The Smith Isomorphism Question: A review and new results(The theory of transformation groups and its applications)

The

Smith

Isomorphism

Question:

_A

_review

_and

new

_results

岡山大学自然科学研究科 鞠先孟 (XianMengJU) *

Faculty of Environmental

Science

and Technology,

Okayama University $\ln$ 1960, P A. Smith [Smi60] raised

an

isomorphism question:

Smith tsomorphism Question. _{Whether the two} $tangentia/G$_{-modules at two fixed}

$\rho oints$ of

an

arbitrary smooth

G-action

on a

sphere with$exactl\gamma$

two

fixed points $are$

/so-morphic

to

each other?

Following [Pet82],

two

real $G$-modulesVand $W$

are

called Smith $equ/va\prime ent$ifthere

existsasmooth action of $G$

on

asphere $S$suchthat $S^{G}=\{x,v\}$fortwo points_$x$and

$V$

at

which $T_{x}(S)\cong V$and $T_{y}(S)\cong w$

as

$rea\uparrow G- modules$.

Let RO(G) denote the real representation ring of G. Define the $Sm/th$ set $Sm(G)$

to

be

$Sm(G)=$

{

$[]-M\in RO(G)|V$_and $W$

are

Smith equivalent}.

TheSmith Isomorphism Question

can

_{be restated}

_as

_foIlows. Smith lsomorphlsm Question. _{Is it true tha}$t$Sm(G) $=0$?

ltis

easy

to

showthatthe

answer

is affirmative if $G$ is

agroup

suchthateach element

has the order 1,

2or

4.

$\ln$ nineteen sixties, the first breakthrough

was

due to M. F. Atiyah and R. Bott

$|AB68$, Theorem 7.15]:

Theorem 1 (Atlyah-Bott). $/fG=C_{p},$$p$

an

odd prime, thenSm(G) $=0$.

Shortlythereafter, J. Milnor[Mi166, _{Theorem 12.}

_t$]

extended their result:

Theorem 2 (Mllnor). If$G$ is

a

compa

$ct$

group

andtheaction semi-free, thenSm(G) $=0$.

$\ln$ nineteen seventies, by using the _$G$-signature theorem, C. $\cup$. Sanchez [San76]

ob-tained astronger result:

Theorem 3(Sanchez). Let X $be$

a

rational-homologysphere$su\rho port/ng$

an a

ctionof$C_{\mathfrak{n}}$

($\mathfrak{n}$ odd)

as

a

group

of diffeomorphisms with onlytwo fixed points_$x$

andv a

$nd$satisfying

thecondition tha$t$

for

every

$\rho ropersubgrou\rho H$ of$C_{\mathfrak{n}}$, either$F(H, X)=\{x,y\}$

or

$F(H,x, X)=$

$F(H,v\cross)$ holds.

$*20\infty$MathmaticsSubiect_{$CIassificat|on.$} Primary; $57S17,57S25,55M35.$ Secondary; _$20C05$.

August21,2007

Then$T_{x}(X)\cong T_{v}(X)$.

$\ln$ fact, bySanchez Theorem andSmith theory,

we

obtain the following Corollary. Corollary 4. $ln$eitherofthefollowingcases, Sm(G) _$=0$

.

$(f)G$ is

a

$grou\rho$ withodd-prime-Powerorder.

(2) $G$ is

a

group

$w/th|G|=pq$ , where_$P$ and_$q$

are

odd

Primes.

Byusing $G$-equivariant

surgery,

T. Petrie[Pet79], [PR85] obtainedthe first

counterex-ample tothe question:

Theorem 5 (Petrle). $lfG$ is

an

odd orderfiniteabelian$grou\rho$ withatleast fournon-cyclic

Sylowsubgroups, then$Sm(G)\neq 0$

.

$\ln$ nineteen eighties, S. E. CaPpell and J. L.

Shaneson

[CS82]

gave

first

counterex-amples

to

the question for $G$ acyclic

group:

Theorem 6 (CaPpeIl-Shaneson). $lfG$ is the cyclic

group

of order$4m$ such that$m\geq 2$ then$Sm(G)\neq 0$

.

By character theory,

we

have $Sm(D_{6})=0$ _and $Sm(C_{6})=0$

.

So, $C_{8}$ is the smal[est

group

with $Sm(G)\neq 0$

.

T. Petrie and his Collaborators Obtained alot of results about $s$-Smith equivalenCe,

See [Pet83], [PR84], [Cho85], [CSu85], [Suh85], [Cho88]. K. H. Dovermann and T. Petrie

[DP85]constructed non-isomorphicSmith equivalent representations OfOdd order cyclic

groups.

Theorem7(Dovermann-Petrle). Let$G$ be

an

odd-ordercyclic

group

suchthattheorder

of$G$ has

at

least

3

prime divisors. $lf$ there existreal G-modulesA and$B$ satisfying the

followingconditions, _thenSm$(G)\neq 0$

.

(0) $A\not\cong B$,

(\dagger )

$A^{9}G,$

$=B^{9}=0$ fOreach $g\in G$ which generates

a

subgrOuPof prime

power

index in

(2) dim$A^{K}=\dim B^{K}wh.enever|G/K|$ is divisiblebyatmost 3 distinct$\rho rimes$,

(3) ${\rm Res}_{P}^{G}A\cong{\rm Res}_{P}^{G}B$ whenever_$|P|$ is

a

Prime power,

(4) $\gamma(A^{P}-B^{P})(9)=\pm 1$ whenever $|P|$ iS

a

prime Power and 9 $\in G$ generates

a

subgroupof prime

power

indexin G.

The

groups

exhibited in that Paper

were

very

large. As J. $Ewi$

ng

computed, their

order

were

at

[east $10^{2812917}$. K. H. Dovermann and L. C. _{Washington [DW89] showed}

thatsuch non-isomorphicSmith equivalent representationsalso existforodd ordercyclic

groups

ofsmall order. Forexample, their orders

can

be5.11.19.29and

3.13.17.23.

K. H. Dovermann and D. YSuh [DS92] _{constructed non-isomorphic Smlth equivalent} representations in the following

cases.

Theorem 8(Dovermann-Suh). If$G$ is

a

group

withrealG-modulesA and$B$

as

in

The-orem

7, then$Sm(G\cross C_{2^{k}})\neq 0$.

Theorem 9 (Dovermann-Suh). $lfG$ is

a

finite abelian _{$groUp$ with} at_least

3

_non-cyclic

Sylow$s$ubgrouPs, withreal$G$-representationsAand$B$ satisfying the followingconditions,

then$Sm(G)\neq 0$,

(0) $A\not\cong B$,

(\dagger ) $A^{K}=B^{K}=0whenever|G/K|$ _is

a

_prime

power,

(2) dim$A^{K}=\dim B^{K}$ for all$K\subset G$,

(3) ${\rm Res}_{P}^{G}A\cong{\rm Res}_{P}^{G}B$ whenever_$|P|$ is

a

prime

power.

A $fini\dagger e$ grOup $G$ is

an

$O\prime iver$

group

if and only if $G$

never

admits anormal series

$P\triangleleft H\triangleleft G$

Such$that|P|$and $|G/H|$

are

prime

powers

and $H/P$ is acyclic

group.

Forafinite

group

$G$,

the following three claims

are

equivalent ([Oli75], [LM98]). (1) $G$ is

an

OIiver

group.

(2) $G$ has

a

smooth

one-fixed-Point

action

on a

sphere.

(3) $G$ has

a

smooth fixed-point-free action

on

a

disk.

For

an

element $9\in G$, [et (9) denote the $con|ugacy$ class of

9in

G. The union $(g)^{\pm}=$

(g) $\cup(g^{-1})$ is called the real_{$conjugac\gamma$}class of $g$ in G. Let $\mathfrak{a}_{G}$ denote the number of the

real $con|ugacy$classes $(g)^{\pm}$ in $G$ suchthat the order of

9is

notaprime

power.

$\ln$ 1996, in the

case

where $G$ is

an

Oliver

group,

E. Laitinen [LP99, APpendix] Iighted the question again with

an coniecture.

Lalttnen COnjecture. If$G$ iS

an

Oliver

group

With $a_{G}\geq 2$, then_{$Sm(G)\neq 0$}.

E. Laitinenand K. Pawatowski proved

two

theorems [LP99]:

Theorem 10 (Laitlnen-Pawalowskl). If $G$ is

a

finite perfect

group

with $a_{G}\geq 2$, then $Sm(G)\neq 0$

.

Theorem 11 (Laltlnen-Pawalowski). If $G\cong A_{\mathfrak{n}},$ $SL(2,p)$

or

$PSL(2, q)$ with $a_{G}\geq 2$,

where$\mathfrak{n}$ is

a

natural numberand

$P$ and$q$

are

primes, then$Sm(G)\neq 0$.

A real G-module Vis called

a

gap

moduleif itsatisfies

(1) $\dim V^{P}>2\dim V^{H}$_for

any

_subgroup$P\subset G$ of prime

power

orderand

any

subgroup

$H\subset G$ with $P\subset\wedge H$, and

Afinite

group

$G$ is called

agap group

if $G$ admits

agap

module. We refer

to

[MSYOO],

[SumOl] and [Sum04] for

more

information about

gap

groups.

Let $P\Sigma L(2,27)$ denote the

splitting extension of $PSL(2,27)$ bythe

group

$Aut(F_{27})$

.

K. $Pawa\dagger owski$ and R. Solomon

$[PaS02]$ _{answered the Smith} isomorphism question in _various

cases:

Theorem 12(Pawalowski-Solomon). $/n$either ofthefollowing cases,Sm_{$(G)\neq 0$}holds. $(f)G/s$

a

finite Oliver$grou\rho$ ofoddorder(thus $a_{G}\geq 2$, and$G$ is

a

gap

group).

(2) $G$ is

a

finiteOliver

group

with

a

cyclicquotientoforderpq fortwodistinct odd

Primes

$p$ and$q$ (thus $a_{G}\geq 2$, and$G$ is

a

gap

group).

(3) $G$ is

a

finitenon-solvable

gap

grouP with$a_{G}\geq 2$, and$G\not\cong P\Sigma L(2,27)$.

Theorem 13 (Pawalowskl-Solomon). In either of the $to/low/ng$ cases, if $a_{G}<2$ then $Sm(G)=0$

.

$(f)G$ is

a

finite non-abelian

slmPle group.

(2) $G\cong PSL(\mathfrak{n}, q)$

or

$SL(\mathfrak{n}, q)$ for

any

$\mathfrak{n}\geq 2$ and

any

primePower$q$

.

(3) $G\cong PSp(2\mathfrak{n}, q)$

or

$Sp(2\mathfrak{n}, q)$ for

any

$\mathfrak{n}\geq 1$ and

any

Prime

Power$q$

.

(4) $G\cong A_{\mathfrak{n}}$

or

$S_{\mathfrak{n}}$ for

any

$\mathfrak{n}\geq 2$

.

We refer

to

the articles [PR84], [CS85], $|DPS85]_{1}|MaP85$], [PawOO] for

survey

of

related results. K. $Pawa\dagger owski$ and R. Solomon _[$PaS02$, Theorem _A.3] pointed out that

$Aut(A_{6})$ is anon-solvableOliver

group

such that $a_{G}=2$. $\ln$ 2006, M. Morimoto $[Mor07a]$

gave

acounterexample to Laitinen $Con|ecture$

:

Theorem 14(Morlmoto). If$G=Aut(A_{6})$ then $a_{G}=2$, andSm(G) $=0$

.

K. $Pawa\dagger owski$ and T. Sumi _$|PaS07$] claim _{$Sm(G)\neq 0$} for

many

Oliver

groups

$G$ such

that $a_{G}\geq 2$ and $G$ is not

agap

group,

although onlythe sketchiest ideas of proofs

are

given. Let $G^{ni1}$ denote the smallest normal subgroup _$N$ of _$G$ such that _$G/N$ is nilpotent. Announce 15 (Pawatowski-Suml). Let $G$ be

a

finite Oliver

group

such that $G/G^{ni1}$ is

$isomorph/c$ to_neither$p$

-group

for

a

prime$p,$ $C_{2}\cross P$ for

an

odd prlme$p$ and

a

$p$

-group

$P$,

nor

$P_{2}\cross C_{3}$ for

a

2-group

$P_{2}$ such thatall elements of$P_{2}$

are

self-conjugate: $(g)=(g^{-1})$

.

Then$Sm(G)\neq 0$

.

Announce 16 (Pawalowskl-Suml). If

a

finite Oliver

group

$G$ has

an

element of order pqr for distinct$pr/mesp,$ $q$ andr, then$Sm(G)\neq 0$

.

Announce 17$(Pawalowskl\cdot Suml)$

.

Let$G$ be

a

finite Oliver

group

with$non\cdot tr/v/al$

center.

$lf$the order of$G^{\mathfrak{n}I\dagger}$ is

Many authors have studied the Smith equivalence for various finite

groups.

But the Smith sets Sm(G)

were

_{rarely determined. In particular, when} $G$ is anon-solvable,

non-perfect

group,

Smith set Sm(G)

was

not determined except the

case

Sm(G) $=0$. We

have interested in the

group

$S_{5}\cross C_{2}$, because it is not

agap

group,

but it’s subgroup $A_{5}\cross C_{2}$ is

agap group.

For aprime$p$, [et $G^{\{p\}}$ denote the smallest normal subgroup $H$ such that the order of

$G/H$ is

apower

of$p$ (possibly1). Let$\mathcal{P}(G)$ denote the

set

of $aII$ subgroups of $G$ of prime

power

order(possibly1). Define $\mathcal{L}(G)$ by

{

$H\leq G|H\geq G^{\{p\}}$ for

some

prime_$p$

}.

A real G-module Vis saidto be $\mathcal{L}(G)$-free if $V^{L}=0$for

any

$L\in \mathcal{L}(G)$

.

Define $RO(G)_{\mathcal{P}}^{\mathcal{L}}$

to be the

set

{

$[]-M\triangleleft\in RO(G)|V$and $W$

are

$\mathcal{L}(G)$-free and ${\rm Res}_{P}^{G}V\cong{\rm Res}_{P}^{G}W$for all $P\in \mathcal{P}(G)$

}.

Announce 18. The following$equa\prime it/es$hold for$G=S_{5}\cross C_{2}$ and$K=A_{5}\cross C_{2}$

.

(\dagger ) Sm(K) $\cong \mathbb{Z}^{2}$

and$Sm(G)\cong \mathbb{Z}$

.

(2) $1nd_{K}^{G}(Sm(K))=Sm(G)$

.

Here the maP $1nd_{K}^{G}$ : $RO(K)arrow RO(G)$ isthe inductionhomomorphism;

$[V]-[\mathbb{R}[G]\otimes_{R[K]}V]$

.

By

means

of GAP [GAP06],Thecomplex character of $G=S_{5}\cross C_{2}$ is

as

in Table 1,

$\ovalbox{\tt\small REJECT}\xi_{1\mathbb{C}}11I111111111111a2a2b2c3a6a2d2e4a4b6b6c5a10a$

$\xi_{2\mathbb{C}}$ 1 $-1$ $-1$ 1 $\rceil$ $-1$ $\rceil$ $-1$ $-1$

1

$-1$ 1 1 $-1$

$\xi_{3\mathbb{C}}$

1

$-1$ 1 $-1$ 1 $-1$ 1 $-1$ 1 $-1$ 1 $-1$ 1 $-1$ $\zeta_{4C}$

1

1 $-1$ $-1$ 1

1

1

1

$-1$ $-1$ $-1$ $-1$

1

1

$\xi_{5\mathbb{C}}$ 4 4 $-2$ $-2$

1

1 $0$ $0$ $0$ $0$

1

1

$-1$ $-1$

a4

$-4$ $-2$

2

1 $-1$ $0$ $0$ $0$ $0$ 1 $-1$ $-1$ 1 $\zeta_{7\mathbb{C}}$

4

4

2

2

1

1 $0$ $0$ $0$ $0$ $-1$ $-1$ $-1$ _$-1$ $\xi_{8\mathbb{C}}$ 4 $-4$

2

$-2$ 1 $-1$ $0$ $0$ $0$ $0$ $-1$ 1 $-1$ 1 $\xi_{10\mathbb{C}}\xi_{9\mathbb{C}}$ $55$ _$-55$ $11$ $-11$ _{$-1-1$ $-11$} $11$ $-11$ $-1-1$ $-1t$ $11$ $-\dagger 1$ $00$ $00$ $\xi_{11\mathbb{C}}$ 5 5 $-1$ -I $-1$ $-1$ 1 1 1 1 $-1$ $-1$ $0$ $0$ $\xi_{12\mathbb{C}}$ 5 $-5$ $-1$ 1 $-1$ 1 1 $-1$ 1 $-1$ $-1$ 1 $0$ $0$ $\xi_{13\mathbb{C}}$

6

6 $0$ $0$ $0$ $0$ $-2$ $-2$ $0$ $0$ $0$ $0$ 1 1 $\xi_{14\mathbb{C}}$ 6 $-6$ $0$ $0$ $0$ $0$ $-2$

2

$0$ $0$ $0$ $0$

1

$-1$

and the complexcharacterof $K=A_{5}\cross C_{2}$ is

as

in Table 2. $\ovalbox{\tt\small REJECT}\delta_{1\mathbb{C}}11111111111a2a3a6a2b2c5a10a5b10b$ $\delta_{2\mathbb{C}}$ 1 $-1$ 1 $-1$ 1 $-1$ 1 $-1$ 1 $-1$ $\delta_{3\mathbb{C}}$ 3

3

$0$ $0$ $-1$ $-1$ AA $\hat{A}$ $\hat{A}$ $\delta_{4\mathbb{C}}$

3

3

$0$ $0$ $-1$ $-1$ $\hat{A}$ $\hat{A}$ AA $\delta_{5\mathbb{C}}$

3

$-3$ $0$ $0$ $-1$

1

A-A $\hat{A}$ $-\hat{A}$ $\delta_{6\mathbb{C}}$

3

$-3$ $0$ $0$ $-1$

1

$\hat{A}$ $-\hat{A}$ A-A $\delta_{7\mathbb{C}}$ 4 4

1

1

$0$ $0$ $-1$ $-1$ $-1$ $-1$ $\delta_{8\mathbb{C}}$ 4 $-4$ 1 $-1$ $0$ $0$ $-1$ 1 $-1$ 1 $\delta_{9\mathbb{C}}$

5

5

$-1$ $-1$

1

1

$0$ $0$ $0$ $0$ $\delta_{10\mathbb{C}}$ 5 $-5$ $-1$

1

1 $-1$ $0$ $0$ $0$ $0$

Table 2: The complexcharacter of $K=A_{5}\cross C_{2}$

where $\omega=\exp\frac{2\pi\sqrt{-1}}{5},$ $A=-\omega-\omega^{4}=\frac{1-\sqrt{5}}{2},\hat{A}=-\omega^{2}-\omega^{3}=\frac{1+\sqrt{5}}{2}$

.

By Morimoto’s Surgery Theory ([Mor95] [Mor98]),

we can

prove

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

Sm(G) and $RO(K)_{\mathcal{P}}^{\mathcal{L}}=$ Sm(K). Lel $\{\xi_{i}, 1\leq i\leq 14\}$ be the $\mathbb{Z}$-basis of RO(G) such

that the complification of $\xi_{i}$ is $\xi_{i\mathbb{C}}$, and $\{\delta_{i}, 1\leq\iota’\leq 8\}$ the $\mathbb{Z}$-basis of $RO(K)$ such

that the complification of $\delta_{i}$ is $\delta_{i\mathbb{C}}$

.

By calculation,

a

$\mathbb{Z}$-basis of _{$RO(G)_{p}^{\mathcal{L}}$} is

$\{v\}_{1}$ where

$v=2\xi_{5}-2\xi_{6}+2\xi_{7}-2\xi_{8}-\xi_{9}+\xi_{10}-\xi_{11}+\xi_{12}-\xi_{13}+\xi_{14}$, and the$\mathbb{Z}$-basis of $RO(K)_{P}^{\mathcal{L}}$

is$\{x_{I},x_{2}\}$, where$x_{1}=\delta_{3}-\delta_{5}-2\delta_{7}+2\delta_{8}+\delta_{9}-\delta_{10},$ $x_{2}=\delta_{4}-\delta_{6}-2\delta_{7}+2\delta_{8}+\delta_{9}-\delta_{10}$

.

Since the equalities

$|nd_{K}^{G}\delta_{1}=\xi_{1}+\xi_{4)}$ $|nd_{K}^{G}\delta_{2}=\xi_{2}+\xi_{3}$, $|nd_{K}^{G}\delta_{3}=\xi_{13}$, $|nd_{K}^{G}\delta_{4}=\xi_{13}$, $|nd_{K}^{G}\delta_{5}=\xi_{14)}$ $|nd_{K}^{G}\delta_{6}=\xi_{14}$,

$|nd_{K}^{G}\delta_{7}=\xi_{5}+\xi_{7}$

,

$|nd_{K}^{G}\delta_{8}=\zeta_{6}+\xi_{8)}$

$|nd_{K}^{G}\delta_{9}=\xi_{9}+\xi_{t1}$, $1nd_{K}^{G}\delta_{10}=\xi_{10}+\xi_{12)}$

hold,

we

obtain $1nd_{K}^{G}(x_{1})=Ind_{K}^{G}(x_{2})=-v$_, which determines the induction map $1nd_{K}^{G}$

:

$Sm(K)arrow Sm(G)$

.

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