数理解析研究所講究録
第 1922 巻 2014 年 154-165

154

TENTATIVE STUDY ON
EQUIVARIANT SURGERY OBSTRUCTIONS:FIXED POINT SETS OF SMOOTH$A_{5}$-ACTIONS

Masaharu Morimoto
Graduate School of Natural Science and Technology, Okayama University

Abstract. Let be the alternating group on 5letters and let be aclosed smooth manifold diﬀeomorphic to the fixed point set of a smooth-action on a disk. Marek Kaluba proved that if $F$ is even dimensional
then there exists a smooth -action on a closed manifold $X$ being homotopy equivalent to a complex projective space such that the fixed
point set of the -action is diﬀeomorphic to $F$ . In this paper we discuss whether series of manifolds diﬀeomorphic or homotopy equivalent
to complex projective spaces, real projective spaces, or lens spaces, admitsmooth -actions with fixed point set diﬀeomorphic to $F.$
$F$

$G$

$G$

$G$

$G$

$G$

1. INTRODUCTION
be a finite group throughout this paper. For a smooth manifold $M$ , letdenote the family of all manifolds $F$ such that $F=M^{G}$ for some smooth-action on $M$ . For a familyof smooth manifolds, letdenote the union
, and
. Let
of
with
denote the families of disks, spheres,and complex projective spaces, respectively. B. Oliver [19] completely determined
the familyfor
not of prime power order. K. Pawa owski and the author[18, 14] studied
for various Oliver groups $G.$
In order to quote a part of Oliver’s result on, we adopt the notationand for the families of all finite groups satisfying the following properties,
respectively.

Let

$G$

$\mathfrak{F}_{G}(M)$$G$$\mathfrak{M}$$M\in \mathfrak{M}$$\mathfrak{F}_{G}(M)$$\mathfrak{F}_{G}(\mathfrak{D})$$\mathfrak{D},$$\mathcal{F}_{G}(\mathfrak{M})$$\mathfrak{S}$$\mathfrak{P}_{\mathbb{C}}$$G$$\mathfrak{F}_{G}(\mathfrak{S})$$\mathfrak{F}_{G}(\mathfrak{D})$$\mathcal{G}_{\mathbb{C}}^{\sigma},$$\mathcal{G}_{\mathbb{C}}$$\mathcal{E}$2010 Mathematics Subject Classification. Primary $57S17$ ; Secondary $20C15.$
Key words and phrases. equivariant manifold, equivariant framed map, fixed point set.
This research was partially supported by JSPS KAKENHI Grant Number 26400090.

$\mathcal{G}_{\mathbb{R}},$

155$\bullet$possesses a subquotient $K/H$ isomorphic to a dihedral group oforder $2pq$ for some distinct primes and , wherecontains an element being conjugate to its inverse of order $pq$
$G\in \mathcal{G}_{\mathbb{R}}:G$$p$$\bullet$$G\in \mathcal{G}_{\mathbb{C}}^{\sigma}:G$$g$

for some distinct primes andcontains an element$p$$\bullet$$H\triangleleft K\leq G.$$q$$G\in \mathcal{G}_{\mathbb{C}}:G$

$q.$

$g$

of order $pq$ for some distinct primes

$p$

and

$q.$

$\bullet$$G\in \mathcal{E}$: A Sylow 2-subgroup of $G$ is not normal in $G$ , and any element of $G$

is of prime power order.
Note that$\mathcal{G}_{\mathbb{R}}\subset \mathcal{G}_{\mathbb{C}}^{\sigma}\subset \mathcal{G}_{\mathbb{C}}$belongs toto$\mathcal{E}$. Let$A_{5}$

denote the alternating group on 5letters. Then

. B. Oliver [19] says that for$G\in \mathcal{F}_{\mathbb{C}}\cup \mathcal{E}$, a closed manifold$G\in \mathcal{G}_{\mathbb{R}}\Rightarrow no$restrictions on $T(F)$ ,$\bullet G\in \mathcal{G}_{\mathbb{C}}^{\sigma}\backslash \mathcal{G}_{\mathbb{R}}\Rightarrow c_{\mathbb{R}}([T(F)])\in c_{\mathbb{H}}(\overline{KSp}(F))+Tor(\overline{KU}(F))$(1.1)$\bullet G\in \mathcal{G}_{\mathbb{C}}\backslash \mathcal{G}_{\mathbb{C}}^{\sigma}\Rightarrow[T(F)]\in r_{\mathbb{C}}(\overline{KU}(F))+Tor(\overline{KO}(F))$$\bullet G\in \mathcal{E}\Rightarrow[T(F)]\in Tor(\overline{KO}(F))$$G\in \mathcal{E}$belongs

if and only if $\chi(F)\equiv 1mod n_{G}$ and$\mathfrak{F}_{G}(\mathcal{D})$$\bullet$If$F$$A_{5}$

and$F\in \mathfrak{F}_{G}(\mathfrak{D})$

,

,

.

then each connected component of $F$ has same dimension.

is nonsolvable.
Marek $Ka1_{b1}ba[5]$ obtained the next two theorems concerned withThe Oliver number$n_{G}$

above is equal to 1 whenever$G$$\mathfrak{F}_{G}(\mathfrak{P}_{\mathbb{C}})$.

andbe a nontrivial perfect group in the class, suppose that some
. In the casebe a closed manifold in
let.
connected component of is even dimensional. Then $F$ belongs to

Theorem. [5, Theorem 2.6] Let
$F$

$G$

$\mathcal{G}_{\mathbb{C}}$$G\in \mathcal{G}_{\mathbb{C}}\backslash \mathcal{G}_{\mathbb{R}}$$\mathfrak{F}_{G}(\mathcal{D})$$F$$\mathfrak{F}_{G}(\mathfrak{P}\mathbb{C})$.
and $F$ a closed manifold in
Suppose that $F$ is even dimensional. Then $F$ is diﬀeomorphic to the fixed point set

Theorem. [5, Theorem 4.11] Letof a smooth

$G$

$G$

be

$A_{5}$$\mathfrak{F}_{G}(\mathfrak{D})$-action on a closed manifold $X$ which is homotopy equivalent to somecomplex projective space.

) denote the complex (resp. real) projective space of complex. The
of order(resp. real) dimension , and let be a cyclic subgroup ofLet,$P_{\mathbb{C}}^{k}$(resp.

$P_{\mathbb{R}}^{k}$$k$$\Gamma$$\mathbb{C}^{\cross}$

orbit space $L^{2k+1}=S(\mathbb{C}^{k+1})/\Gamma$ is a lens space of dimension $2k+1$ . Let$\geq 3$$\mathfrak{L}$be the156$k=2$ , 3, 4, . . .. By examining and improving the proof
family of lens spacesof [5, Theorem 4.11] by M. Kaluba, we obtain the next result.
$L^{2k+1},$

and $F$ a closed manifold in
an integer $N>0$ possessing the property that for any $k\geq N,$

Theorem 1.1. Let(1)$F\in \mathfrak{F}_{G}(D^{k})$(2)$F\in \mathfrak{F}_{G}(S^{k})$

$G$

be

$A_{5}$$\mathfrak{F}_{G}(\mathfrak{D})$,,(3) if $\dim F\equiv 0$ mod2 then

(4)

$F\in \mathfrak{F}_{G}(X_{k})$to. Then there exists

such that$X_{k}$$F\in \mathfrak{F}_{G}(P_{\mathbb{C}}^{k})$,is a smooth closed manifold homotopy equivalent$P_{\mathbb{R}}^{k},$(5) if $\dim F\equiv 1$ mod2 then$F\in \mathfrak{F}_{G}(Y_{k})$manifold homotopy equivalent tosuch that$Y_{k}$

is a smooth closed$L^{2k+1}.$

This result follows from Theorem 3.4. In Theorem 1.1, one may conjecture that
$P_{\mathbb{R}}^{k}$

and $L^{2k+1}$ can be chosen as$X_{k}$

and$Y_{k}$

respectively, but the author cannot provethe conjecture so far.

Acknowledgment. The author would like to express his gratitude to Marek Kaluba
and Krzysztof Pawa owski for their information related to this research.

2. DIMENSION CONDITIONS OF FIXED POINT SETS

be a finite group. Let be a -manifold and $(H, K)$ a pair of subgroups$H<K\leq G$ . We say thatsatisfies the gap condition, cobordism gap condition, or
strong gap cond.tion for $(H, K)$ if the inequality

Let

$G$

$U$

$G$

$U$

(2.1)

$2 \dim(U_{i}^{H})^{K}<\dim U_{i}^{H},$(2.2)$2 (\dim\{(U_{i}^{H})^{K}\backslash (U_{i}^{H})^{N_{G}(H)}\}+1)\leq\dim U_{i}^{H},$

or
(2.3)

$2\{\dim(U_{i}^{H})^{K}+1\}<\dim U_{i}^{H},$holds, respectively, for any connected component$U^{H_{i}}$of$U^{H}.$

157
Proposition 2.1. Let be a perfect group having a cyclic subgroupthe complex projective space associated with the complex -module$G$of order 2,$C_{2}$$G$$Y$$V=\mathbb{C}^{\oplus m+1}\oplus(\mathbb{C}[G]-\mathbb{C})^{\oplus n},$where $m\geq 0$ and $n\geq 1$ , and

$U$

the

$G$

-tubular neighborhood of$Y^{G}.$(1) $Y$ satisfies the gap condition for $(\{e\}, C_{2})$ if and only if $m+1=n.$

if and only if $m+1\leq n.$satisfies the gap condition for(3) If $m+1\leq n$ thensatisfies the strong gap condition for $(H, K)$ such that
(2)

$U$

$(\{e\}, C_{2})$$U$$\{e\}\neq H<K\leq G$ and $|K:H|\geq 3.$

Proof. We readily see that

$Y^{G}=P_{\mathbb{C}}(\mathbb{C}^{m+1})=P_{\mathbb{C}}^{m}$and$Y^{C_{2}}$has two connectedcomponents$Y_{a}^{C_{2}}=P_{\mathbb{C}}(\mathbb{C}^{m+1}\oplus((\mathbb{C}[G]-\mathbb{C})^{C_{2}})^{\oplus n})=P_{\mathbb{C}}^{m+n(|G|/2-1)}$

and$Y_{b}^{C_{2}}=P_{\mathbb{C}}(((\mathbb{C}[G]-\mathbb{C})_{C_{2}})^{\oplus n})=P_{\mathbb{C}}^{n|G|/2-1}$

Thus we have $\dim Y=2m-2n+2n|G|,$ $\dim Y_{a}^{C_{2}}=2m-2n+n|G|$ and$\dim Y_{b}^{C_{2}}=$$n|G|-2$ . Note the equivalences$\bullet 2(2m-2n+n|G|)<2m-2n+2n|G|\Leftrightarrow m<n$$\bullet 2(n|G|-2)<2m-2n+2n|G|\Leftrightarrow n-2<m.$satisfies the gap condition for $(\{e\}, C_{2})$ if and only if $n-2<m<n$ , namely
satisfies the gap condition for $(\{e\}, C_{2})$ if$m+1=n$ . Since $\dim U^{C_{2}}=\dim Y_{a}^{C_{2}},$and only if $m<n$ , namely $m+1\leq n.$
For any $H\leq G,$
is connected. Let denote the point $[1, 0, . . . , 0]$ in $Y=$
Thus

$Y$

$U$

$U^{H}$$y$$P_{\mathbb{C}}(\mathbb{C}\oplus \mathbb{C}^{\oplus m}\oplus(\mathbb{C}[G]-\mathbb{C})^{\oplus n})$$\mathbb{C}^{\oplus m}\oplus(\mathbb{C}[G]-\mathbb{C})^{\oplus n}$. The tangential representation $T_{y}(Y)$ is isomorphic toas complex$G$-modules. Since $\dim U^{H}=\dim T_{y}(Y)^{H}$ , we get$\dim U^{H}=2\{m+n(|G|/|H|-1)\}=2m-2n+2n|G|/|H|$and$\dim U^{H}-2(\dim U^{K}+1)=2\{n-(m+1)\}+2n|G|\{|K|-2|H|\}/(|H||K|)$ .

In the case $n\geq m+1$ and $|K|\geq 3|H|$ , we conclude $2(\dim U^{K}+1)<\dim U^{H}.$

Theorem 2.2 (Disk Theorem). Let

a connected component of

$F$

with

$G$

be a nontrivial perfect group,$m=\dim F_{0}$ , and $W$ a real

$x_{0}\in F_{0},$$F\in \mathfrak{F}(\mathfrak{D})$$G$$\square$,$F_{0}$-module158
with $\dim W^{G}=m$ . Then there exists a smooth -action on the disk$\dim W+N(|G|-1)$ for some integer $N\geq 0$ satisfying the following

$D$

$G$

of dimension$condition\mathcal{S}.$(1) $D^{G}=F.$(2)

as real -modules.
satisfies the strong gap condition for arbitrary pair $(H, K)$ such that $H<$
is isomorphic to$T_{x_{0}}(D)$(3) $D$$G$$W\oplus(\mathbb{R}[G]-\mathbb{R})^{\oplus N}$$K\leq G.$

and $N$ be as
be an arbitrary integer $\geq N.$ Then there exists a smoothof dimension $\dim W+n(|G|-1)$ satisfying the following

Corollary 2.3. Let $G,$

in Theorem 2.2. Let$n$-action on the disk

$S$

$G$

$F\in \mathfrak{F}(\mathfrak{D})$,$F_{0},$$x_{0}\in F_{0},$$m=\dim F_{0},$ $W,$

$D$

conditions.
(1) $S^{G}=FDF’$ and $F’$ is diﬀeomorphic to $F.$

as real -modules.
satisfies the strong gap condition for arbitrary pair $(H, K)$ such that $H<$(2) $T_{x0}(S)$ is isomorphic to

(3)

$S$

$G$

$W\oplus(\mathbb{R}[G]-\mathbb{R})^{\oplus n}$$K\leq G.$

Proof. We set$X=D\cross D((\mathbb{R}[G]-\mathbb{R})^{\oplus(n-N)})$

. Let

$S$

be the double of $X$ . Thensatisfies the desired conditions.

$S$$\square$

3. DELETING THEOREM AND REALIZATION THEOREM
In this section we will give a deleting theorem and a realization theorem. The
latter is obtained from the formar and Corollary 2.3. Our main result Theorem 1.1

follows from the realization theorem.
denoteLetdenote the alternating group on letters 1 , 3, 4, 5, and let
the alternating group on letters 1 , 3, 4. Unless otherwise stated, we use the$5$$A_{5}$$4$

notation:$\bullet C_{2}=\langle(1,2)(3,4)\rangle$$\bullet$$D_{4}=\langle(1,2)(3,4)$, $(1,3)(2,4)\rangle$$\bullet C_{3}=\langle(1,2,5)\rangle$$\bullet$$D_{6}=\langle(1,2,5)$, $(1,2)(3,4)\rangle$$\bullet C_{5}=\langle(1,3,4,2,5)\rangle$$\bullet$$D_{10}=$ $\langle(1,3,4,2,5)$, $(1,2)(3,4$

$,2$

$,2$

$A_{4}$

159
These groups and$A_{4}$

are regarded as subgroups of$A_{5}.$

Then $N_{G}(C_{2})=D_{4},$ $N_{G}(D_{4})=A_{4},$$N_{G}(C_{3})=D_{6},$ $N_{G}(C_{5})=D_{10}$ , and any maximal proper subgroup of
is conjuThroughout this section, let

be

$G$

$A_{5}$.

$G$

gate to one of$A_{4},$$D_{10},$$D_{6}$. We can readily show

Proposition 3.1. Let $H$ be a maximal proper subgroup of G. Then any two sub-groups of $H$ are conjugate in
Proposition 3.2. Let$\alpha$$G$

if and only if they are conjugate inbe the element of the Burnside ring$\Omega(G)$$H.$

given by$\alpha=[G/A_{4}]+[G/D_{10}]+[G/D_{6}]-[G/C_{3}]-2[G/C_{2}]+[G/\{e\}].$

Then for any proper subgroup $H<G,$

coincides with $[H/H]$ in$res_{H}^{G}\alpha$$\Omega(H)$.

Theorem 3.3 (Deleting Theorem). Let $G=A_{5}.)$ Let $Y$ be a compact connected, with $|\pi_{1}(Y)|<\infty_{f}$ and with a decompositionsmooth -manifold of dimension
$($

$G$

$\geq 5$-tubular neighborhood ofsatisfies the gap condition for $(\{e\}, C_{2})$ , $(\{e\}, C_{3})$ and $(\{e\}, C_{5})$ , and thesuch that$Y^{G}=Y_{0}^{G}DY_{1}^{G}$$\partial Y_{0}^{G}=\emptyset$

Supposecobordism gap condition for
$U$

. Let

$(C_{2}, D_{6})$,$U$be the$G$

and$(C_{2}, D_{10})$$Y_{0}^{G}.$$(C_{3}, A_{4})$. Then there exists asmooth $G$ -manifold $X$ possessing the following properties.

(1) $X^{G}=Y_{1}^{G}.$(2) $X$ is homotopy equivalent to $Y.$
(3)

$X^{H}$

is diﬀeomorphic to$Y^{H}$(4) In the case that $\dim Y\equiv 0$

for any

such that $\{e\}\neq H<G.$
mod2 and $\pi_{1}(Y)=1,$ $X$ is diﬀeomorphic to $Y.$
$H$

Guideline for Proof. There exists a compact connected smooth -submanifold
and the inclusion induced
withsuch that freely acts onofis an isomorphism. First, we construct a -framed
homomorphism$G$$Y\backslash \partial Y$$G$$U\subset U_{1}$$U_{1}\backslash U$$G$$\pi_{1}(U_{1})arrow\pi_{1}(Y)$map $f_{1}:X_{1}arrow Y$ rel.

$Y\backslash \mathring{U}_{1}$(i.e.,$U_{1}$$X_{1}\supset Y\backslash \mathring{U}_{1},$$f_{1}|_{Y\backslash U_{1}^{\circ}}:Y\backslash \mathring{U}_{1}arrow Y\backslash \mathring{U}_{1}$is the identity map, and$f_{1}(X_{2})\subset U_{1}$, where$X_{2}=X_{1}\backslash (Y\backslash \mathring{U}_{1})^{o}\backslash \partial Y)$

Next we convertthat$f_{3}$$f_{2}=f_{1}|_{X_{2}}$: $X_{2}arrow U_{1}$ , to a

is a homotopy equivalence by

$G$

$G$

.

-framed map-surgeries rel.

$\partial U_{1}$$f_{3}$: $X_{3}arrow U_{1}$ suchof isotropy types (H)160
for $H<G$ . The construction of a -framed map is discussed in Section 4. We
perform the -surgeries of isotropy types (H) with $\{e\}<H<G$ by means of thereflection method in [8]. We do the -surgeries of isotropy type $(\{e\})$ by showing
triviality of the algebraic -surgery obstruction in the relevant Bak group describedin [9] with the -cobordism invariance property given in [10] and the inductionrestriction property presented in [13, 4].
$G$

$G$

$G$

$G$

$G$

$\square$

Theorem 3.4 (Realization Theorem). Let $G=A_{5}.)$ Let $W$ a real -module with$\dim W^{G}=m,$ $N$ an integer possessing the property described in Theorem 2.2, andsuch thata compact connected smooth -manifold of dimension. Supposethe -tubular neighborhood ofa point in , and an integer$|\pi_{1}(Z)|<\infty$ and. Further suppose
is isomorphic to,, and the cobordism
andsatisfies the gap condition for. Let $F,$
be as in, and,gap condition for
Theorem 2.2. Then there exists a compact -manifold $X$ satisfying the following
conditions.
$G$

$($

$G$

$Z$

$G$

$Z^{G},$$\partial Z^{G}=\emptyset,$$\geq 5$$Z^{G}$$\geq N$$n$$z_{0}$$V$$W\oplus(\mathbb{R}[G]-\mathbb{R})^{\oplus n}$$T_{z_{0}}(Z)$$(\{e\}, C_{2})$$(\{e\}, C_{5})$$(\{e\}, C_{3})$$(C_{2}, D_{10})$$(C_{2}, D_{6})$$V$$(C_{3}, A_{4})$$F_{0},$$x_{0}\in F_{0}$

$G$

(1)

$X^{G}$

is diﬀeomorphic to $F.$(2) $X$ is homotopy equivalent to $Z.$(3) In the case that $\dim Z\equiv 0$ mod2 and $\pi_{1}(Z)=0,$ $X$ is diﬀeomorphic to $Z.$

-action on the sphere$x_{0}\in F_{0}\subset F$ and $S^{G}=FUF’$ such that
$F\cong F’$ . Let
be the connected sum of and$Y_{0}^{G}=F\# Z^{G}$ and $Y_{1}^{G}=F’$ we have

Proof. Take the smooth

$G$

$S$

described in Corollary 2.3 with

$T_{x_{0}}(S)\cong W\oplus(\mathbb{R}[G]-\mathbb{R})^{\oplus n}$

$S$

$Y$

$Z$

at points
and . By setting. Noteis without boundary.
$x_{0}$$Y^{G}=Y_{0^{G}}\coprod Y_{1}^{G}$and$z_{0}$$Y_{0^{G}}$,satisfies the gap condition for
.
, and
,
and $(\{e\}, C_{5})$ , and the cobordism gap condition forby means of Theorem 3.3, there
fromDeleting the fixed point submanifold
exists a -manifold $X$ satisfying the desired conditions in the theorem, where $X^{G}=$The tubular neighborhood

$U$

of

$(\{e\}, C_{2})$$Y_{0^{G}}$$(C_{2}, D_{6})$$F_{0}^{G}$$(C_{2}, D_{10})$$(\{e\}, C_{3})$$(C_{3}, A_{4})$

$Y$

$G$

$F’\cong F.$

4. EQUIVARIANT COHOMOLOGY THEORY$\square$$\omega(\bullet)_{G}^{*}$

Equivariant surgeries are operated on smooth

on

$G$

$G$

AND $G$-FRAMED MAPS

-manifolds, but more precisely-framed maps. T. Petrie gave an idea to construct$G$-framed maps by using

161. In order toand the Burnside ring
the equivariant cohomology theory
construct our -framed maps, we employ a modification given in [11] of Petrie’s$\Omega(G)$$\omega_{G}^{*}$$G$

construction.

Let$\Omega(G)$denote the Burnside ring, i.e.
$\Omega(G)=${ $[X]|X$ is a finite G-CW complex},

where $[X]=[Y]$ if and only if $\chi(X^{H})=\chi(Y^{H})$ for all $H\leq G$ . The next proposition

is well known, see [2, 20, 1].
Proposition 4.1. Let

potnent$\beta$in thebe a nontrivial perfect group. Then there exists an idemringsuch that $\chi_{G}(\beta)=1$ and $\chi_{H}(\beta)=0$ for all$G$$Burn\mathcal{S}ide$$\Omega(G)$$H\neq G.$

Let $S(G)$ and $S(G)_{\max}$ denote the set of all subgroups and all maximal proper
subgroups of , respectively. For a subgroup $H$ of , (H) stands for the -conjugacy
$G$

$G$

$G$

class containing $H$ , i.e.
$(H)=\{gHg^{-1}|g\in G\}.$

The$\beta$in Proposition 4.1 has the form

(4.1)$\beta=[G/G]-\sum_{(K)\subset S(G)_{\max}}[G/K]-\sum_{(H)\subset \mathcal{F}}a_{H}[G/H],$

for some$G$-invariant lower closed $\mathcal{F}\subset S(G)\backslash (S(G)_{\max}\cup(G))$ and

Proposition 4.2. Let $G=A_{5}$ and let$\alpha$$a_{H}\in \mathbb{Z}.$be the element of the Burnside ring$\Omega(G)$given by$\alpha=[G/A_{4}]+[G/D_{10}]+[G/D_{6}]-[G/C_{3}]-2[G/C_{2}]+[G/\{e\}].$

Then for any proper subgroup $H<G,$

Proposition 4.3. Let

$G$

be

$A_{5}$$res_{H}^{G}a$coincides with $[H/H]$ in$\Omega(H)$.

. Then there exists a finite G-CW complex $Z$ fulfillingthe following conditions.
(1) $Z^{G}=\{z_{0}, z_{1}\}.$(2) $\bigcup_{H\in S(G)_{\max}}Z^{H}=\{z_{0}, z_{1}\}*S(G)_{\max}$ , where each subgroup in $S(G)_{\max}$ is re-garded as a point.

Thus$Z^{H}$[-1, 1] for each $H\in S(G)_{\max}.$

is homeomorphic to the 1-dimensional disk

162
(3)

$Z^{D_{4}}=Z^{A_{4}}$(4)$Z^{H}$

andis contractible for any subgroup $H<G.$$Z^{C_{5}}=Z^{D_{10}}.$

The G-CW complex $Z$ in this lemma is constructed using Oliver-Petrie’s G-CW-

surgery theory [21] and the wedge sum technique [16].
, hence weLet
and letbe the one-point compactification of. For a finite G-CW complex $X$ with base point inwrite$M_{n}=\mathbb{C}[G]^{\oplus n}$$M_{n}$$M_{n}^{\cdot}$$X^{G},$$M_{n}^{\cdot}=M_{n}\cup\{\infty\}$$\overline{\omega}_{G}^{0}(X)=\lim_{narrow\infty}[X\wedge M^{\cdot}, M^{\cdot}]_{0}^{G},$wherepointed $G$

stands for the set of all homotopy classes of maps in the category of,-spaces. For a finite G-CW complex $Z,$
is defined to be$-]_{0}^{G}$$\omega_{G}^{0}(Z)$$\overline{\omega}_{G}^{0}(Z^{+})$

where$Z^{+}=ZD(G/G)$

and $G/G$ is regarded as the base point of

For the set$S$of all powers$\beta^{k},$$k\in \mathbb{N}$$Z^{+}.$, the restriction$j^{*}:S^{-1}\omega_{G}^{0}(Z)arrow S^{-1}\omega_{G}^{0}(Z^{G})$induced by the inclusion map $j:Z^{G}arrow Z$ is an isomorphism. It is obvious that forany element$\gamma\in\omega_{H}^{0}(Z)$and any proper subgroup $H$ of $G,$

Lemma 4.4. Let be a nontrivial perfect group,
and a finite G-CW complex with$G$$\beta$$res_{H}^{G}\beta\cdot\gamma=0_{Z}$in$\omega_{H}^{0}(Z)$.

the element in Proposition 4.1,

$Z$

(4.2)

$Z^{G}=\{z_{0}, z_{1}\}.$

such that
andin
Then there exists an element. In addition, for any proper subgroup $H<G$ , there exists a ‘homotopy’and$\gamma\in\omega_{G}^{0}(Z)$$\gamma|_{z\mathfrak{o}}=\beta$$\gamma|_{z_{1}}=0_{z_{1}}$$\Omega(G)$$\beta\gamma=\gamma$$\Gamma_{H}\in\omega_{H}^{0}(Z\cross I)$

and

from

$\Gamma_{H}|_{z_{1}\cross I}=0_{z_{1}xI}$to, such that
$res_{H}^{G}\gamma$$0_{Z},$$rel.$$z_{1}\cross I,$$i.e.$$res_{H}^{G}\beta\cdot\Gamma_{H}=\Gamma_{H}$$\Gamma_{H}|_{Z\cross\{0\}}=res_{H}^{G}\gamma,$. Moreover, for any pair of distinct

proper subgroups $H$ and $K$ of , there exists a ‘homotopy’
$G$

from

$res_{H\cap K}^{H}\Gamma_{H}$to$res_{H\cap K}^{K}\Gamma_{K_{Z}}rel.$$z_{1}\cross I\cross I$$\Gamma_{H}|_{Z\cross\{1\}}=0_{Z},$and$\overline{\Gamma}_{H,K}\in\omega_{H\cap K}^{0}(Z\cross I\cross I)$$Z\cross\partial I\cross I.$

As a next step, consider the elements. Recall that an element
by a -map$1_{Z}-\gamma\in\omega_{G}^{0}(Z)_{\}}1_{Z\cross I}-\Gamma_{H}\in\omega_{H}^{0}(Z\cross I)$$1_{Z\cross I\cross I}-\overline{\Gamma}_{H,K}\in\omega_{H\cap K}^{0}(Z\cross I\cross I)$$\alpha\in\omega_{G}^{0}(Z)$$G$$Z^{+}\wedge M^{\cdot}arrow M^{\cdot}$

preserving the base point$\infty$, where$M=\mathbb{C}[G]^{\oplus n}$

for some

$n.$

, and

is represented

163
Lemma 4.5. Let be a nontrivial perfect group, the element in Proposition 4.1,
, andand a finite G-CW complex with $Z^{G}=\{z_{0}, z_{1}\}$ . Then there exist maps
satisfying the following conditions (1) $-(3)$ , where $H$ and $K$ range all proper$G$$\beta$$Z$$\alpha,$$A_{H}$$\overline{A}_{H,K}$

subgroups of(1)$\alpha$such that $H\neq K.$

is a map of pointed

the
(2)

$G$

$A_{H}$

above, and$\gamma$$G$-spaces$Z^{+}\cross M^{\cdot}arrow M^{\cdot}$$\alpha|_{\{z_{1}\}^{+}\wedge M}\cdot=id_{\{z_{1}\}\wedge M}+\cdot\cdot$

is a homotopy of pointed $H$ -spaces

where$1_{Z}$

such that $[\alpha]=1-\gamma$ for$(Z\cross I)^{+}\wedge M^{\cdot}arrow M^{\cdot}$

from$\alpha$to$1_{Z},$: $Z^{+}\wedge M^{\cdot}arrow M$ and $1_{Z}(z, v)=v$ for all $z\in Z$ and $v\in M,$ $rel.$$\{z_{1}\}^{+}\wedge M^{\cdot}$(3)$\overline{A}_{H,K}$

from

is a homotopy of pointed $H\cap K$ -spaces $((Z\cross I)\cross I)^{+}\wedge Marrow M^{\cdot}$$A_{H}$

to $A_{j}rel.$and$(z_{1}\cross I)^{+}\wedge M$$(Z\cross\partial)^{+}\wedge M^{\cdot}$

Proposition 4.6. Let $G=A_{5}$ and $\mathcal{K}=(A_{4})\cup(D_{10})\cup(D_{6})\cup(D_{4})\cup(C_{5})$ . Let $Z$

be the finite G-CW complex in Proposition 4.3. Then there exist maps

$A_{H}$, and$\alpha^{-1}(0)$con-$\alpha,$of Lemma 4.5 satisfying the additional conditions:
(1)
and
for

$\overline{A}_{H,K}$$|(\alpha|_{z0}^{-1}(0))^{H}|=1$$\alpha|_{z0}^{-1}(0)^{G}=\emptyset$(2) For each$H\in \mathcal{K}$taining both

$H\in \mathcal{K}.$, there exists a connected component $X(H)$ of$(\alpha|_{z0}^{-1}(0))^{H}$

and $(z_{1},0)$ such that$\alpha$is transversal on $X(H)$ to

on $X(H)$ is the identity, and the projection $Z\cross Marrow Z$ diﬀeomorphically maps $X(H)$ to(3) For each pair of $H\in S(G)_{\max}$ andwith $L\leq H$ , there exists a$0\subset M$, the normal derivative of$\alpha$$Z^{H}.$$L\in \mathcal{K}$connected component $W(H, L)$ of $A_{H}^{-1}(O)$ containing $X(H)^{L}\cross\{0\}(\subset Z\cross$$M\cross I)$

and $Z^{L}\cross 0\cross\{1\}(\subset Z\cross M\cross I)$ such that

$A_{H}$

is transversal on

on $W(H, L)$ is the identity,
and the projection $Z\cross I\cross Marrow Z\cross I$ diﬀeomorphically maps $W(H, L)$ to$W(H, L)$ to $0\subset M$ , the normal derivativeof$A_{H}$$Z^{L}\cross I.$: $Xarrow Y$ such that $X$ and:, and an isomorphism
are compact smooth -manifold andof real -vector bundles for some integer$m\geq$ O. In the following we supposeisis connected and :of degree 1,
A

$Y$

$G$

-framed map $f=(f, b)$ consists of a -map
$G$

$G$

$f$

$b$

$f(\partial X)\subset\partial Y$$T(X)\oplus\epsilon_{X}(\mathbb{R}^{m})arrow f^{*}T(Y)\oplus\epsilon_{X}(\mathbb{R}^{m})$

$Y$

$G$

$f$

$(X, \partial X)arrow(Y, \partial Y)$

164
Lemma 4.7. Let be a compact smooth -manifold with a decomposition $Y^{G}=$such that. Let be the -tubular neighborhood of. Then there
exist a -framed map $f=(f, b)$ , $H$ -framed cobordisms $F_{H}=(F_{H}, B_{H})$ : $f\sim id_{Y},$$rel.$:
for $H<G$ , and $H\cap K$ -framed cobordisms$Y$$Y_{0^{G}}DY_{1}^{G}$

$G$

$U$

$\partial Y_{0^{G}}=\emptyset$$G$$Y_{0^{G}}$$G$$Y\backslash U$$\overline{F}_{H,K}=(\overline{F}_{H,K}, \overline{B}_{H,K})$$F_{H}\sim F_{K}rel.$$((Y\backslash \mathring{U})\cross I)\cup(Y\cross\partial I)$, for $H,$ $K<G$ such that $H\neq K$ , where$f:Xarrow Y,$$b:T(X)\oplus\epsilon_{X}(\mathbb{R}^{m})arrow f^{*}(Y)\oplus\epsilon_{X}(\mathbb{R}^{m}))$$F_{H}:W_{H}arrow Y\cross I,$$B_{H}$:$T(W_{H})\oplus\epsilon_{W_{H}}(\mathbb{R}^{m})arrow F_{H}^{*}T(Y\cross I)\oplus\epsilon_{W_{H}}(\mathbb{R}^{m})$,$\overline{F}_{H,K}:\overline{W}_{H,K}arrow Y\cross I\cross I,$$\overline{B}_{H,K}:T(\overline{W}_{H,K})\oplus\epsilon_{\overline{W}_{H,K}}(\mathbb{R}^{m})arrow\overline{F}_{H}^{*},{}_{K}T(Y\cross I\cross I)\oplus\epsilon_{\overline{W}_{H,K}}(\mathbb{R}^{m})$,for some integer $m>0.$This lemma is obtained by the arguments in [11].

Lemma 4.8. Let $G=A_{5}$ and $\mathcal{K}=(A_{4})\cup(D_{10})\cup(D_{6})\cup(D_{4})\cup(C_{5})$ . Then theandin Lemma 4.7 can be chosen so that
andframed maps $f,$
, respectively, for all $H_{f}K\in S(G)_{m}$are $N_{H}(L)$ -diﬀeomorphic toandand allwith $L\leq H.$$F_{H}$$X^{L}$$\overline{F}_{H,K}$$Y^{L}$$W_{H}^{L}$$Y^{L}\cross I$$L\in \mathcal{K}$This modification is achieved by using Proposition 4.6 and the reflection method

in [8].

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Graduate School of Natural Science and Technology
Okayama University
Tsushimanaka 3-1-1, Kitaku
Okayama, 700-8530 Japan-mail address: morimoto@ems.okayama-u.ac.jp$E$