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
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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).
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
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.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 15
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).
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 17
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
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 19
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.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 21
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
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.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 25
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
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 $i is the i-th fundamental weight.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 27
We have∫G/T×T
ch(p∗(a)) =
∫G/T×T
ch
l∏i=1
mi∑j=1
α(λij)⊗ β(λij)
=
∫G/T×T
ch
(α⊗ Id)
l∏i=1
mi∑j=1
(λij ⊗ β(λij))
=
∫G/T×T
ch
((α⊗ Id)
(∑w∈W
det(w)w · ρ⊗l∏
i=1
β($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,
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS 29
[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