HASHIMOTO, SHIN

Osaka City University Repository

Hashimoto, S.
Osaka J. Math.
23 (1986), 299-305
ON Ad -COHOMOLOGY GROUPS OF THE
CLASSIFYING SPACES OF COMPACT LIE GROUPS
SHIN HASHIMOTO
Dedicated to Professor Nobuo Shimada on his sixtieth birthday
(Received February 12, 1985)
1. Introduction
Let k be a positive integer and R a proper subring of O which contains
k~ι. We consider the complex j£*-cohomology theory, the Bott element β
and the Adams operation ψk. Since ψk(β)=k β it induces the stable Adams
operation ^r\
m
 in K2m(X; R), the i?-coefficient i£*-cohomology theory. We
consider the secondary cohomology theory [9] associated to the stable operation
1—ψ| and denote the theory by Ad*(X; R). That is, Ad* is a cohomology
theory satisfying the following exact sequence
- -> Ad*(X; R) - * K*(X; R) -Ξ%. K*(X; R) -> Ad*+1(X; R) - * ••• .
When k is a prime power, the associated connective theory of Ad*(X; R) co-
incides with the algebraic iC-cohomology theory defined by Quillen [6] and
Seymour [8]. We consider a compact lie group G and its classifying space
BG. As is well known, K2m(BG; R) is a torsion free group and K2m+\BG; R)
= 0 . So it is easy to see that Ad2m(BG; R) is a torsion free group. We denote
the connected component of the unity by G
o
, which is a closed normal sub-
group of G and we consider the group of connected components G/GQ. Main
result is the following.
Theorem, (i) For any integer m Φ 0, Ad2m(BG R)=0.
(ii) When \ GjGQ\ is invertible in Ry then Ad°(BG; R)=0.
In section 2, we prove the Theorem for finite groups by considering their
cyclic subgroups. In section 3, we prove the Theorem for compact connected
Lie groups by considering their maximal tori. In section 4, we prove the
Theorem for general compact Lie groups by considering their Cartan sub-
groups.
The author expresses his hearty gratitude to Professors H. Minami, Z.
Yosimura and A. Kono for their invaluable suggestions.
300 S. HASHIMOTO
2. Cyclic groups and finite groups
Let R(G) be the complex representation ring of the compact Lie group
G. The augmentation ideal I=I(G) is the kernel of the induced ring homo-
morphism R(G)->R({1}). Let/) be a prime number. To compute Ad*{BG\
R), we use
Proposition 2.1 (Atiyah). There holds the natural isomorphism
And when G is a p-group, I(G)-adic completion coincides with the p-adic comple-
tion.
Let r be a positive integer. We consider the case G=Zjpr, the cyclic
group of order/. Since R(Z/pr)=Z[
v
]l(
v
pr
-l), K(BZ/pr) is a free ^-module
with basis {η' — l; l^j^pr—l}y where η is the canonical line bundle over
BZ/pr. In cases p is not invertible in R>
f; R) =
and in cases p is invertible in R>
So we have
Proposition 2.2. Let p be a prime number which is invertible in R. Then,
Ad2m(BZlpr;R)=0.
As is well known, the &-th Adams operation ψk acts on line bundles as
Λ-times tensorial product. So when k is prime to p, ψk acts on the basis
{rf—1 l^j?ίpr— 1} as a permutation. So we have
Proposition 2.3. Let p be a prime number which is not invertible in R.
In this case, p is prime to k. Then
(i) Ad\BZ\pr R) is a free Zp-module of rank r.
(ii) For any integer mΦO, Ad2m(BZjpr; JR)=O.
For any non-negative integer /, there exist prime numbers ply •••,/?* which
are not invertible in R and an integer l
x
 which is invertible in R satisfying /
=pril*p22—prttmli. We consider a cyclic group C=Zjl. Since Zjl^
X ••• xZ/p'txZ/li, we use the results of Mahammed [5] and have
Lemma 2.4. K(BZfi)=@ K(BZ\p\i)®K(BZ\l^)
» = 1
Since K((BZ/l)(n)) is a finite group, the lim^term is vanished. Thus
^4d*-CoHOMOLOGY GROUPS 301
K(BZβ)=\im. K((BZIiy>) and K{BZ\l; R)=Yιm K^BZjϊ)^; R). So we have
Corollary 2.5. R{BZ\l; R)=φK(BZIpri R)
ί = l
Proposition 2.2, 2.3 and Corollary 2.5 induces
Corollary 2.6. (i) For any finite cyclic group C, Ad2m(BC; R)=0 for
any integer mΦO.
(ii) When \C\ is invertible in R, then Ad°(BC; R)=0.
Thus we prove the Theorem for finite groups as follows.
Proposition 2.7. (i) For any finite group G, Ad2m(BG; R)=0 for any
integer /wφO.
(ii) When \G\ is invertible in R} then Ad°(BG; R)=0.
Proof. Let {C
λ
; λEΛ} be the representative of the conjugacy class of
cyclic subgroups of G. We consider the following commutative diagram
where horizontal lines are exact:
0 > Ad2m (BG; R) > K2m (BG; R)
I I
0 -> 0 Ad2m (BC
λ
; R) -> 0K 2 m (BC
λ
;R).
λ λ
Since the map
K2m (BG; R)->® K2m (BC
λ
; R)
is an injection, [2], the left vertical homomorphism is also an injection. This
completes the proof.
3. Tori and compact connected Lie groups
Let S2n+1 be the unit sphere in Cn+1y the (n+l)-dimensional complex vector
space. Then S1 is an abelian group. The complex vector space structure
on Cn+1 induces an ^-action on S2n+1 by
s (*o> —>**) = (* *o> — , * • * , ) •
We consider the w-dimensional complex projective space CPn=S2n+1 /S1. The
infinite complex projective space CP°°=colim CPn is the classifying space of S1.
Let Ts be the ^-dimensional torus. Then its classifying space BTS is the
s-times cartesian product of CP°°. As is well known,
K(BT°; Q) = Q[[X1} -, Xs]],
where ξ{ is the canonical line bundle over CP°° and Xi=ξi—1 is the euler class
302 S. HASHIMOTO
of ξi. Let J=(j\, ~',js) be a sequence of non-negative integers. We denote
V(ί,«)=if{/; \J\=m}.
Lemma 3.1. (i) For any non-negative integer m, the groups Ad2m((CPn)s;
Q) and Ad2m+1{{CPn)5; Q) are N(s, m)-tίnιes direct sum of Q for any n^m. More-
over, the inclusion map i: {CPn)s-> {CPn+1)s induces the isomorphism of those groups.
(ii) For any negative integer m, the group Ad2m {{CPn)s: Q)=0.
Proof. K*((CFγ',Q) = QIβ, β~\ Xi, "'> X
s
]l((Xi)n+2', l^i^s). We
consider the basis {β~m-XJ\ l^j^n} of the module K2m{{CPn)s\ Q), where
XJ is the multi index. We put the lexicographical order on the basis and re-
present the operation \—ψk2m by a matrix. This matrix is a triangular matrix
with diagonal entries l—kuι~m. Clearly, the diagonal entry equal zero if and
only if I / | =m.
Corollary 3.2. (i) For any non-negative integer m, the group Ad2m(BTs;
Q) is an N(s, m)-dimensional Q-module generated by {β~m YJ\ \J\=m}y where
(ii) For any negative integer m> the group Ad2m(BTs; Q)—0.
Proof. The space BT is a colimit of the direct system i(CPn)s}. Since
the lirr^-term is vanished, Ad2m(BTs; Q) is an N(s, m)-dimensional O-module.
Easy computation shows that ΛJrk(Yi)=k Yi. So when \J\=my (1— ψk2m)
(β~m YJ)=0. It is easy to prove that {β~m YJ\ \J\ —m} are linearly indepen-
dent. This completes the proof.
Proposition 3.3. For any integer my the group Ad
2m(BTs; i?)=0.
Proof. We consider the following commutative diagram where horizontal
lines are exact:
0 -* Ad2m (BT\ R) -* K2m (BTS\ R)
I I
0 -* Ad2m {BT Q) -> K2m {BT Q)
Since
is a subgroup of
the left vertical homomorphism is an injection. Thus it is enough to show
Lemma 3.4. Let m be a positive integer. If an element
^4</*-COHOMOLOGY GROUPS 303
is in
R[[XU »., X J ] ,
then aj=0 for all J.
Proof. Since there exists at least one prime p satisfying p is not invertible
in R> it is enough to show the case R—Z^p). Using the fact that the denomi-
nator of the coefficients of the power series log(l+X) contains sufficient large
^>-power, the proof is only an induction.
Clearly, Proposition 3.3 implies
Corollary 3.5. If an element
a<ΞK2*»(BTs; R) = R[[XU •-., X,]],
satisfies k~mψka=a for some m, then a=0. Moreover this is also true when R
is a ring of p-adic integers for some prime p.
The following proposition contains a special case of the Theorem.
Proposition 3.6. For any compact connected Lie group G, the group Ad2m
(BG; R)=0for any integer m.
Proof. Let Ts be the maximal torus of G. We consider the following
commutative diagram where horizontal lines are exact:
0 -> Ad2m (BG; R) -> K2™ (BG; R)
0 -> Ad2m (BT; R)-> K2m (BT5; R).
Since the group K2m(BG;R) is a subgroup of K2m(BT;R) [2], the right vertical
homomorphism is an injection. Thus the left vertical homomorphism is also
an injection. This completes the proof.
4. General cases
First we consider the topological cyclic group T. It is a cartesian product
of a torus Tσ and a finite cyclic group Zjl and its classifying space is BT==(0Poo)σ
X BZjL As same as in section 2, we denote
Buhstaber and Miscenko [3] shows
Lemma 4.1. K(BT) = K(BTσ)®K(BZμ)
- Z®K(BT°)®K(BZIl)®K(BT«)®K(BZIl)
304 S. HASHIMOTO
Considering the i?-coefficient theory, we have
Corollary4.2. K(BT;R)=R®K(BT',R)®K{BZIl;R)®K(BT°)®K(BZI
l R).
When / is invertible in R the group fc(BZjl; R)=0. So we have
Corollary 4.3. When \ T/T
o
\ is invertible in Ry then Aά
2m(BT\ R)=0 for
any integer m.
This is the second part of the Theorem.
Lemma 4.4. For any integer mΦO, the group Ad2m(BT; R)=0.
Proof. We must prove the map 1— ψk2m is injective. We only need to
prove that
l-k-m-ψk: K(BTη®K(BZIl; R) -> K{BTσ)®K{BZβ: R)
is injective.
In section 2, we proved
K(BZIl; R) = θ 4 < £ , ,μ; 1 ^ ^ # < - 1 >
where ηf is the image of the canonical line bundle of the map K{BZjprii) —>
K(BZ\ΐ) andg^=( V i y-l . So
K(BT')®K(BZIl; R) = K(BT°)®ZPi<giy, ί^μ^p^-iy .
Thus any element of this group can be written as
where aiμ are reduced elements of the ring Zp.[[Xl9 •••, Xs]] and M=/>ί* —-1.
We assume that (1— k~m"ψk)a=0. In section 2 we showed that the Adams
operation ψk act on £'s as permutation. By assumption, there exists an in-
teger e such that (ψk)egitμ=gitμ.. So aitV. must satisfies k~emψkβaitμ,=aitμ.. So
using Corollary 3.5, we have aiμ=0.
Let \T
λ
; λGΛ} be the set of all Cartan subgroups of a compact Lie group G.
Remember that K2m(BG) -> ®K2m{BT)) is a monomorphism [7]. We consider
the following commutative diagram with horizontal injective homomorphism :
0 > Ad2m (BG; R) > K2m (BG; R)
1 I
0 -* ®Λd2m (BT
λ
; R) -> @K2m (BT
λ
; R).
x λ
Since the right vertical homomorphism is injective, the left vertical homomor-
^4i/*-CoHOMOLOGY GROUPS 305
phism is also injective. Together with the Lemma 4.4, we have
Proposition 4.5. For any integer mΦO, Ad2m(BG R)=0.
This is the first part of the Theorem.
References
[1] M.F. Atiyah: Characters and cohomology of finite groups, Publ. Math. I.H.E.S.
9 (1961), 23-64.
[2] M.F. Atiyah and G.B. Segal: Equivariant K-theory and completions, J. Differ-
ential Geom. 3 (1969), 1-18.
[3] V.M. Buhstaber and A.S. Miscenco: K-theory on the category of infinite cell
complexes, Math. USSR Izv. 2 (1968), 515-556, and 3 (1969), 227-228.
[4] K. Knapp: Some applications of K-theory to framed bordism: e-invariant and
transfer, Habilitationschrift (1979).
[5] N. Mahammed: A propos de la K-theorie des espaces lenticulairs, C.R. Acad. Sci.
Paris 271 (1970), 639-642.
[6] D. Quillen: On the cohomology and K-theory of the general linear groups over
finite field, Ann. of Math 96 (1972), 552-586.
[7] G.B. Segal: The representation ring of compact Lie groups, Publ. Math. I.H.E.S.
34(1968), 113-128.
[8] R.M. Seymour: Vector bundles invariant under the Adams operations, Quart. J.
Math. Oxford (2) 25 (1974), 359-414.
[9] L. Smith: Secondary cohomology theories, Indiana Univ. Math. J. 23 (1974),
899-923.
Department of Mathematics
Osaka City University
Sugimoto, Sumiyoshi-ku
Osaka 558, Japan



On Ad ∗ -cohomology groups of the classifying spaces of compact Lie groups

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