数理解析研究所講究録
第 1922 巻 2014 年 147-153

147

GROUP ACTIONS ON COMPLEX PROJECTIVE SPACESVIA GROUP ACTIONS ON DISKS AND SPHERESMAREK KALUBA AND KRZYSZTOF PAWALOWSKI

Dedicated to Professors Mikiya Masuda and Masaharu Morimotoon the occasion of their 60th birthdays
Keywords: smooth group action, fixed point set, complex projective space.
Primary $57S15,$ $57S25$ . Secondary $55N15.$$AMS$ Subject Classification (2010):

1. Two QUESTIONS IN TRANSFORMATION GROUPS

When studying classification problems in the theory of transformation groups one usually focuses onsmooth actions of compact Lie groups on specific manifolds $M$ , such as Euclidean spaces, disks, spheres,and complex projective spaces. Consider the following two basic related questions.
in $M$ ?
(1) Which manifolds $F$ are diﬀeomorphic to the corresponding fixed points sets$F$in(2) Which -vector bundles over are isomorphic to the -normal bundles of$G$$M^{G}$$M^{G}$$G$$G$$M^{7}$

Our goal is to discuss results related to (1) obtained so far for actions on Euclidean spaces, disks, andspheres, and then to describe new results for actions on complex projective spaces obtained in [2]. Hence,
every manifold $F$ which occurs as the fixed point set is a second-countable space, i.e., is paracompact
and $F$ has countably many connected components, possibly not of the same dimension.
We do not discuss the Smith theory and the converse related results. Except for Theorem 2.1 below,
the acting group is always a finite group not of prime power order.
$F$

$G$

2. EQUIVARIANT TRIANGULATION AND THICKENING CONCLUSION
For a finite dimensional countable CW-complex $X$ , let $KO(X)$ be the reduced real $K$-theory of $X.$
More generally, if is a compact Lie group and $X$ is a G-CW complex (i.e., a topological space built up
from -equivariant cells), we denote by $KO_{G}(X)$ the -equivariant reduced real $K$-theory of $X.$
$G$

$G$

$G$

be a compact Lie group and let $F$ be a smooth manifold such that $F$ is compact
Theorem 2.1. Let. Let be a real -vector bundle over $F$ such that $\dim\nu^{G}=0$ . Then the following two
statements are equivalent.
(1) There exists a finite (resp., finite dimensional countable) contractible G-CW complex $X$ suchstably extends to a real -vector bundle over $X$ , i. e.,that $X^{G}=F$ , and the Whitney sum
lies in the image of the restriction map
the class$G$$(resp., \partial F=\emptyset)$

$G$

$\nu$

$G$

$\tau_{F}\oplus v$$[\tau_{F}\oplus\nu]$$\overline{KO}_{G}(X)arrow\overline{KO}_{G}(F)$.

(2) There exists a smooth action of on a disk (resp., Euclidean space) $M$ such that (i) the fixed
point set $M^{G}$ is diﬀeomorp $hic$ to $F$ , and (ii) the $G$ -equivariant normal bundle of $M^{G}$ in $M$ isstably isomorphic to
$G$

$v.$

has the structure of a real
For a smooth -manifold $M$ with fixed point set $F$ , the tangent bundle, where is the -equivariant normal bundle of $F$ in $M.$
-vector bundle over $M$ such thatand $\dim\nu^{G}=0$ . Moreover, by the Equivariant
In particular, acts trivially on the tangent bundle$M$
Triangulation Theorem [1],has the structure of a G-CW complex containing $F$ as a subcomplex.
Therefore, in Theorem 2.1, if (2) is true, so is (1). The converse implication, (1) implies (2), follows by
the Equivariant Thickening Theorem [11].
$G$$\tau_{M}$$G$$\tau_{M}|_{F}\cong\tau_{F}\oplus v$

$G$

$v$

$G$

$\tau_{F}$This is a research report based on a talk given by K. Pawalowski at the RIMS conference “‘ Topology and Algebraic, Kyoto, May 26-30, 2014.
Structures of Vansformation G$\mathcal{I}$$roups’\rangle$

148
M. KALUBA AND K. PAWALOWSKI

3. SIX$G$-FIXED POINT SET BUNDLE CONDITIONSthe
The discussion below is based on the work of Oliver [10]. For a finite group , denote byfamily of subgroups of $G$ of prime power order. Two real (resp., complex) -modules and are called.
-matched if $U$ and $V$ are isomorphic as real (resp., complex) $P$-modules for every
$G$

$G$

$\mathcal{P}(G)$

$V$

$U$

$P\in \mathcal{P}(G)$$\mathcal{P}(G)$of finite groups not of prime power order,
Definition 3.1. The Oliver three-class series
is defined as follows.
if there exist two real-matched -modules and such thatand
if there exist complex-matched -modules and such that, and moreover andare self-conjugate.1
and-matched -modules and such that
if there exist complex
and$G$$\mathcal{G}_{R}\subset C_{\mathbb{C}}\subset \mathcal{G}_{\mathbb{C}}$$G$$\mathcal{P}(G)$$G\in \mathcal{G}_{\mathbb{R}}$

$U$

$V$

$\dim_{\mathbb{R}}(U^{G})=0$

$U$

$V$

$\dim_{\mathbb{C}}(U^{G})=0$

$U$

$V$

$\dim_{\mathbb{C}}(U^{G})=0$$\dim_{\mathbb{R}}(V^{G})=1.$$G\in C_{\mathbb{C}}$$G$$\mathcal{P}(G)$

$V$

$U$

$\dim_{\mathbb{C}}(V^{G})=1$$G$$\mathcal{P}(G)$$G\in \mathcal{G}_{\mathbb{C}}$$\dim_{\mathbb{C}}(V^{G})=1.$

Lemma 3.2. The follovnng three statements are true.
such that $H/N$ is isomorphic to the(1)
if and only if there exist subgroups
dihedral group of order $2pq$ for some two distinct primes andan element $g\in G$ such that is not of prime power order, and(2)
if and only if there
is conjugate to its inverse(3)
if and only if has an element not of prime power order.
$N\underline{\triangleleft}H\leq G$$G\in \mathcal{G}_{R}$

$q.$

$p$

$ex\iota sts$

$G\in C_{\mathbb{C}}$$g$$g^{-1}.$

$g$

$G$

$G\in \mathcal{G}_{\mathbb{C}}$$g$

be the class of groups
Let be the class of finite groups not of prime power order. Let
i.e.,
,
.
if has an element
Note
that
.
2-SylowanormalSet
subgroupwithis not normal in $G.$of prime power order such that is conjugate to its inverse, then$\mathcal{G}_{2}^{\triangleleft}\subset \mathcal{G}$$\mathcal{G}$$G$$C_{\mathbb{C}}\subset \mathcal{G}_{2}^{\sqrt{}}$$\mathcal{G}_{2}^{i}=\mathcal{G}\backslash \mathcal{G}_{2}^{\triangleleft}$$G_{2}$$G\in \mathcal{G}$

$g$

not

$G_{2}$$g$

Definition 3.3. The Oliver six-class splitting of the class$\mathcal{G}$by the Oliver three-class series$\mathcal{G}_{\mathbb{R}}\subset C_{\mathbb{C}}\subset \mathcal{G}_{\mathbb{C}}\subset \mathcal{G}=\mathcal{G}_{2}^{\triangleleft}\cup \mathcal{G}_{2}^{4}$and the two classes(1)and$\mathcal{G}_{R}$

(2)
(3)

$\mathcal{G}_{2}^{\triangleleft}$$C_{\mathbb{C}}\backslash \mathcal{G}_{\mathbb{R}}$$(\mathcal{G}_{\mathbb{C}}\backslash C_{\mathbb{C}})\cap \mathcal{G}_{2}^{4}$and$\mathcal{G}_{2}^{\sqrt{}}$, is the following decomposition of, both contained inandand$\mathcal{G}$into six mutually disjoint classes:$\mathcal{G}_{2}^{i},$$(\mathcal{G}\backslash \mathcal{G}_{\mathbb{C}})\cap \mathcal{G}_{2}^{\sqrt{}},$$(\mathcal{G}\backslash \mathcal{G}_{\mathbb{C}})\cap \mathcal{G}_{2}^{\triangleleft}.$$(\mathcal{G}_{\mathbb{C}}\backslash C_{\mathbb{C}})\cap \mathcal{G}_{2}^{\triangleleft}=\mathcal{G}_{\mathbb{C}}\cap \mathcal{G}_{2}^{\triangleleft}$

Consider the following maps (group homomorphisms):-the complexification of real bundles$c_{R}:\overline{KO}(F)arrow\overline{KU}(F) , [\xi]\mapsto[\xi\otimes \mathbb{C}],$-the quaternization of complex bundles$q_{\mathbb{C}}:\overline{KU}(F)arrow\overline{KSp}(F) , [\xi]\mapsto[\xi\otimes\mathbb{H}],$-the complexification of symplectic bundles$c_{\mathbb{H}}:\overline{KSp}(F)arrow\overline{KU}(F) , [\xi]\mapsto[c_{\mathbb{H}}(\xi)],$-the realification of complex bundles$r_{\mathbb{C}}:\overline{KU}(F)arrow\overline{KO}(F) , [\xi]\mapsto[r_{\mathbb{C}}(\xi)].$For an abelian group$A$, the subgroup Div $A$ of quasidivisible elements of$DivA=\bigcap_{\varphi}Ker(\varphi)$$A$

is defined as,where varies within homomorphisms mapping into free abelian groups. Note that if is finitely
generated then quasidivisible elements are simply torsion elements. In particular, if $F$ is a compact smoothmanifold, the $K$ -theory groups of $F$ are finitely generated, and therefore$A$$\varphi$$DivK(F)=TorK(F)$for the real, complex, and symplectic$K$-theory groups of $F.$

1 A complex $G$-module is self-conjugate if it is isomorphic to its complex conjugate.

$A$

149
GROUP ACTIONS ON COMPLEX PROJECTIVE SPACESFIGURE 1. Oliver six-class splitting of$\mathcal{G}$with six G-fixed point set bundle conditions

The six G-fixed point set bundle conditions defined below depend on the classes in the Oliver six-classsplitting of , the class of finite groups $G$ not of prime power order, described in Definition 3.3.
$\mathcal{G}$Definition 3.4. The G-fixed point set bundle conditions. Letof a smooth manifold $F$ is said to be well-G-located inbundle: there is no restriction on the class(1) if$G\in \mathcal{G}$. Then the class$\overline{KO}(F)$$\tau_{F}$$G\in \mathcal{G}_{\mathbb{R}}$$[\tau_{F}]\in\overline{KO}(F)$(2) if$G\in C_{\mathbb{C}}\backslash \mathcal{G}_{\mathbb{R}}$.

:$[\tau_{F}\otimes \mathbb{C}]\in c_{\mathbb{H}}(\overline{KSp}(F))+Div\overline{KU}(F)$(3) if$G\in(\mathcal{G}_{\mathbb{C}}\backslash C_{\mathbb{C}})\cap \mathcal{G}^{\oint_{2}}$:$[\tau_{F}]\in r_{\mathbb{C}}(\overline{KU}(F))+Div\overline{KO}(F)$(4) if$G\in(\mathcal{G}\backslash \mathcal{G}_{\mathbb{C}})\cap \mathcal{G}_{2}^{4}$:$[\tau_{F}]\in Div\overline{KO}(F)$(5) if$G\in \mathcal{G}_{\mathbb{C}}\cap \mathcal{G}_{2}^{\triangleleft}$$[\tau_{F}]\in r_{\mathbb{C}}(\overline{KU}(F))$(6) if.

:$(G\in \mathcal{G}\backslash \mathcal{G}_{\mathbb{C}})\cap \mathcal{G}_{2}^{\triangleleft}$.

:$[\tau_{F}]\in r_{\mathbb{C}}(Div\overline{KU}(F))$

.

.

.

, provided:$[\mathcal{T}_{F}]$of the tangent

150
$M$ ,

KALUBA AND K. PAWALOWSKI

4. $0$ LIVER NUMBER AND EQUIVARIANT EXTENSION OF BUNDLES
The results presented in this section have been obtained by Oliver [9, 10].

not of prime power order, the set{ $\chi(X^{G})-1:X$ is a finite contractible G-CW complex}
forms a subgroup of the group of integers.
Proposition 4.1. For a finite group$G$$\mathbb{Z}$Therefore, if $X$ is a finite contractible G-CW complex, then(mod$\chi(X)\equiv 1$$n_{G}$)

as to the Oliver number of $G.$
for a unique integer $n_{G}\geq 0$ . We refer to
If $n_{G}=1,$ is called an Oliver group. In algebraic terms this means that contains no normal series
such that $P$ and $G/H$ are of prime power order and $H/P$ is cyclic. Examples
of subgroupsof Oliver groups include finite nonsolvable groups, as well as finite nilpotent groups with three or morenoncyclic Sylow subgroups.
$n_{G}$$G$$G$$P\underline{\triangleleft}H\underline{\triangleleft}G$

Theorem 4.2. Let be a finite group not of prime power order, and let $F$ be a finite $CW$-complex. Thenis homeomorphic to $F$
there exists a finite contractible G-CW-complex $X$ such that the fixed point set
if and only if $\chi(F)\equiv 1(mod n_{G})$ .
$G$$X^{G}$

denote the subgroup of consisting of the infinitely
For an abelian group and a prime , letdenote the localization of atdivisible elements of . Moreover, let
Let be a finite group not of prime power order. For a finite dimensional, countable CW-complex $F,$
consider the abelian group$A$$Div_{p}^{\infty}$

$p$

$A$

$p$

$A$

$A$

$A$

$A_{(p)}$

$p.$

$G$

$\overline{KO}_{\mathcal{P}(G)}(F)=\overline{KO}(F)\oplus\oplus\overline{KO}_{P}(F)_{(p)}/Div_{p}^{\infty}\overline{KO}_{P}(F)_{(p)}$$P\neq\{e\}$.
where $P$ varies within the family
According to Theorem 4.2, the Euler characteristic is the only obstruction for a finite CW-complex$F$ to occur as the fixed point set of a finite contractible G-CW complex $X$ .
The possibility of stable
extension of a -vector bundle over $F$ to a -vector bundle over $X$ is obstructed by the location oflies in the kernel of), namely, the stable extension of exists if and only if
the classinthe canonical map $KO_{G}(F)arrow KO_{\mathcal{P}(G)}(F)$ .
In the case where $X$ is not finite, the stable extension of is obstructed in the same way, but there is
no restriction on the Euler characteristic of $F.$$\mathcal{P}(G)$$G$$[\eta]$$\eta$

$G$

$\xi$

$\xi$

$KO_{G}\underline{(F}$$[\eta]$$\eta$$\xi$$\eta$

Theorem 4.3. Let be a finite group not of prime power order, and let be a real -vector bundle overa smooth manifold $F$ , such that $\dim\nu^{G}=0$ . Assume also that $F$ is compact and the Euler characteristic$\chi(F)\equiv 1(mod n_{G})$ . Then the following three statements are equivalent.
$G$(1) The class(2) The class$[\tau_{F}]$

$\nu$

$G$

.
is well-G-located inlies in the kernel of the canonical map$\overline{KO}(F)$$[\tau_{F}\oplus\nu]$$\overline{KO}_{G}(F)arrow\overline{KO}_{\mathcal{P}(G)}(F)$.

(3) There exists a finite contractible G-CW complex $X$ such that $X^{G}=F$ and the classin the image of the restriction map$\overline{KO}_{G}(X)arrow\overline{KO}_{G}(F)$$[\tau_{F}\oplus v]$

lies

.

Theorem 4.4. Let be a finite group not of prime power order, and let be a real -vector bundle overa smooth manifold $F$ , such that $\dim\nu^{G}=0$ . Then the following three statements are equivalent.
.
(1) The classis well-G-located in(2) The classlies in the kernel of the canonical map
$G$

$\nu$

$G$

$\overline{KO}(F)$$[\tau_{F}]$$[\tau_{F}\oplus\nu]$$\overline{KO}_{G}(F)arrow\overline{KO}_{\mathcal{P}(G)}(F)$.

(3) There exists a finite $dimensional_{Z}$ countable, contractible G-CW complex $X$ such that $X^{G}=F$and the classlies in the image of the restriction map$[\tau_{F}\oplus\nu]$$\overline{KO}_{G}(X)arrow\overline{KO}_{G}(F)$.

151
GROUP ACTIONS ON COMPLEX PROJECTIVE SPACES

5. GROUP ACTIONS ON DISKS AND EUCLIDEAN SPACES
Some of the results of this section were obtained in [11, 12], and the complete classification theorems
presented here go back to Oliver [10].
Theorem 5.1. Let be a group not of prime power order. Then there exists a smooth action ofthat the fixed point set is diﬀeomorphic to a smooth manifold $F$ if and only ifsome disk(i) $F$ is compact, $\chi(F)\equiv 1(mod n_{G}$ and(ii) the classis well-G-located in $KO(F)$ .
$G$

$G$

on

$\mathcal{S}uch$$[\tau_{F}]$

Theorem 5.2. Let be a group not of prime power order. There exists a smooth action of $G$ on someis diﬀeomorphic to a smooth manifold $F$ if and only if
Euclidean space such that the fixed point(i) the boundary of $F$ is empty, and.
(ii) the classis well-G-located in$G$$\mathcal{S}et$$\overline{KO}(F)$$[\tau_{F}]$

Theorems 5.1 and 5.2 follow from Theorems 4.3 and 4.4, respectively, and Theorem 2.1.

6. GROUP ACTIONS ON SPHERESThe results of this section have been obtained in the series of papers $[4]-[8]$ . If is a finite non-trivial
perfect group, then any Sylow 2-subgroup of $G$ is not normal in . Therefore, the union of the classes
$G$

$G$

$\mathcal{G}_{\mathbb{R}}, C_{\mathbb{C}}\backslash \mathcal{G}_{\mathbb{R}}, (\mathcal{G}_{\mathbb{C}}\backslash C_{\mathbb{C}})\cap \mathcal{G}_{2}^{4}, (\mathcal{G}\backslash \mathcal{G}_{\mathbb{C}})\cap \mathcal{G}_{2}^{4}$(cf. Definition 3.3) contains all finite non-trivial perfect groups. Moreover, every of the four classes above
contains an infinite family of perfect groups.
be a finite perfect group, and let $F$ be a smooth manifold. There exists a smooth
Theorem 6.1. Let
is diﬀeomorphi.c to $F$ and $S^{P}\neq S^{G}$ for every
action of $G$ on a sphere such that the fixed point set, if and only if(i) $F$ is closed and.
(ii) the classis well-G-located in$G$$S^{G}$$S$$P\in \mathcal{P}(G)$$\overline{KO}(F)$$[\tau_{F}]$Let be a finite group with an element not of prime power order. Assume that has a normal Sylow
by Lemma 3.2. Unravelling the notion of well-G-location, we see that
. Then2-subgrouplies in the image of the map
is well-G-located in $KO(F)$ if and only if
$G$

$G$

$G_{2}$$G\in \mathcal{G}_{\mathbb{C}}\cap \mathcal{G}_{2}^{\triangleleft}$$[\tau_{F}]$$[\tau_{F}]$$r_{\mathbb{C}}:\overline{KU}(F)arrow\overline{KO}(F)$.

This amounts to $F$ being a stably complex manifold, i.e., the stable normal bundle of $F$ admits a complex
structure. In particular, the dimensions of the connected components of $F$ are of the same parity.
Theorem 6.2. Let
be a finite Oliver group with a quotient isomorphic to the cyclic group of order $pqr$
primes $p,$ , and . Moreover, suppose
is normal in G. Then there exists a smooth
is diﬀeomorphic to $F$ and $S^{P}\neq S^{G}$ for every
on a sphere such that the fixed point set, if and only if(i) $F$ is closed and(ii) $F$ is stably complex.
$G$for threeaction of$di_{\mathcal{S}}tinct$

$q$

$G$

$G_{2}$$r$$S^{G}$$S$$P\in \mathcal{P}(G)$In particular, Theorem 6.2 holds for any finite abelian, more generally, finite nilpotent group with threeor more noncyclic Sylow subgroups.

7. GROUP ACTIONS ON COMPLEX PROJECTIVE SPACES
The results of this section are obtained in the $PhD$ Thesis of Marek Kaluba [2].
Theorem 7.1. Let be a finite perfect group and let $F$ be a smooth manifold. Assume also that either(1) or (2) below holds.
and (i) $F$ is closed and (ii) there is no restriction on(1)(2)
and (i) $F$ is closed, the connected components of $F$ all are even dimensional, and (ii) the.
class
is well-G-locatel inon a complex projective space such that the fixed point set isThen there exists a smooth action of
diﬀeomorphic to $F.$
$G$$[\tau_{F}].$$G\in \mathcal{G}_{\mathbb{R}}$$G\in \mathcal{G}_{\mathbb{C}}$$[\tau_{F}]$$\overline{KO}(F)$

$G$

152
M. KALUBA AND K. PAWALOWSKI

The idea of the proof Consider a smooth action of on the sphereof dimension $2n$ for some integer$n\geq 1$ , with the given fixed point set $F$ , obtained by Theorem 6.1. Next, modify the action so that the
fixed point set consists of $F$ and an isolated point . Aside, create the complex projective spaceequipped with the linear action of coming from the projectivisation
of $V=T_{x}(S^{2n})$ , thetangent -representation space at the point . Finally, perform the -equivariant connected sum$S^{2n}$$G$$\mathbb{C}P^{n}$

$x$

$G$

$P_{\mathbb{C}}(V\oplus \mathbb{C})$

$G$

$G$

$x$

$S^{2n}\#\mathbb{C}P^{n}=S^{2n}\# P_{\mathbb{C}}(V\oplus \mathbb{C})$around the points

$x$

in

$S^{2n}$and $[0 :. . . : 0 : 1]$ in$P_{\mathbb{C}}(V\oplus \mathbb{C})$

Theorem 7.1 does not include perfect groups infive and six letters, respectively.

$\mathcal{G}\backslash \mathcal{G}_{\mathbb{C}}$, to obtain the required action of $G$ on, such as$A_{5}$

and$A_{6}$$\mathbb{C}P^{n}\square$, the alternating groups on

Theorem 7.2. Let $G=A_{5}$ . Let $F$ be a smooth manifold. Assume also that(i) $F$ is closed, the components of $F$ are of the same, even dimension, and(ii), i. e.,.
is well-G-located incomplexprojective space such that the fixed point set ison aThen there exists a smooth action of
diﬀeomorp $hic$ to $F.$$[\tau_{F}]\in Tor\overline{KO}(F)$$\overline{KO}(F)$$[\tau_{F}]$$G$

In this setting, we are not able to repeat the arguments from the proof of Theorem 7.1, because if, the lack of the appropriate real -modules (cf. Definition 3.1) means that there is no smoothaction of on a spherewith fixed point set $F\sqcup\{x\}$ for $\dim F>0.$$G$$G\in \mathcal{G}\backslash \mathcal{G}_{\mathbb{C}}$$S^{2n}$$G$

The idea of the proof Consider a smooth action of on the sphereof dimension $2n$ for some integer
with the given fixed point set , obtained by Theorem 6.1. Next, modify the action so that the, where $2d=\dim F.$
fixed point set consists of $F$ and the spherearound two
Following the construction above, perform the -equivariant connected sum
points, one chosen from
This yields a smooth action ofand one chosen fromonsuch that the fixed point set consists of $F$ and a number of components diﬀeomorphic to complex
projective spaces, possibly of distinct dimesions.
The new step of the construction is to use the -equivariant surgery to modify the action of onso that the fixed point set is just $F$ , i.e., the extra components diﬀeomorphic to complex projective spaces
are deleted. More specifically, construct an appropriate -equivariant normal map of degree 1,$S^{2n}$$G$$F$$n\geq 1_{\rangle}$$S^{2d}$$G$$S^{2d}\subset S^{2n}$$S^{2n}\#\mathbb{C}P^{n}$$G$$\mathbb{C}P^{d}\subset \mathbb{C}P^{n}$$\mathbb{C}P^{n}$

$G$

$G$

$\mathbb{C}P^{n}$$G$$f:Xarrow \mathbb{C}P^{n}$

To convert$f$into a homotopy equivalence$Marrow \mathbb{C}P^{n}$, the intermediate surgery obstructions for the maps$f^{H}:X^{H}arrow(\mathbb{C}P^{n})^{H}, H<G,$

are killed by means of the (geometric) reflection method due to Morimoto [3]. The final surgery obstruction
vanishes (algebraically) by the Dress Induction. As a result, one obtains a smooth action of on a closed, with fixed point set diﬀeomorphic tosmooth manifold $M$ homotopy equivalent to$G$$F^{2}$$\mathbb{C}P^{n}$$\square$

We expect that similar arguments are true and Theorem 7.2 holds for any finite perfect group
We wish to pose the following problem, where we assume that is a finite group not of prime power$G\in \mathcal{G}\backslash \mathcal{G}c.$$G$

order, such that $n_{G}=1$ $(i.e., G is an$ Oliver group) and $G$ is not a perfect group.

Problem 1. Let $F$ be a smooth manifold such that (i) $F$ is closed, the connected components of all are. Is it true that there exists a smootheven dimensional, and (ii) the classis well-G-located inaction of on some complex projective space, such that the fixed point set is diﬀeomorphic to $F$ ?
$F$$\overline{KO}(F)$$[\tau_{F}]$$G$

Answering the following question seems to be a challenging project.
with fixed point set $F$ , what are the closed
Problem 2. Given a smooth action of a finite group on
which admit a smooth action of with fixed point set
smooth manifolds homotopy equivalent todiﬀeomorphic to$G$$\mathbb{C}P^{n}$$G$$\mathbb{C}P^{n}$$F^{7}$

2 We are grateful to Masaharu Morimoto for bringing to our attention the fact that the resulting manifold $M$ is also
normally cobordant toand therefore, $M$ is actually diﬀeomorphic to$\mathbb{C}P^{n}$$\mathbb{C}P^{n}.$153
GROUP ACTIONS ON COMPLEX PROJECTIVE SPACESREFERENCES[1] Illman, S., The Equivariant Thangulation Theorem for actions of compact Lie groups, Math. Ann., Vol. 262, Issue 4(1983), pp. 487-501.
[2] Kaluba, M., Constructions of smooth exotic actions on homotopy complex projective spaces and products of manifolds,
PhD Thesis, UAM Pozna\’{n}, 2014.
in Transformation Groups, Lecture[3] Morimoto, M., Most of the standard spheres have one fixed point actions ofNotes in Mathematics, Vol. 1375, pp. 240-259, Springer-Verlag, 1989.
[4] Morimoto, M., Equivariant surgery theory: deleting-inserting theorems of fixed point manifolds on spheres and disks,$K$ -Theory, Vol. 15, Issue 1 (1998), pp. 13-32.
[5] Morimoto, M., Fixed-point sets of smooth actions on spheres, Journal of $K$-Theory, Vol. 1, Issue 1 (2008), pp.
$A_{5_{\rangle}}$95-128.
[6] Morimoto, M., Pawalowski, K., Equivamant wedge sum construction of finite contractible G-CW-complexes with$G$
-vector bundles, Osaka J. Math., Vol. 36, Issue 4 (1999), 767-781.
[7] Morimoto, M., Pawatowski, K., The equivariant Bundle Subtraction Theorem and its applications, Fundamenta
Math. Vol. 161, Issue: 3 (1999), pp. 279-303.
[8] Morimoto, M., $p_{awa}\}_{owski}$ , K., Smooth actions of finite Oliver groups on spheres, Topology, Vol. 42, Issue 2 (2003),

pp. 395-421.

[9] Oliver, R., Fixed-point sets of group actions on fnite acyclic complexes, Comment. Math. Helvetici 50 (1975), pp.

155-177.
, Topology, Vol. 35, Issue[10] Oliver, R., Fixed point sets and tangent bundles of actions on disks and Euclidean
3 (1996), pp. 583-615.
[11] Pawalowski, K., Fixed point sets of smooth group actions on disks and Euclidean spaces, Topology, Vol. 28, Issue 3(1989), pp. 273-289.
[12] Pawalowski, K., Chern and Pontryagin numbers in perfect symmetrees of spheres, $K$ -Theory, Vol. 13, Issue 1 (1998),$space\mathcal{S}.$pp. 41-55.

$P$

ADAM MICKIEWICZ UNIVERSITY 1N
FACULTY OF MATHEMATICS AND COMPUTER SCIENCE
UL. UMULTOWSKA 87
61-614POZNA , POLAND$\acute{N}$$E$-mail address, Marek Kaluba: kalmar@amu.edu.pl$E$-mail address, Krzysztof Pawalowski: kpa@amu.edu.pl