数理解析研究所講究録
第 1569 巻 2007 年 52-58

52

Construction of smooth actions on spheres
for Smith equivalent representations
岡山大学大学院自然科学研究科

森本雅治 (MORIMOTO, MASAHARU)

Graduate School of Natural Science and Technology
Okayarna University

1. PROBLEMS AND RESULTS
Throughout this paper, let$G$be a finite group. A real G-representation of finite

dimension is meant by a real G-module, a smooth manifold is meant by a manifold,
and a smooth G-manifold is meant by a G-manifold. For a G-manifold $X$ , let$\mathcal{T}\mathcal{R}(X)$denote the set of all isomorphism classes (as real G-modules) of tangential representations$T_{x}(X)$, where$x$

runs over the G-fixed point set$X^{G}$. We are interested in$\mathcal{T}\mathcal{R}(X)$formanifolds $X$ such thatconsists of exactly two points. In particular, the case where$X$ are homotopy spheres has been studied as Smith Problem.
$X^{G}$

Smith Problem. Let$\Sigma$be a homotopy sphere with G-action such that the G-fixed

point set consists of exactly two points

and$T_{b}(\Sigma)$

$a,$

$b$

isomorphic to each other (namely. Are the tangential representations$|\mathcal{T}\mathcal{R}(\Sigma)|=1$$T_{a}(\Sigma)$) ?

We have affirmative answers (e.g. Atiyah-Bott, Milnor, Sanchez) as well as negative
answers (e.g. Petrie, Cappell-Shaneson, Petrie-R,andall, Petrie-Dovermann, DovermannWashington, Dovermann-Suh, Laitinen-Pawalowski, Pawalowski-Solomon), to Smith Problem under various hypotheses. There are surveys relevant to studies on Smith Problem
in [24], [18] and [6].

To study the problem, we define the following relations

definition below,$V$$\sim\emptyset,$$\sim e$and$\sim\emptyset \mathfrak{S}$. In theand $W$ are real G-modules.

(1) $V\sim DW$ if there exists a disk $D$ with G-action such that $D^{G}=\{a, b\}$ and$\{[V], [W]\}=\mathcal{T}\mathcal{R}(D)$(2)$V\sim \mathfrak{S}W$and.

if there exists a homotopy sphere.

$\Sigma$with G-action such that $\Sigma^{G}=\{a, b\}$$\{[V], [W]\}=\mathcal{T}\mathcal{R}(\Sigma)$(3) $V\sim DSW$ if $V\sim\otimes W$ and $V\sim e^{W}$ .
2000 Mathema tics Subject Classification. Primary: $57S17,57S25,55M35$ . Secondary: $20C05$ .

53$Here\sim oand\sim \mathfrak{D}\mathfrak{S}$may not be equivalence relations, although they stably yield equiva-

lence relations. We have been interested in the$relation\sim \mathfrak{S}$(namely the Smith equiva-

lence), but in the present paper we will mainly pay our attention to the$relation\sim \mathfrak{D}\mathfrak{S}$Let $RO(G)$ denote the real representation ring. We define the subsets
andof RO $(G)$ by$\mathfrak{D}(G),$.

$\mathfrak{S}(G)$$\mathfrak{D}\mathfrak{S}(G)$$\mathfrak{D}(G)=\{V-W\in RO(G)|V\sim \mathfrak{D}W\}$$\mathfrak{S}(G)=\{V-W\in RO(G)|V\sim \mathfrak{S}W\}$$\mathfrak{D}\mathfrak{S}(G)=\mathfrak{D}(G)\cap \mathfrak{S}(G)$was usually denoted by $Sm(G)$ . By R. Oliver [16], there exists a disk with
G-action with $|D^{G}|=2$ if and only if is an Oliver group (namely,
is not a mod
The set$\mathfrak{S}(G)$

$G$

$G$

$\mathcal{P}$hyperelementary group). Thus it is worthwhile to study

Oliver groups .
If $M$ is a subset of $RO(G)$ then for families
define$\mathfrak{D}(G)$and$\mathfrak{D}\mathfrak{S}(G)$only for$G$$\mathcal{A},$$\mathcal{B}$consisting of subgroups of

$G$

we

$\Lambda I_{A}^{def}=\{x\in M|res_{H}^{G}x=0\forall H\in \mathcal{A}\}$$M^{\mathcal{B}^{def}}=\{x=V-W\in M|V^{K}=0=W^{K}\forall K\in \mathcal{B}\}$$A’I_{A}^{\mathcal{B}^{de}f}=\{x=V-W\in M_{A}|V^{K}=0=W^{K}\forall K\in \mathcal{B}\}$.

Using the notation with the families$\mathcal{P}=\mathcal{P}(G)^{def}=${ $P\leq G||P|=p^{a}(p$ a prime)}$\mathcal{N}_{2}=\mathcal{N}_{2}(G)^{def}=\{N\underline{\triangleleft}G||G/N|=1,2\}$$\mathcal{N}=\mathcal{N}(G)^{def}=$$\mathcal{L}=\mathcal{L}(G)^{def}=$

we study the subsets
smallest normal subgroup of$\mathfrak{D}(G),${{$N\underline{\triangleleft}G||G/N|=1$$L\leq G|L\supseteq G^{\{p\}}$$\mathfrak{S}(G)$

$G$

and

$\mathfrak{D}\mathfrak{S}(G)$or a prime}

for some prime },$p$of $RO(G)$ . Here the group$G^{\{p\}}$is thewith prime power index, namely$G^{\{p\}}=$$H$$\cap$$H\underline{\triangleleft}G:|G/H|=p^{a}$for some.

$a$defined above is called a large $su$ bgroup of .
Many authors (e.g. Petrie-Randall, Petrie-Dovermann, Dovermann-Washington,
Dovermann-Suh, Laitinen-Pawalowski, Pawalowski-Solomon) found various pairs (V, $W$ )An element in$G$$\mathcal{L}$-related real G-modules $V,$ $W$ . But their (V, $W$ ) withsatisfy $I^{rN}=0=W^{N}$ for allwith prime index. In other words, they showedof nonisomorphic$V\sim\emptyset \mathfrak{S}W$$\mathfrak{D}\mathfrak{S}$$N\triangleleft G$$\mathfrak{D}\mathfrak{S}(G)^{\mathcal{N}}\neq 0$

54
for various$G$. Now we recall the next proposition.

Proposition 1 ([12], [13]). The implications$RO(G)_{P}^{N_{2}}$$\mathfrak{S}(G)\subseteq RO(G)_{Q}^{N_{2}}$and$\mathfrak{D}\mathfrak{S}(G)\subseteq$

hold.

These facts motivate us to study the following problem.
Problem A. Does there exist a finite group$G$satisfying$\mathfrak{D}\mathfrak{S}(G)\neq \mathfrak{D}\mathfrak{S}(G)^{N}$?

The notion gap module is convenient to study this problem as well as Smith Problem.

A real G-module$V$is called a gap module if it satisfies the following conditions.

(1) $V^{L}=0$ for all$L\in \mathcal{L}(G)$(2) dim $V^{P}>$ dim$V^{H}$.

for all pairs $(P, H)$ of subgroups of$G$such that$P\in \mathcal{P}(G)$and $H>P$ .

A finite group is called a gap group if admits a gap real G-module. PawalowskiSolomon showed in [18] that for an arbitrary nonsolvable gap group with
and$G\not\cong P\Sigma L(2,27)$ ,
$G$

$G$

$G$

$\mathfrak{D}\mathfrak{S}(G)\supseteq RO(G)_{P}^{\mathcal{L}}\neq 0$$a_{G}\geq 2$.

Since the appearance of this result, the next problem has been asked.

Problem B. Are the sets

and$\mathfrak{S}(G)$$\mathfrak{D}\mathfrak{S}(G)$nontrivial in the case $G=P\Sigma L(2,27)$ ?

The purpose of the present paper is to answer to Problems A and$B$, and we obtained

the following results.

Theorem 2. For each odd prime , there exist a gap Oliver group
$p$

$V$

and $W$ such that$|G/N|=p$ , hence$V\sim \mathfrak{D}\mathfrak{S}W$$G$

and real G-modules, dim $V^{N}>0$ and dim $W^{N}=0$ for some$\mathfrak{D}\mathfrak{S}(G)\neq \mathfrak{D}\mathfrak{S}(G)^{N}$$N\triangleleft G$

with

.

Let $SG(m, n)$ denote the small group of order $m$ and type$n$

appearing in the computer

software GAP [5].
Theorem 3. Let $G=P\Sigma L(2,27),$ $SG(864, 2666)$ , or $SG(864, 4666)$ . Thenbut

$RO(G)_{P}^{\mathcal{L}}=0$.

$\mathfrak{S}(G)=\mathfrak{D}(G)=\mathfrak{D}\mathfrak{S}(G)=RO(G)_{\mathcal{P}}^{\{G\}}\cong \mathbb{Z}$

2. ADDITIONAL INFORMATION
For $g\in G$ , let $(g)$ denote the conjugacy class ofof is the union of $(g)$ and$g$$(g^{-1})$. Let$a_{G}$

$g$

in

$G$

. The real conjugacy class$(g)^{\pm}$denote the number of all real conjugacy classes

55
of elements

$g$

of

$G$

such that$g$does not have prime power order. By the representation

theory, we have.

$a_{G}=rankRO(G)_{\mathcal{P}}$Let$\delta$denote the homomorphism from

$RO(G)_{\mathcal{P}}$to$\mathbb{Z}$given by$\delta([V]-[W])=\dim V^{G}-\dim tV^{G}$.

$RO(G)_{P}^{\{G\}}=Ker[\delta :.

Then by definition,
B. Oliver [17] showed that if$a_{G}\geq 1$$Image[\delta :RO(G)_{\mathcal{P}}arrow \mathbb{Z}]$then

RO(G)_{\mathcal{P}}arrow \mathbb{Z}]\supseteq 2\mathbb{Z}$.

Thus the next proposition immediately follows.
Proposition (Laitinen-Pawalowski [8]). If $a_{G}\geq 1$ then rank $RO(G)_{P}^{\{G\}}=a_{G}-1$ .

In addition, B. Oliver [17] implies the next result.

Theorem (Oliver). If$G$is an Oliver group then$\mathfrak{D}(G)=RO(G)_{\mathcal{P}}^{\{G\}}$.

Viewing these facts, E. Laitinen conjectured the next.

Laitinen’s Conjecture. If$G$is an Oliver group with $a_{G}\geq 2$ then$\mathfrak{D}\mathfrak{S}(G)\neq 0$.

This conjecture had been positively expected until 2006. We, however, have a negative
example.

Theorem 4 ([12], [13]). Let $G=Aut(A_{6})$ . Then Laitinen’s Conjecture fails, in fact$a_{G}=2$ and.
$\mathfrak{S}(G)=0=\mathfrak{D}\mathfrak{S}(G)$Most finite Oliver groups are gap groups, but neither
where$S_{5}$

is the symmetric group on five letters and$A_{6}$$S_{5}$

nor Ant$(A_{6})$

is a gap group,is the alternating group on six

letters. Pawalowski-Solomon [18] showed the next theorem using a deleting-inserting
theorem of G-fixed point sets to disks ([10], [15, Appendix]).
Theorem (Pawalowski-Solomon [18]). If$G$

is a gap Oliver group then

$RO(G)_{P}^{\mathcal{L}}\subseteq \mathfrak{D}\mathfrak{S}(G)$.

On the other hand, they also showed the next result using the finite group theory.
Theorem (Pawalowski-Solomon [18]). Let

If$G\not\cong P\Sigma L(2,27)$$G$

be a nonsolvable gap group with $a_{G}\geq 2$ .

then
$RO(G)_{\mathcal{P}}^{\mathcal{L}}\neq 0$.

56
Putting these results together, we obtain a corollary.
Corollary (Pawalowski-Solomon [18]). Let

If

then

$G\not\cong P\Sigma L(2,27)$

Since$S_{5}\cross C_{2}$$\mathfrak{D}\mathfrak{S}(G)\neq 0$, where$C_{2}$$G$be a nonsolvable gap group with $a_{G}\geq 2$ .

.

is the cyclic group of order 2, is not a gap group, the next

result is also interesting.

Theorem (X.M. Ju [6]). In the case $G=S_{5}\cross C_{2}$ , the equalities$\mathfrak{S}(G)=\mathfrak{D}\mathfrak{S}(G)=RO(G)_{\mathcal{P}}^{\mathcal{L}}\cong \mathbb{Z}$hold.
We obtained a deleting-inserting theorem [14] of new kind by employing an equivariant

interpretation of Cappell-Shaneson’s surgery obstruction theory for getting homology(possibly, not homotopy) equivalences as well as employing the induction theory of Wall’s

surgery obstruction groups. We state here the theorem in a simplified form.
Theorem 5. Let$G$

be an Oliver group and$Y$a disk with G-action. Supp ose the following

conditions are satisfied.
(1) $Y^{G}=\{y_{1}, \ldots, y_{m}\}$ , where $m\geq 1$ .

.
for all(3) dim
for all mod cyclic subgroups $H,$ .
(4) dim $Y^{P}>2(\dim Y^{H}+1)$ for all
and $H>P$ .
(5)
and $(|\pi_{1}(Y^{P})|, |P|)=1$ for all.
(2)$\partial Y^{L}=\emptyset$$L\in \mathcal{L}(G)$$Y^{H}\geq 5$$i.e$$\mathcal{P}$$1\triangleleft P\triangleleft H\mathcal{P}cyclic$$P\in \mathcal{P}(G)$$|\pi_{1}(Y^{P})|<\infty$$P\in \mathcal{P}(G)$(6) The inclusion induced maps$\mathcal{P}(G)$$\pi_{1}(\partial Y^{P})arrow\pi_{1}(Y^{P})$are isomorphisms for all $P\in$.

Then there exists a disk $X$ with G-action such that$\partial X=\partial Y$

and$X^{G}=\emptyset$.

Remark that the union $\Sigma=XU_{\partial}Y$ identified along the boundaries of $X$ andthe theorem above is a homotopy sphere such that$\mathcal{T}\mathcal{R}(\Sigma)=\mathcal{T}\mathcal{R}(Y)$

$Y$

in

. Since various

G-actions on disks are constructed by Oliver’s theory [17], we would obtain G-actions
on homotopy spheres from those on disks. In fact, the next result is an outcome ofTheorem 5.
$Y$$\Sigma$

Theorem 6. Let$p$be an odd prime. Let

all premes $q\neq p$ and $|G/G^{\{p\}}|=p$ . If

distinct primes

$q$

and

$r$

$G$

$G$

be an Oliver group such that $G=G^{\{q\}}$ for

has a dihedral subquotientand further that$G$

$D_{2qr}$(order $2qr$ ) withcontains distinct real G-conjugacy classes

57
not of prime power order, thenof elementssummand of $RO(G)$ of rank 1.
$(x)^{\pm},$$(y)^{\pm}$

$x,$

$y$

$\mathfrak{D}\mathfrak{S}(G)$contains a direct

Theorems 2 and 3 follow from Theorem 6. In addition, we conclude the next.
Theorem 7. Laitinen’s Conjecture is affirmative for any finite nonsolvable gap group.

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