HASHIMOTO, SHIN

Hashimoto, Shin

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A note on some periodicity of Ad-cohomology groups of lens spaces

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Osaka City University Repository

Hashimoto, S.
Osaka J. Math.
23 (1986), 307-312
A NOTE ON SOME PERIODICITY OF
Ad-COHOMOLOGY GROUPS OF LENS SPACES
SHIN HASHIMOTO
(Received February 12, 1985)
1. Introduction
Let p be an odd prime and q=pr. We choose a positive integer k such
that the class of k in Zjp2 generates the group of units (Z/p2)x.
Let K* be the i^-cohomology theory and Kfp) its />-localized theory.
The Adams operation ψk on K induces a stable operation ψk on Kfp). We
denote by K(p) the spectrum which represents the i£(%-cohomology theory.
Since stable operations induce maps of spectra, we have the following cofibra-
tion of spectra
We define a spectrum Ad as Σ^CΊ-ψ* and its associated cohomology theory
Ad*. When k is a prime power, the associated connective theory of Ad* co-
incides with the cohomology theory defined by Seymour [9] and Quillen [8].
Let m and n be positive integers. We identify Z\m with the set of m-th
root of 1 in C, and S2n+1 with the unit sphere in Cn+1. The complex vector
space structure on Cn+1 induces a Z/m-action on S2n+1 and we define the stand-
ard Lens space mod m as S2n*\Z\m). As is well known, the standard Lens
space Ln{m) has a CPF-complex structure
L > * ) = 2 l j V
and we denote its 2/z-skeleton by Ll(m). Since the canonical inclusion Cn+1d
Cn+2 induces a cellular inclusion L"(m)cL"+1(m), we have a CH^-complex
L°°(m)=colim Ln{m). This space L°°(m) is a classifying space BZjm of Zjm.
We consider the case m=pr. Main results are the following.
Theorem 1.1. Let M(ri)=r-{-[(n— l)/(p— 1)]. For any integers iy j satis-
fying i—j=0 (mod (p—l)pMin)"1)9 there holds the following isomorphisms.
Ad2\Ll(pr)) « Ad2\Ll{pr))
Ad2i+XL»o(pr))
308 S. HASHIMOTO
Theorem 1.2. We put N=r-\-vp(i). Let j be an integer which satisfies
Vp(i)=vp{j) and i—j=0 (mod (p— l ) ^ " 1 ) . Then, there holds the following iso-
morphism:
Ad2i(L»o(pr))
We used the computer of Osaka City University computer center to cal-
culate the samples of the group structure of Ad*(Lo(pr)).
The author expresses his hearty gratitude to Professors S. Araki, Z. Yosi-
mura and A. Kono.
2. Preliminaries
Let G be a compact Lie group and R(G) be its complex representation
ring. The augmentation ideal / is the kernel of the induced ring homomor-
phism R(G)->R({1}).
Proposition 2.1 (Atiyah, Atiyah-Segal). There holds the following natural
isomorphism:
a:R(G)ϊ->K(BG).
In case G=S1, R(S1)=Z[Hy H~ι] where H is the canonical representation
of S\ Let x be e(H)=H-l, the euler class of H. Since I=(x), K{BSι)=Z
[x]x=Z[[x]] and K\BSl)=0. We consider the case G=Z\q. There is the ca-
nonical inclusion i: ZjqClS1, and we write i*(H)=H and i*(x)=x. Then R(Z]q)
=Z[H]I(H9-1) and I=(x). Thus K(BZIq)=(Z[x]l((x+l)9-l)Y
x
 and Kι
(BZ/q)=0. We denote R=K(p)(BZIq)=(Z(p)[x]l((x+iy-l)Yx. The definition
of the completion induces
Lemma 2.2. There exists a continuous surjection
The i£-ring of finite Lens spaces is given by Mahammed [7]. That is
(2.3) K(U(q)) = Z[*]/(
where x is the restriction of xy the euler class of the canonical line bundle.
Lemma 2.4. The ring K(p)(Lo(q)) is isomorphic to R/(xn+1) and the in-
clusion Lo(q)czL°°(q) induces the projection R->RI(xn+1).
We denote Lo'n(q)=L°°(q)ILno(q) and L%;Z(q)=LZ(q)ILn0(q). Then we have
the following exact sequence.
(2.5) 0 -* £(L? %)) -> K{L~(q)) - K(Ll(q)) - 0 .
^[J-COHOMOLOGY GROUPS 309
Thus we have
Lemma 2.6. K(Lo'n(q)) = xn+1 R
K\Lf»(q)) = 0.
It is easy to see that Ln0%
ι
'
n{q) = S2n+ι U e?n+2. Thus we have
Q
"Lemma 2.Ί.
= 0 .
On the other hand, the exact sequence
0 -> K(Lo'n+1(q)) -> K(Lo'n(q)) -> K(U:oUn(q)) - 0
shows that K(Ll^'n(q))=xnRlxn+ιi which is a cyclic group generated by xn.
So we have
Proposition 2.8. When m—n = 0 (mod p— 1), we put t=min(r, vp(m)-{-l).
Then
~~ ~ [Zip1 if n—m = 0 (mod p— 1)
I 0 if n—m^βυ ( m o d £ — 1) .
3. Proof of Theorems
We identify KfpftX) with K(p)(X) using the Bott periodicity. We denote
the stable operation ψk on K2$(X) by ψk'm. Then ψk>m=k-mψk under this
identification.
Lemma 3.1. Under the above identification, when m—WΪ'ΞO (mod p—1),
lm((l-ψk>η-(l-ψk>m'))czp1+Vm-m') Im(ψ*).
Kobayashi, Murakami and Sugawara [6] have computed the explicit abe-
lian group structure of K(Lo(q)). One corollary of their results is
Proposition 3.2.
Let M(n)=r+[(n-l)l(p-l)] and Λ f ^ ^ - l ) ^ ^ " 1 . Then
Corollary 3.3. Under the above identification
When i—j=0 (mod M r), we consider the following commutative diagram
where horizontal lines are exact:
310 S. HASHIMOTO
0 - Ad2i{U{q)) -* K2\L%{q)) - K2i(L«0(q)) - Ad"+\L%{q)) -* 0
1 - I -
0 -> Ad2\Ll{q)) -> K*(Lftq)) - K*(Lϊ(q)) - AP>+1(Ljj(?)) - 0 ,
where the vertical isomorphism are induced from the Bott periodicity. This
completes the proof of Theorem 1.1.
We took k as a generator of (Zip2)*. It is easy to see the following lem-
mata.
Lemma 3.4. The class of k in Z\pr is a generator of (Z/pr)x for every r.
Lemma 3.5. Let W={p-\)pr-ιβ. Then kw = -l (mod/) .
Let U=(p—l)/2. Lemma 3.5 induces
Corollary 3.6.
DEFINITION 3.7.
A = ff (H^-1)tr'2-l) = Π (^(ί-1)ίΓ~V (H-ί)
i-0 ί=0
We compute the action of the Adams operation on the element A, and
using Corollary 3.6 we have
Lemma 3.8. There exists a natural number t such that
(ψγ-V-2/2 A = -HfA.
The proof is only a computation. The natural number t=— Σ kj{p~l)pr~2,
but we don't need it. Let Q=(p—l)pr~2l2. i=1
Lemma 3.9. There exists a natural number s such that
(ψ*)C*-D*r-2/2 (HSA) = -HSA .
Proof. Using Lemma 3.8, we need to solve the following equation in Zjq.
Since Q is not a multiple of p—1, vp{\— kQ)=0. This implies that 1— kQ
is invertible in Zjpr. Thus the above equation has a solution.
DEFINITION 3.10. We put B=HSA and C = Σ ( Ψ * y # . This element
yo
(
C is a polynomial of H, so we write C=C(H). It is easy to see that ψkB= —B,
Σ^-COHOMOLOGY GROUPS 311
φ*C=-C and B and C are in the ideal (H-ί)p. So we define D(H)={H-\yp
XC(H).
Lemma 3.11. The integer D{\) is prime to p.
Proof. Letf{X)=(Xu-l)l(X-l), thenf(l)=u. Thus
D(ί) =
i=
= kυ
Since z/^(l—&Q)=0, the proof is completed.
Lemma 3.12. The ideal generated by C coincides with the ideal (H—1)*
inR.
Proof. We write D as a polynomial of x=H— 1. Then D=D(ί)+higher.
So D is an invertible element in Z(P)[x]Ax. By Lemma 2.2 Z) is invertible in R.
Since C=(H—l)pD, the Lemma is proved.
Proposition 3.13. When n=0, —1 (mod^>), then
Ad2m+\Lo'n(q)) = Θ Z f ^
Proof. By Lemma 2.6 K(L%'n(q))=xn+1R. The assumption implies w=
tp—Sy where t is a positive integer and £ is 0 or — 1. We choose an additive
basis of x"+1R as {g
Λ
(i,j); 1^/^r, l ^ ^ ( ^ - l ) / - ] - l } where g
n
(hj)=(-Cy
χ(Hpr~Jkt-\-S—1). As same as in computation in [4], we have the required
result.
Consider the following exact sequence
0 -> Ad2m{LlT\q)) -> Ad2m+1(Lo'n(q))
-* Ad2m+\Lo'n'\q)) -> Ad2m+\Ln0:n0-\q)) -> 0 .
By Proposition 2.8, we have
Corollary 3.14. If n—m = 0 (mod p— 1), we have an isomorphism:
Ad2m+\Lo'n(q))
Proposition 3.13 and Corollary 3.14 imply
Lemma 3.15. Let N(m)=rJrvp(m). Then
pN^Άd2m+\Lo'n(q)) = 0 .
The exact sequence
312 S. HASHIMOTO
0 -> Ad2m(Lno(q)) -> Ad2m+\Lo'n(q)) -
induces
Corollary 3.16. pN™ 2d 2 " + 1 (LS(ί))=0 /or tf/ry mfcgβr rc.
Lemma 3.1 and Corollary 3.16 imply Theorem 1.2. This completes the
proof.
References
[1] J.F. Adams: On the groups J(X)-Πy Topology 3 (1965), 137-171.
[2] M.F. Atiyah: Characters and cohomology of finite groups, Publ. Math. I.H.E.S.
9 (1961), 23-64.
[3] M.F. Atiyah and G.B. Segal: Equivariant K-theory and completions, J. Differ-
ential Geom. 3 (1969), 1-18.
[4] S. Hashimoto: Note on Ad*-cohomology groups of Lens spaces, to appear.
[5] K. Knapp: Some applications of K-theory to framed bordism: e-invariant and
transfer, Habilitationschrift (1979).
[6] T. Kobayashi, S. Murakami and M. Sugawara: Note on J-groups of Lens spaces,
Hiroshima Math. J. 7 (1977), 387-409.
[7] N. Mahammed: A propos de la K-theorie des espaces lenticulaires, C.R. Acad.
Sci. Paris 271 (1970), 639-642.
[8] D. Quillen: On the cohomology and K-theory of the general linear groups over
a finite field, Ann. of Math. 96 (1972), 552-586.
[9] R.M. Seymour: Vector bundles invariant under the Adams operations, Quart.
J. Math. Oxford (2) 25 (1974), 359-414.
Department of Mathematics
Osaka City University
Sugimoto, Sumiyoshi-ku
Osaka 558, Japan
