗ 1 + (β ⊗ 1) · (1⊗ β) + (1⊗ β) · (β ⊗ 1) + 1⊗ β2

= β ⊗ β − β ⊗ β

So ∆ indeed defines a comultiplication, and hence A is a Hopf algebra.

Example 3.4. Let A1 and A2 be Hopf algebras over R. Then so is A = A1 ⊗R A2, with

comultiplication being

∆A = ∆A1 ⊗R ∆A2

As a result, the polynomial algebra R[α1, · · · , αn] with degree of αi even for all i, and∧R(β1, · · · , βm) with degree of βj odd for all j, are Hopf algebras over R. So is their

tensor product R[α1, · · · , αn]⊗R∧R(β1, · · · , βm).

The following theorem is a partial converse of Example 3.4

Theorem 3.5. Let A be a Hopf algebra over a field F of characteristic 0 such that An is

finite dimensional over F for each n. Then A must be one of the following.

(1) A polynomial algebra F [α1, · · · , αn] with degree of αi even for all i.

(2) An exterior algebra∧F (β1, · · · , βm) with degree of βj odd for all j.

(3) The tensor product a polynomial algebra as in (1) and an exterior algebra as in (2).

Proof. Since each graded piece An is finite dimensional over F , there exist x1, · · · , xn, · · ·such that they generate A and |xi| < |xj | if i < j. Let An be the subalgebra generated by

x1, · · · , xn. We may assume that xn /∈ An−1. An is a Hopf subalgebra because ∆(xi) ∈An ⊗F An for 1 ≤ i ≤ n. Consider the multiplication map

f : An−1 ⊗F F [xn]→ An if |xn| is even

or f : An−1 ⊗F∧F

(xn)→ An if |xn| is odd

14 CHI-KWONG FOK

f is surjective by the definition of An. If we can show that f is injective, then An is a

tensor product of a polynomial algebra and an exterior algebra, as An−1 is by inductive

hypothesis.

Let us consider the case where |xn| is even. Suppose that f is not injective, that is,

there is a nontrivial relation∑k

i=0 αixin = 0, with αi ∈ An−1. We may assume that k is

the minimal degree of the equations of any nontrivial relation between xn and elements of

An−1. Let I be an ideal in An generated by the positive degree elements in An−1 and x2n,

and q : An → An/I. Note that xn /∈ I. Consider the composition of maps

An∆−→ An ⊗F An

Id⊗q−→ An ⊗F An/I

We have

(Id⊗ q) ∆(αi) = αi ⊗ 1

(Id⊗ q) ∆(xn) = 1⊗ q(xn) + xn ⊗ 1

=⇒ 0 = (1⊗ q) ∆(k∑i=0

αixin) =

k∑i=0

(αi ⊗ 1) · (1⊗ q(xn) + xn ⊗ 1)i

=k∑i=0

i∑j=0

(i

j

)αix

jn ⊗ q(xn)i−j

=k∑i=0

(αixin ⊗ 1 + iαix

i−1n ⊗ q(xn)) (q(xin) = 0 for i ≥ 2)

=

(k∑i=0

iαi ⊗ xi−1n

)⊗ q(xn)

Thus we have another relation∑k

i=0 iαixi−1n = 0 since q(xn) 6= 0. This relation is nontrivial,

because xn 6= 0, and iαi 6= 0 if αi and i 6= 0 as F is a field of characteristic 0. We get

another nontrivial relation of lower degree, contradicting the minimality of k. Hence the

multiplication map is in fact an algebra isomorphism. The case where |xn| is odd can be

dealt with in a similar way.

Theorem 3.6. If G is a compact connected Lie group of rank l, then H∗(G,R) is an

exterior algebra on l generators of odd degrees.

Proof. Note that H∗(G,R) satisfies the conditions in Theorem 3.4, and that it is finite

dimensional over R because G is compact. Thus H∗(G,R) must be an exterior algebra on

odd degree generators. It remains to show that there are l generators.


Consider the squaring map

f : G→ G

g 7→ g2

Let g0 ∈ G be a regular element, i.e. 〈g0〉 is a maximal torus of G. Then ZG(g0) = 〈g0〉.Any preimage of g0 under f commutes with g0 and so f−1(g0) ⊂ 〈g0〉. It follows that

|f−1(g0)| = 2l, and deg(f) = 2l, which implies that f∗ amounts to multiplication by 2l on

Htop(G,R). Suppose H∗(G,R) is an exterior algebra on m generators β1, · · · , βm of odd

degrees. Assume that |βi| < |βj | if i < j. Note that f∗ = d∗ ∆, where

d∗ : H∗(G×G,R) ∼= H∗(G,R)⊗H∗(G,R)→ H∗(G,R)

is induced by the diagonal embedding d : G→ G×G and amounts to the wedge product

map. So

f∗(βi) = d∗ ∆(βi)

= d∗(1⊗ βi + βi ⊗ 1 +∑j

γ′ij ⊗ γ′′ij) (|γ′ij |, |γ′′ij | < |βi|)

= 2βi +∑j

γ′ijγ′′ij

Note that f∗(βi) = 2βi for i = 1, 2, 3. Since β1β2 · · ·βm ∈ Htop(G,R), we have

2lβ1β2 · · ·βm = f∗(β1β2 · · ·βm)

= (2β1)(2β2)(2β3)(2β4 +∑j

γ′4jγ′′4j) · · · (2βm +

∑j

γ′mjγ′′mj)

= 2mβ1β2 · · ·βm

We conclude that m = l and the proof is complete.

We will give another proof of Theorem 3.6 in Section 3.2.

3.2. The map p∗. Recall that, if g is a Lie algebra of a compact Lie group G, then gC is

a complex reductive Lie algebra. Let t be the Lie algebra of T , and tC its complexification.

The maps adξ : gC → gC, ξ ∈ gC are simultaneously diagonalizable and give the eigenspace

decomposition of gC

gC = tC ⊕⊕α∈∆

gα

where α ∈ ∆ ⊂ t∗C are roots of g satisfying [H,Xα] = α(H)Xα for H ∈ tC, Xα ∈ gα. Note

that dimgα = 1. There exists Hα ∈ tC, α ∈ ∆ and Xα ∈ gα such that

16 CHI-KWONG FOK

(1) α(Hα) = 2

(2) [Xα, X−α] = Hα

(c.f. [Se], Ch. VI, Thm. 2), and g is the real span of iHα, Xα−X−α, i(Xα+X−α)α∈∆(c.f.

[H2], Ch. 3, proof of Thm. 6.3). It follows that t = spanRiHαα∈∆, and

g = t⊕⊕α∈∆+

mα

where ∆+ is the set of positive root, and mα = g ∩ (gα ⊕ g−α) is the real span of the basis

i(Xα + X−α), Xα −X−α. We will denote⊕

α∈∆+ mα by m. The matrix representation

of the action ad(H) on mα with respect to this basis is(0 −iα(H)

iα(H) 0

)whereas that of the adjoint action Adexp(H) is(

cos iα(H) − sin iα(H)

sin iα(H) cos iα(H)

)

Theorem 3.7. If TgTG/T is identified with m, TtT with t and Tgtg−1G with g by left

translation, then

dp(gT,t) : m⊕ t→ g

(X,T ) 7→ Adgt−1g−1(X)−X + Adg(T )

In matrix form,

dp(gT,t) =

(Adgt−1g−1 − Idm 0

0 Adg|t

)

Proof. Identify (X,T ) ∈ m⊕ t with (Lg∗X,Lt∗T ) ∈ TgTG/T ⊕ TtT . Then

dp(gT,t)(Lg∗X,Lt∗T ) =d

ds

∣∣∣∣s=0

gexp(sX)texp(sT )g−1exp(−sX)

= Rtg−1∗Lg∗X +Rg−1∗Lgt∗T − Lgtg−1∗X

= Lgtg−1∗(Adgt−1g−1X −X + AdgT )

The last line is identified with Adgt−1g−1X −X + AdgT .

Corollary 3.8. det dp(gT,t) = det(Adt − Idm).


Proof. Note that det(Adg) = 1 for all g ∈ G because µ : G → R× defined by µ(g) =

det(Adg) is a group homomorphism and G is compact connected. By Theorem 3.7,

det dp(gT,t) = det(Adgt−1g−1 − Idm) det(Adg|t)

= det(Adg(Adt−1 − Idm)Adg−1)

= det(Adt−1 − Idm)

= det(Adt(Adt−1 − Idm))

= det(Idm −Adt)

= det(Adt − Idm) (dimm is even)

Lemma 3.9. If t = expH, H ∈ t, then det(Adt − Idm) =∏α∈∆+ 4 sin2 iα(H)

2 .

Proof.

det(Adt − Idm) =∏α∈∆+

det

(cos iα(H)− 1 − sin iα(H)

sin iα(H) cos iα(H)− 1

)=∏α∈∆+

2(1− cos iα(H))

=∏α∈∆+

4 sin2 iα(H)

2

Suppose g0 ∈ G is a regular value of p. It is well-known that p−1(g0) consists of |W |points. By Lemma 3.9, the determinant of dp at each point in the pre-image must be

positive. Hence degp = |W |. The pull-back formula for integration gives

Theorem 3.10 (Weyl integration formula). Let ωG, ωG/T and ωT be the normalized volume

form of G, G/T and T respectively, and f : G→ C a continuous complex-valued function

on G. Then ∫Gf(g)ωG =

1

|W |

∫G/T×T

f p(gT, t) det(Adt − Idm)ωG/T ∧ ωT

In particular, if f is a class function on G, then∫Gf(g)ωG =

1

|W |

∫Tf(t) det(Adt − Idm)ωT

Lemma 3.11. dimRH∗(G,R) = 2l

18 CHI-KWONG FOK

Proof. By Corollary 2.22, dimH∗(G,R) = dim(∧∗ g)∗G = dim(

∧∗ g)G. Note that (∧∗ g)G

is the trivial subrepresentation of the adjoint representation of G on∧∗ g. Let χ∧∗ g be

the character of this representation. Then

dim(

∗∧g)G =

∫Gχ∧∗ g(g)ωG

=

∫Tχ∧∗ g(t) det(Adt − Idm)ωT

=

∫T

det(Adt + Idm) det(Adt − Idm)ωT

=

∫T

det(Adt2 − Idm)ωT

= 2dimT

∫T

det(Ads − Idm)ωs

= 2l

Now that G and G/T × T are manifolds of the same dimension, and degp = |W | 6= 0,

the induced map

p∗ : H∗(G,R)→ H∗(G/T × T,R)

is injective. There is a W -action on G/T × T defined by

w · (gT, t) = (gw−1T,wtw−1)

and it is easy to see that p(gT, t) = p(w·(gT, t)) for w ∈W . Thus Im(p∗) ⊆ H∗(G/T×T )W .

By abuse of notation we also use p∗ to mean the map

p∗ : H∗(G,R)→ H∗(G/T × T,R)W

We claim that

Theorem 3.12. p∗ is a ring isomorphism.

Before giving a proof of Theorem 3.12, we shall examine H∗(G/T,R) more closely. As

a first shot, we shall employ Morse theory to compute the cohomology groups.

Let 〈·, ·〉 be an Ad(G)-invariant inner product on g. This can be obtained by averaging

any inner product on g over G. Let X ∈ t+ be a regular element in the positive Weyl


chamber, i.e. α(X)i > 0 for all α ∈ ∆+. We let

f : G/T → R

gT 7→ 〈Adg(X), X〉

and claim that it is a Morse function. Suppose g0T is a critical point of f . Then for all

Y ∈ m,

0 = df(Lg0∗Y )

=d

ds

∣∣∣∣s=0

f(g0exp(sY )T )

=d

ds

∣∣∣∣s=0

〈Adg0exp(sY )(X), X〉

= 〈Adg0([Y,X]), X〉

= 〈[Y,X],Adg−10X〉

= 〈Y, [X,Adg−10X]〉 by Ad-invariance of the inner product

Note that m⊥ = t, as

〈t,m〉 = 〈t, [t,m]〉 = 〈[t, t],m〉 = 0

Thus [X,Adg−10

(X)] ∈ t, and Adg−10X ∈ t. g−1

0 T and hence g0T must be a Weyl group

element. The critical points of f are therefore all the Weyl group elements.

The Hessian Hw(Y, Z) for Y,Z ∈ m, w = g0T ∈W is

d

dt

∣∣∣∣t=0

〈Y, [X,Ad(g0exp(tZ))−1(X)]〉

=〈Y, [X, [−Z,Adg−10

(X)]]〉

=− 〈[Y,X], [Z,Adg−10

(X)]〉

=− 〈[X,Y ], [Adg−10

(X), Z]〉

For α ∈ ∆+,

Hw(Xα −X−α, Xα −X−α) = α(X)α(Adg−10

(X))〈i(Xα +X−α), i(Xα +X−α)〉 6= 0

Hw(i(Xα +X−α), i(Xα +X−α)) = α(X)α(Adg−10

(X))〈Xα −X−α, Xα −X−α〉 6= 0

So Hw is nondegenerate for all w ∈ W and f is indeed a Morse function. The index of f

at w is twice the number of positive roots α such that

α(X)α(Adg−10

(X)) < 0

20 CHI-KWONG FOK

Since α(X)i > 0, and α(Adg−1

0(X)) = (w · α)(X), the index is also twice the number

of positive roots α such that w · α is also positive. Let Index(w) = 2m(w). Then the

Poincare polynomial of H∗(G/T,R) is P (t) =∑

w∈W t2m(w), and the Euler characteristic

is χ(G/T ) = P (−1) = |W |.

The W -action on G/T given by w · (gT ) = gw−1T induces a W -representation on

H∗(G/T,R). The trace of w on H∗(G/T,R) is just the Lefschetz number of the action

of w because H∗(G/T,R) is concentrated on even degrees. If w 6= 1, then it has no fixed

points on G/T , and therefore its trace is 0 by Lefschetz Fixed Points Theorem. If w = 1,

then the trace is just the Euler characteristic |W |. As a result,

Proposition 3.13. H∗(G/T,R) is a regular representation of W .

Proof of Theorem 3.12. Since p∗ is injective, it suffices to show that dimH∗(G/T×T,R)W =

dimH∗(G,R) = 2l. By Kunneth formula, H∗(G/T×T,R)W = (H∗(G/T,R)⊗H∗(T,R))W .

So

dim(H∗(G/T,R⊗H∗(T,R))W

=1

|W |∑w∈W

χH∗(G/T,R)⊗H∗(T,R)(w)

=1

|W |∑w∈W

χH∗(G/T,R(w)χH∗(T,R)(w)

=1

|W |χH∗(G/T,R)(1)χH∗(T,R)(1)

=dimH∗(T,R)

=2l

Let S be the real polynomial ring S∗(t∗) on t, and I be the ideal in S generated by

W -invariant polynomials. A famous theorem of Borel describes the ring structure of

H∗(G/T,R) using

Theorem 3.14 (Borel). There is a degree-doubling W -equivariant ring isomorphism

c : S/I → H∗(G/T,R)

where c(λ)(X,Y ) = λ([X,Y ]) for X,Y ∈ m and deg(λ) = 1.


We refer the reader to [R] for a proof of Theorem 3.14 using invariant theory. Here we

would like to show that H∗(G/T,R) is isomorphic to S/I using equivariant cohomology.

For a review of equivariant cohomology the reader is refer to the Appendix.

Consider G/T with T acting on it by left translation. Then

H∗T (G/T,R) ∼= H∗T×T (G,R)

where T × T acts on G by

(t1, t2) · g = t1gt2

Next note that G is diffeomorphic to the orbit space of the G-action on G × G given by

g · (g1, g2) = (g1g, g−1g2). We get

H∗T×T (G,R) ∼= HT×T×G(G×G,R)

where T × T ×G acts on G×G by

(t1, t2, g) · (g1, g2) = (t1g1g, g−1g2t2)

So

H∗T×T×G(G×G,R) ∼= H∗G(G/T ×G/T,R)

∼= H∗G(G/T,R)⊗H∗G(pt,R) H

∗G(G/T,R)

∼= H∗T (pt,R)⊗H∗G(pt,R) H

∗T (pt,R)

It is well-known that

H∗T (G/T,R) ∼= H∗T (pt,R)⊗R H∗(G/T,R)

as H∗T (pt,R)-modules, because G/T is a T -Hamiltonian manifold. Therefore

H∗(G/T,R) ∼= R⊗H∗G(pt,R) H

∗T (pt,R)

∼= H∗T (pt,R)/〈r∗HT (pt,R)〉

where r∗ : H∗G(pt,R)→ H∗T (pt,R) is the map induced by restricting G-action to T -action.

By the abelianization principle(c.f. [AB]), r∗ is injective and its image is H∗T (pt,R)W .

Identifying H∗T (pt,R) with S, we get Theorem 3.14. Combining Theorem 3.12 and 3.14,

and regarding H∗(T,R) =∧∗ t∗ as differential forms, we have

Theorem 3.15. H∗(G,R) ∼= ((S/I)(2) ⊗∧∗ t∗)W where the RHS is the space of W -

invariant differential forms with coefficients in S/I. Here (S/I)(2) means S/I with degree

of each polynomial doubled.

22 CHI-KWONG FOK

It is a classical result in invariant theory, due to Chevalley, that the W -invariant poly-

nomial SW is generated by l algebraically independent polynomials F1, · · · , Fl. In other

words

SW = R[F1, · · · , Fl]

Definition 3.16. The exponents mi, 1 ≤ i ≤ l of G are defined to be

mi = degFi − 1

Remark 3.17. It is known that∑l

i=1mi = 12dimG/T and

∏li=1(1 +mi) = |W |.

Theorem 3.18 (Solomon [So]). The space (S ⊗∧∗ t∗)W of W -invariant differential forms

with polynomial coefficients is an exterior algebra over SW generated by dF1, · · · , dFl.

Before proving Theorem 3.18, we need a

Lemma 3.19. Let J = Jac(F1, · · · , Fl). Then w · J = det(w)J . Here det(w) means the

determinant of w as a linear transformation on t∗. If R ∈ S and satisfies w ·R = det(w)R,

then R = SJ for some S ∈ SW .

Proof. A classical result in invariant theory asserts that, if α1, · · · , αl ∈ t∗ are simple roots,

J = cα1α2 · · ·αl

for some c ∈ R. Let u ∈ S such that w · u = det(w)u. As α1, · · · , αl forms a basis

for t∗, those simple roots can be regarded as coordinate functions on t, and we write

u = u(α1, · · · , αl), a polynomial of α1, · · · , α. Note that

u(α1, · · · , αl−1, 0) = u(α1(Hl), · · · , αl−1(Hl), αl(Hl))

= u(α1(sHl(Hl)), · · · , αl−1(sHl

(Hl)), αl(sHl(Hl)))

= (sαl· u)(α1(Hl), · · · , αl(Hl))

= −u(α1, · · · , αl−1, 0)

So u(α1, · · · , αl−1, 0) = 0, and αl divides u. Similar reasoning implies that αi divides u for

1 ≤ i ≤ l. Since α1, · · · , αl are coprime, J divides u.

Proof of Theorem 3.18. Let L = S(0) be the field of fraction of S. We shall first show that

dFi1 ∧ · · ·∧dFir1≤r≤l is linearly independent over S and L. Suppose∑ki1···irdFi1 ∧ · · ·∧

dFir = 0. Multiplying dFj1 ∧ · · · ∧ dFjl−r, where j1, · · · , jl−r = 1, · · · , l \ i1, · · · , ir,

we have

±ki1···irJdx1 ∧ · · · ∧ dxl = 0


Since F1, · · · , Fl are algebraically independent, J 6= 0. Thus ki1···ir = 0 as both S and Lhave no zero divisors.

Let ω ∈ (S ⊗∧p t∗)W . As S ⊂ L, we may regard ω as a W -invariant differential p-form

with coefficients being rational functions on t. Note that L⊗∧p t∗ is an L-vector space of

dimension

(l

p

), and thus dFi1 ∧ · · · ∧ dFip spans L ⊗

∧p t∗ over L. Write

ω =∑ si1···ir

ti1···irdFi1 ∧ · · · ∧ dFir

where si1···ir and ti1···ir ∈ S. By the W -invariance of ω,si1···irti1···ir

is also W -invariant. Multi-

plying dFj1 ∧ · · · ∧ dFjl−rwith j1, · · · , jl−r = 1, · · · , l \ i1, · · · , ir, we get

ui1···irdx1 ∧ · · · ∧ dxl =si1···irti1···ir

Jdx1 ∧ · · · ∧ dxl

for some ui1···ir ∈ S. Comparing coefficients, we have

ui1···ir ti1···ir = si1···irJ

Note that w · ui1···ir = det(w)ui1···ir . By Lemma 3.19, ui1···ir = Pi1···irJ for some Pi1···ir ∈SW . It follows that Pi1···ir =

si1···irti1···ir

and the proof is complete.

Corollary 3.20. (S/I ⊗∧∗ t∗)W is an exterior algebra over R with generators

dFi ∈ ((S/I)mi ⊗1∧t∗)W

where dFi means dFi with polynomial coefficients modulo I.

Theorem 3.21 ([R]). If G is compact and connected, then H∗(G,R) is an exterior algebra

generated by elements of degree 2mi + 1, which correspond to dFi ∈ ((S/I)2mi ⊗R∧1 t)W

with the degree of polynomial coefficients doubled.

Example 3.22. Let G = SU(n), and T be the subgroup of diagonal matrices, which is of

dimension n− 1.

t =

ix1

. . .

ixn

∣∣∣∣∣∣∣∣xi ∈ R,

n∑i=1

xi = 0

S ∼= R[x1, · · · , xn]/〈x1 + · · ·+ xn〉

SW ∼= R[σ2, · · · , σn]

24 CHI-KWONG FOK

where σi =∑

1≤j1<···<ji≤n∏ik=1 xjk is the i-th elementary symmetric polynomial. Hence

the exponents mi are 1, 2, · · · , n− 1. By Theorem 3.14,

H∗(G/T,R) ∼= R[x1, · · · , xn]/〈σ1, · · · , σn〉 ∼= R[x1, · · · , xn]/〈(1 + x1) · · · (1 + xn) = 1〉

If we think of SU(n)/T as the full flag manifold F l(Cn) = 0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂Vn−1 ⊂ Vn = Cn|dimVi = i, then xi = c1(Li/Li−1), where

Li = ((V0, V1, · · · , Vn), v) ∈ F l(Cn)× Cn|v ∈ Vi

and by Whitney Product Formula, the single relation (1 + x1) · · · (1 + xn) = 1 translates

to the fact that the direct sum⊕n

i=1 Li/Li−1 is isomorphic to the trivial rank n complex

vector bundle on F l(Cn). By Theorem 3.21, H∗(SU(n),R) is an exterior algebra on n− 1

generators of degrees 3, 5, · · · , 2n− 1.

4. Elements of K-theory

Let X be a compact topological space and Vect(X) be the category of isomorphism

classes of (finite rank) complex vector bundles over X. Note that Vect(X) is a monoid

with direct sum being binary operation. Let

S(X) = [E]− [F ]|[E], [F ] ∈ Vect(X)

We say [E1] − [F1] ∼ [E2] − [F2] if there exists [G] ∈ Vect(X) such that [E1 ⊕ F2 ⊕ G] =

[E2 ⊕ F1 ⊕G].

Definition 4.1. K(X) := S(X)/ ∼. In other words, K(X) is the Grothendieck group of

Vect(X).

Definition 4.2. Let dim : K(X)→ Z be the group homomorphism which sends the class

of a vector bundle to its rank. Define the reduced K-theory K(X) to be the kernel of dim.

Definition 4.3. K0(X) := K(X), K−q(X) := K(Sq ∧ X), where ∧ means the smash

product.

One can make⊕∞

q=0K−q(X) into a ring using tensor product of vector bundles(c.f.

[H1]). The renowned Bott periodicity states that K−q(X) ∼= K−q−2(X) for all q ≤ 0.

Definition 4.4. K∗(X) := K0(X) ⊕ K−1(X), with ring structure induced by tensor

product of vector bundles.


K∗ is a Z2-graded generalized cohomology theory. Let U(∞) be the direct limit of U(n)

as n tends to infinity, where the morphism U(n) → U(m) for n ≤ m is inclusion. It is

well-known that Z×BU(∞) and U(∞) are classifying spaces of K0 and K−1 respectively.

Example 4.5. K∗(S2) ∼= Z[H]/(H − 1)2 as rings, where 1 ∈ Z represents the trivial line

bundle and H = O(1).

5. K-theory of compact Lie groups

From now on we assume that G is a simply-connected compact Lie group. This section

is mainly taken from [A1] and [H3]. Let ρ1, · · · , ρl be the l fundamental representations of

G. A representation ρ : G → U(n), composed with the inclusion U(n) → U(∞), defines

an element in K−1(G), which we denote by β(ρ). The main result of this section is

Theorem 5.1. If G is a simply-connected compact Lie group, then

K∗(G) ∼=∧

(β(ρ1), · · · , β(ρl))

We postpone the proof of Theorem 5.1 to the end of this section. Consider p∗ : K∗(G)→K∗(G/T×T ). Note that K∗(G/T ) is torsion-free because G/T can be given a CW-complex

structure consisting of only even-dimensional cells. The same is true of K∗(T ) by Lemma

5.8. By Kunneth formula for K-theory, K∗(G/T × T ) ∼= K∗(G/T )⊗K∗(T ).

Lemma 5.2. Consider p∗ : K∗(G)→ K∗(G/T×T ) ∼= K∗(G/T )⊗K∗(T ). Then p∗(β(ρ)) =∑ni=1 α(µj)⊗ β(µj) where µj’s are all the weights of ρ, and

α : R(T )→ K∗(G/T )

is defined by α(µ) = [G×T Cµ].

Proof. Note that K−1(G) ∼= K(S(G)), where S(G) is the unreduced suspension of G. We

would like to construct β(ρ) explicitly as a (virtual) vector bundle over S(G). Consider

the principal G-bundle G∗G→ S(G), where G∗G = sg1 + (1−s)g2|g1, g2 ∈ G, s ∈ [0, 1]and G acts on G ∗ G by g · (sg1 + (1 − s)g2) = sg1g + (1 − s)g−1g2. Note that it is the

pullback of EG→ BG through the canonical embedding

S(G) = G ∗G/G → BG = limn→∞

G ∗ · · · ∗G︸︷︷︸n times

/G

. We have that

β(ρ) = [(G ∗G)×G Vρ]− [Cdim(ρ)]

26 CHI-KWONG FOK

Consider the following diagram

G× (T ∗ T )m //

θ

G ∗G

ψ

G/T × S(T )

f//

h

G ∗G/T

q

S(G/T × T )

Sp// S(G)

where m is defined by (g, st1 + (1− s)t2) 7→ sgt1 + (1− s)t2g−1, the various vertical maps

projection maps of fiber bundles, and f and Sp are defined in such a way that the above

diagram commutes. Note that

f∗[(G ∗G)×T Cµ] = [(G× (T ∗ T )×T×T Cµ] = α(µ)⊗ (1 + β(µ))

h∗(Sp)∗[(G ∗G)×G Vρ] = f∗q∗[(G ∗G)×G Vρ]

= f∗

(n∑i=1

[(G ∗G)×T Cµi ]

)

=

n∑i=1

α(µi)⊗ (1 + β(µi))

= [Cdim(ρ)] +n∑i=1

α(µi)⊗ β(µi)

Definition 5.3. Let a :=∏li=1 β(ρi) ∈ K∗(G).

Proposition 5.4.

∫G/T×T

ch(p∗(a)) = |W |.

Proof. By Lemma 5.2, p∗(β(ρi)) =∑m

j=1 α(λij)⊗ β(λij), where λij are the weights of the

fundamental representation ρi and mi is its dimension. We first prove a

Claim 5.5.l∏

i=1

mi∑j=1

(λij ⊗ β(λij)) =

(∑w∈W

det(w)w · ρ

)⊗

l∏i=1

β($i)

where ρ = 12

∑α∈R+ α and $i is the i-th fundamental weight.


We have∫G/T×T

ch(p∗(a)) =

∫G/T×T

ch

l∏i=1

mi∑j=1

α(λij)⊗ β(λij)

=

∫G/T×T

ch

(α⊗ Id)

l∏i=1

mi∑j=1

(λij ⊗ β(λij))

=

∫G/T×T

ch

((α⊗ Id)

(∑w∈W

det(w)w · ρ⊗l∏

i=1

β($i)

))

=

∫G/T×T

ch

(α

(∑w∈W

det(w)w · ρ

))⊗ ch

(l∏

i=1

β($i)

)

=

∫G/T

ch

(α

(∑w∈W

det(w)w · ρ

))×∫T

ch

(l∏

i=1

β($i)

)= χ(G/T ) · 1

= |W |

Proposition 5.6.

∫G

ch(a) = 1.

Proof.

deg(p)

∫G

ch(a) =

∫G/T×T

ch(p∗(a)) = |W |

and deg(p) = |W |.

Lemma 5.7 (a special case of [H3], Theorem A(i)). K∗(G) is torsion-free.

Before proving Lemma 5.7 we show

Lemma 5.8 ([H3]). If X is a finite CW -complex, and K∗(X) has p-torsion, then so does

H∗(X,Z).

Proof. Let Qp be Z localized at the prime p. If H∗(X,Z) has no p-torsion, then the

homomorphism of spectral sequences for K∗(X)⊗Qp and K∗(X)⊗Q

Er(X,Q)→ Er(X,Q)

is injective when r = 2, as E2(X,Qp) = H∗(X,Qp) and E2(X,Q) = H∗(X,Q). By a result

of Atiyah-Hirzebruch’s(c.f. [AH], p. 19), Er(X,Q) collapses on the E2-page. Induction on

28 CHI-KWONG FOK

r gives that Er(X,Qp) also collapses on the E2-page. Now the associated graded group of

K∗(X)⊗Qp is E2(X,Qp) = H∗(X,Qp) which has no p-torsion. Therefore K∗(X,Qp) has

no p-torsion and so does K∗(X).

Sketch of proof of Lemma 5.7. By virtue of Lemma 5.8, it suffices to show that, even if

H∗(G,Z) has p-torsion, then K∗(G) has no p-torsion. This can be done using a result of

Borel’s which give an exhaustive list of prime p and simple, simply-connected compact Lie

group G such that H∗(G,Z) has p-torsion, and showing that K∗(G) has no p-torsion case

by case. We refer the reader to [H3], III.1 for detailed proof.

Proof of Theorem 5.1. From the proof of Theorem 3.12, we know dimH∗(G,Q) = 2l.

Chern character gives a ring isomorphism between K∗(G) ⊗ Q and H∗(G,Q)(c.f. [AH]).

As K∗(G) is torsion-free by Lemma 5.7, it is a free abelian group of rank 2l. Let Λ =∧(e1, · · · , el) be the exterior algebra generated by e1, · · · , el over Z. Define

j : Λ→ K∗(G)

ei 7→ β(ρi)

Proposition 5.6 implies that j is an injective ring homomorphism. Since both Λ and K∗(G)

have the same rank, j(Λ) has a finite index in K∗(G). Note that one can use the K-theory

pushforward(or the index map)

f! : K∗(G)→ K∗(pt) ∼= Z

defined by f!([E]) =∫G ch(E)td(G) =

∫G ch(E)(td(G) = 1 as G is parallelizable), to define

k : K∗(G)→ Hom(Λ,Z)

such that

k(x)(y) = f!(xj(y)), x ∈ K∗(G), y ∈ Λ

We have k j(∧i∈I ei)(

∧j∈J ej) = ±1 if I ∪ J = 1, · · · , l. So k j is an isomorphism. It

follows that j(Λ) is a direct summand of K∗(G). Being of finite index in K∗(G), j(Λ) is

actually isomorphic to K∗(G). This completes the proof.

References

[A1] M. F. Atiyah, On the K-theory of compact Lie groups, Topology, Vol. 4, 95-99, Pergamon Press,

1965.

[A2] M. F. Atiyah, K-theory, W. A. Benjamin, Inc., 1964.

[AB] M. F. Atiyah, R. Bott,


[AH] M. F. Atiyah, F. Hirzebruch, Vector bundles and homogeneous spaces, Proceedings of Symposia in

Pure Math, Vol. 3, Differential Geometry, AMS, 1961.

[BtD] T. Brocker, T. tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics,

Vol. 98, Springer-Verlag, Berlin-Heidelberg-New York, 1985.

[CE] C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Transactions of the

American Mathematical Society, Vol. 63, No. 1., 85-124, Jan., 1948.

[GKM]

[GS] V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer-Verlag

1999

[H1] Allen Hatcher, Vector bundles and K-theory, available at

http://www.math.cornell.edu/ hatcher/VBKT/VBpage.html

[H2] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathe-

matics, Vol. 34, AMS, 2001

[H3] L. Hodgkin, On the K-theory of Lie groups, Topology Vol.6, pp. 1-36, Pergamon Press, 1967.

[R] M. Reeder, On the cohomology of compact Lie groups, L’Enseignement Mathematique, Vol. 41, 1995.

[Se] J-P. Serre, Complex semisimple Lie algebras, translated from the French by G. A. Jones, Springer

Monograph in Mathematics, Springer-Verlag Berlin Heidelberg, 2001

[So] L. Solomon, Invariants of finite reflection groups, Nagoya J. Math. 22, 57-64, 1963.

COHOMOLOGY AND K-THEORY OF COMPACT LIE...

Documents

Transcript of COHOMOLOGY AND K-THEORY OF COMPACT LIE...