FIXED POINT SETS AND TANGENT BUNDLES
OF ACTIONS ON DISKS AND EUCLIDEAN SPACES
by Bob Oliver
The main result of this paper is the determination, for any given finite gro*
*up G not
of prime power order, of exactly which smooth manifolds can be fixed point sets*
* of
smooth Gactions on disks or on euclidean spaces. General techniques for constr*
*ucting
smooth actions on disks with fixed point set of a given homotopy type were deve*
*loped
in [O1], and the procedure for constructing actions on euclidean spaces is simi*
*lar (but
simpler). What is new here is a way of constructing a Gvector bundle over a G*
*complex
of given homotopy type which extends a given Gbundle over the fixed point set.*
* Such
a Gbundle can then be used to control the process of "thickening up" the Gcom*
*plex to
get a manifold with smooth Gaction; and in particular to control the diffeomor*
*phism
type of the fixed point set. Here "Gcomplex" always means GCW complex: a comp*
*lex
built up of orbits G=Hx Dn of cells (where G acts trivially on the disk Dn).
The main technical result for constructing Gbundles, for a finite group G n*
*ot of
prime power order, is given in Theorem 2.4. Let P(G) denote the set of subgroup*
*s of
G of prime power order. Very roughly, given a finite Gcomplex X, a Gvector bu*
*ndle
j over XNP def=[H2=P(G)XH , and P vector bundles P #X for all P 2 P(G), Theor*
*em
2.4 gives conditions for being able to combine j and the P (after stabilization*
*) to get
a Gbundle over a Gcomplex X0 of the same (nonequivariant) homotopy type as X,
and with (X0)NP = XNP . This result can then be combined with the equivariant
thickening theorem of Edmonds & Lee [EL] and Pawalowski [Pa2] (see Theorem A.12
below), to construct manifolds with smooth Gaction having given homotopy type *
*and
given tangential structure on the fixed point sets. Note that this procedure do*
*es not (di
rectly) apply to construct closed manifolds with Gaction, but only open (nonco*
*mpact)
manifolds, or compact manifolds with boundary.
Instead of trying to formulate a general (and very messy) theorem about the *
*con
struction of manifolds with smooth actions, we concentrate our applications her*
*e to
the case of smooth actions on disks and euclidean spaces. The study of this pr*
*oblem
goes back to P. A. Smith [Sm], who showed that the fixed point set of any conti*
*nuous
action of a pgroup (for any prime p) on a finite dimensional Fpacyclic space *
*is itself
Fpacyclic. A converse to Smith's theorem was proven by Lowell Jones [Jo], who *
*showed
among other things that any compact smooth stably complex Fpacyclic manifold c*
*an
be the fixed point set of a smooth action of the group of order p on a disk. T*
*hus, if
G is any nontrivial pgroup, then a compact smooth manifold can be the fixed po*
*int
set of a smooth Gaction on a disk if and only if it is stably complex and Fpa*
*cyclic.
A similar (but simpler) construction can be used to prove the corresponding res*
*ult for
smooth pgroup actions on euclidean spaces.
Typeset by AM S*
*TEX
1
Later examples by various authors (cf. [Br, xI.8]) showed that when G does n*
*ot have
prime power order, then the situation is much less rigid. For example, one can*
* find
a smooth action of any such G on some euclidean space with fixed point set havi*
*ng
the homotopy type of any given countable finite dimensional complex. However, *
*the
situation for actions on disks is more complicated. The main result in [O1] sa*
*ys that
there is an integer nG 0 with the property that a finite CW complex F is homot*
*opy
equivalent to the fixed point set of some smooth Gaction on a disk if and only*
* if
O(F ) 1 (mod nG ). The results here now make it possible to determine exactly*
* which
manifolds can be fixed point sets of smooth actions on disks or on euclidean sp*
*aces;
and also to describe (at least stably) the possibilities for the normal bundle *
*of the fixed
point set. These results are summarized in the following theorem:
Theorem 0.1. Let G be any finite group not of prime power order. Fix a smooth
manifold F and a Gvector bundle j#F satisfying the following three conditions:
(1) j is nonequivariantly a product bundle;
fi
(2) for each prime pfiG and each psubgroup P G, [jP ] is infinitely pdivi*
*sible in
gKOP (F )(p)(where gKO P(F ) = KOP (F )=KOP (pt)); and
(3) jG ~= o(F ) (the tangent bundle of F ).
Then there is a smooth action of G on a contractible manifold M such that MG =*
* F ,
and such that o(M)F ~= j (V xF ) for some Grepresentation V with V G = 0. *
*If
@F = ;, then M can be chosen to be a euclidean space. If F is compact and O(F )*
* 1
(mod nG ), then M can be chosen to be a disk.
Conditions (1)(3) in Theorem 0.1 are also necessary: if G acts smoothly on*
* any
contractible manifold M, then they hold for the pair (F; j) = (MG ; o(M)MG ). *
*Note in
particular point (2): [jP ] is infinitely pdivisible in gKOP (F ) since jP i*
*s the restriction
of a P bundle over the Fpacyclic manifold MP (hence the group gKOP (MP ) is u*
*niquely
pdivisible).
Theorem 0.1 still leaves it rather unclear exactly which manifolds can be th*
*e fixed
point set of a Gaction on a disk or euclidean space. In order to make this mor*
*e precise,
we first need some definitions. Let MC MC+ MR be the classes of finite groups
for which there exist Grepresentations V and W which are complex, selfconjuga*
*te, or
real, respectively, such that V P ~=W P for any P G of prime power order, an*
*d such
that dim (V G) = 1 and dim (W G) = 0. By Lemma 3.1 below, G 2 MC if and only if
it contains an element not of prime power order, G 2 MC+ if and only if it con*
*tains
an element not of prime power order which is conjugate to its inverse, and G 2 *
*MR
if and only if it contains a subquotient which is dihedral of order 2n for some*
* n not a
prime power. (This condition for a group to be in MR was pointed out to me by E*
*rkki
Laitinen.)
In the following theorem, for any abelian group A, we let qdiv(A) (the subgr*
*oup
of "quasidivisible" elements) denote the intersection of all kernels of homomor*
*phisms
from A into free abelian groups. When A is finitely generated, this is just the*
* torsion
subgroup of A. Also, the standard induction and forgetful maps between the grou*
*ps of
real, complex, and quaternion vector bundles over a space X are denoted as foll*
*ows:
0
gKO (X) ________wcu________eK(X)]________u________cKSp(X):
r q w
2
Theorem 0.2. Let G be a finite group not of prime power order. Let Fix(G) be
the class of smooth manifolds F (without action) for which there is a Gbundle *
*j#F
satisfying conditions (1)(3) in Theorem 0.1. Then a smooth manifold F is the *
*fixed
point set of a smooth action of G on a euclidean space if and only if F 2 Fix(G*
*) and
@F = ;; while F is the fixed point set of a smooth action of G on a disk if and*
* only if
F is compact, F 2 Fix(G), and O(F ) 1 (mod nG ). Furthermore, Fix(G) is descr*
*ibed
as follows (where Syl2(G) is a Sylow 2subgroup of G):
F 2Fix(G) ()  Syl2(G) 6C G  Syl2(G) C G *
* 
___________________________________________________________________________*
*___
G2 MR  (A) (no restriction)  ___ 
_________________________________________________________________________*
*__(B)
G2 MC+ rMR  c([o(F )]) 2 c0(]KSp (F ))+ qdiv(Ke(F)) ___ 
_________________________________________________________________________*
*__(C)(D)
 e g  [o(F )] 2 r(Ke(F )*
*)
G2 MCr MC+  [o(F )] 2 r(K (F ))+ qdiv(KO (F )) (F is stably complex*
*)
___________________________________________________________________________*
*__(E)(F)
G62MC  [o(F )] 2 qdiv(KgO (F ))  [o(F )] 2 r qdiv(Ke(F*
* )) 
___________________________________________________________________________*
*___ 
This theorem extends results of Edmonds & Lee [EL, Theorem A] and Pawalowski
[Pa2, Theorems 5.6 & 5.9]. However, all of their constructions give fixed point*
* sets with
stably complex tangent bundle, and the possibility of having other fixed point *
*sets (when
Syl2(G) 6C G) is new. Note that if G =2MC, then all connected components of the*
* fixed
point set of a Gaction on a disk or euclidean space must have the same dimensi*
*on;
while if G 2 MCr MR then the dimensions of the components can be different but
must have the same parity (see [Pa1, Theorem A]). In contrast, if G 2 MR, then *
*the
components of the fixed point set can have arbitrary dimensions.
Examples of groups in the above classes include: (A) D(2n), (B) Q(4pa), (*
*C)
D(2pa)x Cqb, (D) Cn, (E) D(2pa), (F) F2io Cpa where pa2i 1. Here, in all c*
*ases,
n denotes any integer not a prime power, p and q denote distinct odd primes; an*
*d Cm ,
D(m), and Q(m) denote cyclic, dihedral, and quaternion groups of order m.
The conditions on F in cases (B) and (C) above are very similar, and it is *
*not
immediately clear that they give distinct classes Fix(G). To see that they do,*
* set
X = S5[ j2e8: the complex obtained by attaching an 8cell to S5 via the nontri*
*vial
element j2 2 ss7(S5). We leave it as an exercise to check that gKO (X) ~=Z, and*
* that
the maps
KgO (X)  c! eK(X)  c0 ]KSp(X)
~= ~=
are isomorphisms. Thus, if F is a compact manifold with the homotopy type of X *
*such
that [o(F )] generates gKO(F ), then F 2 Fix(G) for G of type (B), but not for *
*G of type
(C).
So far, we have only discussed the case of actions of finite groups. If G is*
* a compact
Lie group with identity component G0 which acts smoothly on a contractible mani*
*fold
M with fixed point set F , then one can show (using [JO, Proposition 4.6] and t*
*he
definition of Fix()) that F 2 Fix(G=G0). More precisely, (o(M)F )G0 is a G=*
*G0
bundle which satisfies conditions (1)(3) in Theorem 0.1. In particular, if G i*
*s connected
3
and nonabelian, then F can be the fixed point set of a smooth action of G on a *
*disk
if and only if it is stably parallelizable (see [O2, Theorems 3 & 5]). This is*
*, however,
still far from answering the question of which manifolds can be fixed point set*
*s, since in
general more homotopy types can occur as fixed point sets of Gactions on disks*
* than
of G=G0actions on disks.
Since the numbers nG play such a key role in the above theorem, we summarize*
* here
their computation in [O1, Corollary to Theorem 5] and [O3, Theorem 7]. Let G1 *
*be
the class of all finite groups G which contain a normal subgroup P C G of prime*
* power
order such that G=P is cyclic. For each prime p, let Gp denote the class of al*
*l finite
groups G which contain a normal subgroup in G1 of ppower index.
Theorem 0.3. Fix a finite group G not of prime power order. For any prime p, p*
*nG
if and only if G 2 Gp. Thus, nG = 0 if and only if G 2 G1, and nG = 1 if and o*
*nly
if G =2[ pGp. In general, nG is equal to 0, 1, a product of distinct primes, *
*or 4; and
nG = 4 if and only if
(1) G lies in an extension 1 ! Cm ! G ! C2k ! 1, where Cm is cyclic of od*
*d order
m and C2k is cyclic of order 2k,
(2) G =2G1, but its subgroup of index 2 does lie in G1, and
(3) there is no unit u 2 (Zim )* such that ff(u) = u, where im is a primitive *
*mth root
of unity, and ff 2 Gal(Qim =Q) is induced by the conjugation action of a gen*
*erator
of G=Cm ~= C2k.
As another special case of Theorem 0.1, we note the the following theorem ab*
*out
tangential representations at isolated fixed points. This generalizes results o*
*f Edmonds
& Lee [EL, Theorem B] and Pawalowski [Pa1, Theorem B]:
Theorem 0.4. Let G be any group not of prime power order. Let V0; V1; : :;:Vm *
* be
(real) Grepresentations such that V0P ~=V1P ~=. . .~=Vm P for any P G of p*
*rime
power order, and such that ViG = 0 for all i. Then there exists a Grepresentat*
*ion W
with W G = 0, and a smooth action of G on a euclidean space (or a disk if nG m*
*) with
exactly m+ 1 fixed points x0; : :;:xm , such that the tangential representation*
* at xi is
Vi W .
The paper is organized as follows. In Section 1, a space B*GO is constructed*
* which
has the following property (Proposition 1.3): for any finite Gcomplex X, [X; B*
**GO]G
is (roughly) the inverseflimitiof the groups KOP (X)(p), taken over all psubgr*
*oups
P G and all primes pfiG. The problem of lifting maps X ! B*GO to BG O is th*
*en
handled in Section 2, and this leads to a general criterion (Theorem 2.4) for c*
*onstructing
Gbundles over Gcomplexes of given (nonequivariant) homotopy type and with giv*
*en
fixed point data. The proofs of Theorems 0.1 and 0.2, and some examples, are t*
*hen
given in Section 3. Finally, in an appendix, some technical results are listed*
*, most of
which are well known but seem hard to find in the literature. The equivariant t*
*hickening
theorem in the version of Pawalowski is also stated there (Theorem A.12).
The proof of these results _ more precisely the constructions in Section 1 _*
* were to
a great extent motivated by my joint work with Stefan Jackowski on vector bundl*
*es over
classifying spaces of compact Lie groups [JO]. The connections with [JO] have l*
*argely
4
disappeared while this work has evolved, but it probably would not have been po*
*ssible
without the discussions we had while writing that paper.
1. An approximation to the classifying spaces for Gbundles
Throughout this section, G will be a fixed finite group. Let O(G) denote the*
* orbit
category of G: the category whose objects are the orbits G=H for all subgroups *
*H G,
and where Mor O(G)(G=H; G=K) is the set of all Gmaps G=H ! G=K. For each prime
p, Op(G) O(G) will denote the full subcategory whose objects are the orbits G=*
*P for
psubgroups P G. Also, O1(G) denotes the full subcategory with one object G=1.
For any full subcategory C O(G), we define
EC = hocolim!G=H :
G=H2C
This can be regarded as the nerve of the category whose objects are the cosets *
*aH
for G=H 2 C, and where there is one morphism from aH to bK for each Cmorphism
G=H ! G=K which sends aH to bK. (In particular, there is at most one morphism
between any pair of objects.) From this definition, EC is seen to be a Gcomple*
*x all of
whose orbit types lie in C. Also, ECH is contractible for any G=H in C, since *
*it is the
nerve of a category with initial object the coset eH. In particular, EO1(G) ~=E*
*G. More
generally, by equivariant obstruction theory, EC is "universal" among Gcomplex*
*es with
orbits in C: for any such X there is a Gmap X ! EC which is unique up to Gho*
*motopy.
In the appendix, BG O is defined to be the infinite mapping cylinder of ma*
*ps
BG O(0) ! BG O(d) ! BG O(2d) ! : :,:where d = G, where BG O(n) is the
base space of the universal ndimensional Gbundle, and where the maps are sta
bilization by the regular representation RG. Bundle direct sum define*
*s product
maps BG O(n)x BG O(m) ! BG O(n+ m), and these combine to define a G*
*map
BG Ox BG O ! BG O which makes BG O into a Gequivariant Hspace.i Alternativel*
*y,j
` 1
one can define BG O as the identity component of the loop space B n=0BG O(n)
(once the BG O(n) have been defined precisely enough to make their disjoint uni*
*on into
a topological monoid); and then the Hspace structure on BG O is automatic.
*
* fi
We will also have need for the plocalization BG O(p)of BG O, for any prime *
*pfiG.
One elementary way to define this is as the infinite mapping cylinder of the ma*
*ps
BG O .n1!BG O .n2!BG O .n3!BG O .n4!:;: :
where BG O .n!BG O is multiplication by n (using the Hspace structure), and w*
*here
n1; n2; : :i:s any sequence of positive integers prime to p such that each prim*
*e dif
ferent from p divides infinitely many of the ni. Thus, for any finite Gcomple*
*x X,
[X; Z(p)xBG O(p)]G ~=KOG (X)(p). We regard BG O as a subcomplex of BG O(p)via
inclusion into the first term of the cylinder. By construction, for each subgro*
*up H G,
(BG O(p))H is the plocalization of (BG O)H (where the group of components ha*
*s also
been plocalized). Hence by equivariant obstruction theory, it is immediate th*
*at the
equivariant Hspace structure on BG O extends to an equivariant Hspace structu*
*re on
BG O(p).
5
Definition 1.1. Define the Gspace B*GO to be the pullback in the diagram:
Y
B*GO _______________w map (EOp(G); BG O(p))

 pG 
 
 
 
u Y u
map (EO1(G); BG O) ________wdiagmap(EO1(G); BG O(p));
pG
where the right hand vertical map is induced by restriction to EO1(G) (regarded*
* as a
subspace of EOp(G)). Let
LG : BG O , ! B*GO
be the Gequivariant map induced by inclusions into constant maps in the above *
*square.
By the homotopy extension property for inclusions of simplicial complexes, t*
*he right
hand vertical map in the above square satisfies the equivariant homotopy liftin*
*g prop
erty. Thus, B*GO is also a homotopy pullback of that square.
We want to study maps from finite Gcomplexes to B*GO. The following lemma, a
special case of a theorem of [JM], will be needed to handle the higher inverse *
*limits
which arise as obstructions.
fi
Lemma 1.2. For any finite Gcomplex X and any prime pfiG,
i j
limj KOG (G=P xX)(p) = 0 for all j > 0.
G=P2Op(G)
Proof. A contravariant functor F : Op(G) ! Ab is called a Mackey functor if t*
*here is
a covariant functor F* : Op(G) ! Ab which takes the same values on objects, an*
*d such
that any pullback square
ak
G=Ki ________wff1G=H1
i=1 
 
ff2 fi1
 
u u
G=H2 _________wfi2G=H
induces a commutative square
Mk
F (G=Ki) ________wF*(ff1)F (G=H1)u
i=1 u 
 
F(ff2) F(fi1)
 
 
F (G=H2) _________wF*(fi2)F (G=H):
6
By a theorem of Jackowski and McClure [JM, Proposition 5.14], for any Mackey fu*
*nctor
F : Op(G) ! Z(p)mod , limj(F ) = 0 for all j > 0.
Now let F be the contravariant functor F (G=H) = KOG (G=P xX)(p). Then any
map f : G=H ! G=H0 in O(G) induces a homomorphism F* : F (G=H) ! F (G=H0):
defined by sending a Gbundle #(G=Hx X) toLthe Gbundle 0#(G=H0x X) such that
the fiber over any (a0; x) 2 G=H0x X is a2f1a0(a;x). This makes F into a Mac*
*key
functor; and hence its higher limits over Op(G) vanish by [JM].
*
*____
It will be convenient, when X is a (finite) Gcomplex, to write *
*KO G (X) =
[X; BG O]G ; i.e., the group of virtual Gbundles which have virtual dimension *
*zero over
all connected components of X. We are now ready to prove the following proposit*
*ion,
which describes how to construct maps from a finite Gcomplex X to B*GO.
Proposition 1.3. For any finite Gcomplex X, the square
Y ____
[X; B*GO]G ___________w lim KO P(X)(p)
 G=P2Op(G)
 pG 
 
  (*
*1)
 
____u Y __u_
KO (X)G ___________________w KO (X)(p) G
pG
is_a_pullback_square._ Here, functoriality on the right is induced by the ident*
*ification
KO P(X) ~=KO G(G=P xX); and the right hand vertical arrow is induced by restr*
*icting
the limit to the subcategory O1(G) Op(G) and identifying limO1(G)() with ()*
*G .
Proof. The basic idea of the proof is to regard map G(X; B*GO) as the homotopy *
*inverse
limit, over an appropriate category, of the spaces map P (X; BG O(p)) (for psu*
*bgroups
P G) and map (X; BG O). We want to show that [X; B*GO]G is the inverse limit o*
*f the
corresponding sets of components; and this follows upon showing that certain hi*
*gher
inverse limits vanish. The argument given here is a more direct version of this*
* idea; and
is similar to the approach used by Wojtkowiak [Wo] to describe maps from a homo*
*topy
direct limit to a space.
Square (1) above is equivalent to the diagram
Y
[X; B*GO]G ______________wXS(X) _______________w lim [x X; BG O(p)]G
 Op(G) 
 pG 
 
 
  (*
*2)
u Y u
lim[Gx X; BG O]G ________w lim[Gx X; BG O(p)]G ;
O1(G) pGO1(G)
where S(X) is defined to be the pullback. We must show that X is a bijection.*
* In
Step 1, certain cochain complexes D*(X; n) are defined, and their homology grou*
*ps are
7
shown to vanish. And in Step 2, the obstructions to constructing maps X ! B*GO*
* (or
to constructing a homotopy between two such maps) are shown to be homology grou*
*ps
of the D*(X; n).
Step 1: For any category C and any contravariant functor F : C ! Ab , let C*(*
*C; F )
denote the cochain complex
i Y Y Y j
C*(C; F ) = 0 ! F (c) ! F (c0) ! F (c0) ! : :;:
c c0!c1 c0!c1!c2
where the differentials are alternating sums of face maps. The homology groups*
* of
C*(C; F ) are the higher limits lim*(F ) (cf. [O4, Lemma 2]).
For each n 1, FnX : O(G) ! Ab will denote the functor
h restr. i
FnX(G=H) = Coker KOG (G=Hx Dn+1 xX) ! KOG (G=Hx Sn xX)
~=KgO H(n(X+ )):
Let D*(X; n) be the cochain complex defined via the short exact sequence
Y
0 ! D*(X; n)  ! C*(O1(G); FnX) C* Op(G); FnX()(p)
pG
Y (*
*3)
  ! C* O1(G); FnX()(p) !0:
pG
For all j > 0, limjOp(G)FnX()(p) = 0 by Lemma 1.2 (applied to the Gcomplexes
Sn xX and X). Also, a functor M : O1(G) !QAb is the same as a Z[G]module, and
lim*O1(G)M ~= H*(G; M). Since Hj(G; M) ~= pGHj(G; M(p)) for any Z[G]module
M, the long exact cohomology sequence for (3) reduces to an exact sequence
n G Y 0 n
0 ! H0 D(X; n) ! gKO( (X+ )) limgKO P( (X+ ))(p)
pGOp(G)
(*
*4)
'! Y KgO (n(X G 1
+ ))(p) ! H D(X; n) ! 0;
pG
and Hj D(X; n) = 0 for all j 2.
For any P G, the composite
gKO (n(X+ )) transfer!gKOP(n(X+ )) restr.!gKO(n(X+ ))
is the norm homomorphism for the action of P , and in particular sends any x 2
KgO (n(X+ ))G to P .x. Here, the transfer map sends a vector bundle over n(X+*
* ) to
the direct sum of its translates under the action of P (considered as a P bund*
*le). Thus,
if pm is the highest power of p dividing G, then
h i
Im lim0gKOP(n(X+ ))  ! KgO (n(X+ ))G pm .gKO(n(X+ ))G :
Op(G)
8
Since gKO(n(X+ ))G maps onto the sum of the Z=pm KgO (n(X+ ))G , this shows th*
*at
the map ' in (4) is surjective, and hence that H1 D(X; n) = 0.
Step 2: We now consider maps from X to B*GO. For each 0 n 1, let Un(X) be
the space defined by the pullback square
Y
Un(X) _________________w map (EOp(G)(n)xX; BG O(p))
 
 pG 
 
 
  (*
*5)
u Y u
map (EO1(G)(n)xX; BG O) ________wdiagmap (EO1(G)(n)xX; BG O(p)) :
pG
Here, EC(n) (for C = Op(G) or O1(G)) denotes the nskeleton of the complex
a i a j OE
EC = (G=H0x n) ~; (*
*6)
n0 G=H0!:::!G=Hn
with the usual identifications induced by face and degeneracy maps. By Definiti*
*on 1.1,
U1 (X) ' map (X; B*GO).
By the pullback square in (2), an element of S(X) corresponds to a choice of*
* G
maps Gx X ! BG O, and G=P xX ! BG O(p)for all p and all psubgroups P G:
maps which agree up to homotopy with respect to morphisms in Op(G). In other
words, by (6), we can identify S(X) with Im [ss0(U1(X)) ! ss0(U0(X))]. To show*
* that
X : [X; B*GO]G ! S(X) is onto, we must thus show that any element of U1(X) can
be lifted to an element of U1 (X) which has the same image in ss0(U0(X)).
Fix an element f1 2 U1(X), and consider the obstructions to lifti*
*ng it to
U2(X). By (6) again, a 2simplex in EOp(G) corresponds to a sequence of maps
G=P0! G=P1! G=P2 in Op(G), and the obstruction to extending f1 to that 2simp*
*lex
lies in the group
h____ ____ i
Coker KO G (G=P0x D2x X)(p) ! KO G(G=P0x S1x X)(p) = F1X(G=P0)(p):
Similarly, the obstruction to extending f1 to any 2simplex in EO1(G) lies in F*
*1X(G=1).
These individual obstructions combine to give an element ff2 2 D2(X; 1) as the *
*total
obstruction to lifting f1 to some f2 2 U2(X). This element is easily seen to be*
* a cocycle,
and hence is a coboundary by Step 1. And if ff1 = ffi(fi1) for fi1 2 D1(X; 1),*
* then f1
can be changed on 1simplices (in a way specified by fi1) to remove the obstruc*
*tion;
after which the "modified" map can be lifted to an element f2 2 U2(X). (Note t*
*hat
the coHspace structure on the suspensions induces the usual addition on the g*
*roups
KgO P((X+ )).) Upon continuing this process, we see that at each stage the obst*
*ruction
to lifting fn 2 Un(X) to fn+1 2 Un+1 (X) (while allowing fn to be changed on n
simplices) lies in Hn+1 (D*(X; n)), which again vanishes by Step 1.
This shows that X : [X; B*GO]G ! S(X) is onto. To show that it is injectiv*
*e, we
start with two elements f; f0 2 map G(X; B*GO) ~=U1 (X), together with a homoto*
*py
9
F0 2 U0(Xx I), and then lift the homotopy one step at a time. For each n 0, t*
*he
obstruction to lifting a homotopy Fn 2 Un(Xx I) to Un+1 (Xx I) (while taking a *
*given
value on Xx {0; 1}) lies in Hn+1 (D*(X; n+1)). And this again vanishes by Step *
*1.
Proposition 1.3 does not in general hold for infinite Gcomplexes. But it do*
*es hold
for countable complexes with fixed Gaction.
The following corollary to Proposition 1.3 will be needed in Section 3.
Corollary 1.4. B*GO can be given the structure of a Gequivariant Hspace in a *
*way
such that LG : BG O ! B*GO is an Hspace homomorphism. Also, for any finite *
*G
complex X, [X; B*GO]G is an abelian group with the property that (Z=n) [X; B*G*
*O]G
is finite for all n > 0.
Proof. The Hspace structure on B*GO, and LG being an Hspace homomorphism,
follow immediately from the pullback square in Definition 1.1, together with th*
*e H
space structures on BG O and its localizations (see the discussion before Defin*
*ition 1.1).
By Proposition 1.3, for any finite Gcomplex X, there is an exact sequence
____ Y i ____ j Y ____ G
0 ! [X; B*GO]G ! KO (X)G limKO P(X)(p) ! KO (X)(p) : (*
*1)
pGOp(G) pG
So [X; B*GO]G is abelian, and (Z=n) [X; B*GO]G is finite for any n > 0 since (*
*Z=n) 
and Tor(Z=n; ) are finite for the other two terms in (1).
2. Construction of Gbundles
Proposition 1.3 describes a procedure for constructing Gmaps from a finite *
*G
complex X to B*GO. What we really are interested in is the construction of Gma*
*ps from
X to BG O. In general, of course, Gmaps from B*GO cannot be lifted to BG O (LG*
* is not
a Ghomotopy equivalence). To get around this, we prove a rather complicated li*
*fting
result (Proposition 2.3); and then apply it in Theorem 2.4 to construct Gvecto*
*r bundles
by "pasting together" bundles over certain subcomplexes and for certain subgrou*
*ps.
The first step is to compare the homotopy groups of fixed point sets in B*GO*
* with
those in BG O. Let RO (G) ~=KOG (pt) denote_the orthogonal representation ring *
*of G,
and let IRO (G) = Ker[RO (G) dim!Z] ~=KO G(pt) be its augmentation ideal.
Lemma 2.1. Let G be any finite group, and let LG : BG O ! B*GO be the map of
Definition 1.1. Then the following hold.
(a) LG is (nonequivariantly) a homotopy equivalence.
fi
(b) Fix a prime pfiG and a psubgroup 1 6= P G. Then ss0((B*GO)P ) ~=IRO *
*(P )(p),
and ss0(LPG) is isomorphic to the inclusion of IRO (P ) into IRO (P )(p). For *
*each x 2
(BG O)P ,
ssi(LPG; x) : ssi (BG O)P ; x ! ssi (B*GO)P ; LG (x)
is an isomorphism for i = 1; and for i > 1 its kernel and cokernel are torsion *
*prime to
p with the additional property that their mtorsion subgroups are finite for al*
*l m.
10
fi
Proof. Fix a prime pfiG and a psubgroup P G. By Proposition 1.3 (applied wi*
*th
X = G=P ) there is a pullback square
Y ____
[G=P; B*GO]G ~=ss0((B*GO)P ) _________w lim KO Q (G=P )(q)
 G=Q2Oq(G)
 qG 
 
  (*
*1)
 
____ u Y ____ u
KO (G=P )G _______________w KO (G=P )(q)G :
qG
Similarly, for any i > 0, Proposition 1.3 yields a pullback square
[i(G=P+_);_B*GO]G_ ss~((B* O)P ) __________wY lim KOi(G=P )
[pt; B*GO]G = i G G=Q2Oq(G) Q (q)
 qG 
 
  (*
*2)
 
u Y u
KOi(G=P )G ______________w KOi(G=P )(q)G :
qG
For any i 0 and any Q G,
KOiQ(G=P ) ~=KOiG(G=P x G=Q) ~=KOiP(G=Q):
In general, if FP is any contravariant functor from P complexes to abelian gro*
*ups such
that FP (Xq Y ) = FP (X) FP (Y ), then
aeF (pt) if q = p
lim FP (G=Q) ~= lim(FP ) ~= P P (*
*3)
G=Q2Oq(G) Oq(P) FP (P ) if q 6= p.
The first isomorphism holds since for each P orbit P .gQ in any G=Q, there is *
*an Oq(G)
morphism G=Q0 ! G=Q (where Q0 = P \gQg1 ) which sends the P orbit P=Q0 in
Oq(P ) isomorphically to P .gQ. Also, Oq(P ) has a final object if p = q, and *
*contains
only the free orbit if p 6= q.
____
If we now apply (3) with FP = KO P ()(q), then square (1) is reduced to an *
*isomor
phism ~ ____
ss0 (B*GO)P  =! KO P(pt)(p)~=IRO (P )(p)
____
(note that KO (G=P ) = [G=P; BO] = 0). Thus, ss0(LPG) is isomorphic to the inc*
*lusion
of IRO (P ) into IRO (P )(p). And if (3) is applied with FP = KOiP()(q), then*
* square
(2) reduces to a pullback square
ssi((B*GO)P ) ___________wKOiP(pt)(p)
 
  (*
*4)
u u
KOi(pt) ____________wKOi(pt)(p):
11
When P = 1, this shows that ssi(B*GO) ~=KOi(pt) ~=ssi(BG O) for all i > 0, and*
* hence
that LG is nonequivariantly a homotopy equivalence.
i forget
Now set Mi = Ker KOP (pt)  i KOi(pt) . The forgetful map is a surjec
tion: split by regarding a bundle over Si as a P bundle with trivial action. S*
*o by (4),
the kernel and cokernel of the homomorphism
P ssi(LPG) * P
KOiP(pt) ~=ssi (BG O)   ! ssi (BG O)
are given by
P
Ker(ssi(LPG)) ~=Ker Mi!Mi (p) and Coker (ssi(LG )) ~=Coker Mi!Mi (p): (*
*5)
In particular, since Miis finitely generated, Ker ssi(LPG) and Coker ssi(LPG) *
* are torsion
of order prime to p, and have finite mtorsion for any m > 0.
It remains to show that ss1(LPG) is an isomorphism. By Propositio*
*n A.2(b),
KO1P(pt) ~= ss1((BG O)P ) is a sum of one copy of Z=2 for each irr*
*educible P 
representation of real type. So if p = 2, then ss1(LPG) is an isomorphism by (5*
*), since M1
is a finite 2group. If p is odd, then the only irreducible P representation o*
*f real type is
the trivial one (see Proposition A.1(c)); so KOiP(pt) ~=KOi(pt) ~=Z=2, M1 = 0*
*, and
ss1(LPG) is again an isomorphism.
When f : X ! Y is a given map between spaces, we will frequently write ssi(*
*Y; X; x)
(for x 2 X) to denote the relative homotopy group ssi(Zf; X; x). Here, Zf is th*
*e mapping
cylinder of f. Also, we write ssi(Y; X) when X is connected and the basepoint n*
*eed not
be specified.
The next lemma will provide the induction step in our construction of Gbund*
*les.
Lemma 2.2. Fix a finite group G and a prime p. Let X ff!Z fi!Y be Gmaps, whe*
*re
(1) X and Y are countable finite dimensional (nonempty) Gcomplexes;
(2) Z and Y are connected, ss1(Y; Z) = 1, ss2(Y; Z) is abelian, and ssi(Y; Z) *
* Z(p)= 0
for all i 2; and
(3) for any nontrivial psubgroup 1 6= P G, (fiff)P : XP ! Y P is an Fphom*
*ology
equivalence.
__
Then there exist a countable finite dimensional Gcomplex X X and an extension
_ff: __X! Z of ff, such that G acts freely on __XrX and such that fif_f: __X!*
* Y is an
Fphomology equivalence. If in addition,
(4) X and Y are finite Gcomplexes,
(5) Ker(ss1(fi)) has finite ntorsion for all n, and
(6) O(XH ) = O(Y H) for all cyclic subgroups H G of order prime to p,
__
then X can be chosen to be a finite Gcomplex.
Proof. Finite case: Assume that all points (1)(6) hold. By attaching some f*
*inite
number of free orbits of 1cells to X if necessary (and extending ff accordingl*
*y), we can
12
assume that X is connected and that ss1(ff) : ss1(X) i ss1(Y ) is onto. Set
ss1(fiff) ss1(fi)
KX = Ker ss1(X)  i ss1(Y ) and KZ = Ker ss1(Z)  i ss1(Y )*
* :
Since ss1(X) and ss1(Y ) are both finitely presented, KX is finitely generated *
*as a normal
subgroup of ss1(X) (cf. [Ro, Lemma 1.43(i)]). By (2), KZ is abelian and torsion*
* prime to
p, and by (5) its ntorsion subgroup is finite for any n. Then Im[KX ff*!KZ ] *
*is finite (it
has bounded torsion since KX is finitely generated); and hence Im[ss1(X) ss1ff*
*!ss1(Z)] is
finitely presented (an extension of one finitely presented group by another). S*
*o by [Ro]
again, Ker (ss1(ff)) is finitely generated as a normal subgroup of ss1(X). We *
*can thus
attach finitely many free orbits of 2cells to X, to obtain a finite Gcomplex *
*X1 X
and a map ff1 : X1 ! Z extending ff, such that ss1(ff1) is injective, and Ker(*
*ss1(fiff1))
is finite abelian of order prime to p.
Set d = max {dim (X); dim(Y ); 2}. We next construct a sequence of finite co*
*mplexes
X1 X2 X3 . . .Xd;
together with Gmaps ffm : Xm ! Z extending ff1, such that each Xm (for 2
m d) is constructed from Xm1 by attaching free orbits of mcells, and such t*
*hat
ssi(Ze; eXm) Z(p)= 0 for each i m. Assume that Xm1 has been constructed. Th*
*en
by Lemma A.7,
ssm (Ze; eXm1) Z(p)~=ssm (Ye; eXm1) Z(p)
is finitely generated as a Z(p)[ss1(Xm1 )]module. We can thus attach some fin*
*ite number
of free orbits of mcells to Xm1 (while extending ffm1 ) to get a new comple*
*x Xm such
that ssm (Ze; eXm) Z(p)~=ssm (Ye; eXm) Z(p)= 0.
Now consider the sequence of maps Xd ffd! Z fi! Y . By assumpt*
*ion,
ssi(Y; Xd) Z(p) = 0 for all i d = dim (Xd) dim (Y ). Also, by Lemma A.7, ap
plied to the pairs (Y; Xd) and (Y; Z), Hi(Y; Xd; Z(p)) ~= Hi(Z; Xd; Z(p)) = 0 f*
*or all
i d, and the Hurewicz homomorphism
ssd+1(Z; Xd) Z(p)~=ssd+1(Y; Xd) Z(p)  i Hd+1(Y; Xd; Z(p))
is surjective. By Lemma A.10, together with conditions (3) and (6), Hd+1(Y; Xd;*
* Fp) is
free as an Fp[G]module. We can thus choose elements of ssd+1(Z; Xd) which repr*
*esent
a basis of Hd+1(Y; Xd; Fp), and attach accordingly_free orbits of_(d + 1)cells*
* to Xd,
extending ffd, to get a finite Gcomplex X Xd and a map _ff: X ! Z which is*
* an
Fphomology equivalence.
Countable case: The proof is similar in the countable case (i.e., when we on*
*ly
assume conditions (1)(3)), but much simpler. All homotopy groups of Y and the*
* Xi
are countable by Lemma A.6. So at each stage, the relevant homotopy elements ca*
*n be
eliminated by attaching only countably many cells. And in the last step, Hd+1(Y*
*; Xd; Fp)
is stably free as a countably generated projective Fp[G]module by Lemma A.10, *
*and
so free orbits of d and_(d+_1)dimensional cells can be attached to Xd to cons*
*truct the
Fphomology equivalence X ! Z.
We are now ready lift maps from B*GO to BG O.
13
Proposition 2.3. Assume that G is a finite group not of prime power order. Let
X ________wfBG O

'  
 LG (*
*1)
u u
Y ________wfYB*GO
be a homotopy commutative square of Gmaps, wherefXiand Y are countable finite
dimensional Gcomplexes. Assume, for each prime pfiG and each psubgroup 1 6=*
* P
G, that
P (fPY)* * P P (LPG)* * P
Im ss0(Y ) ! ss0((BG O) ) Im ss0((BG O) ) ! ss0((BG O) ) :*
*(2)
__
Then there is_a_countable finite dimensional Gcomplex X X such_that all isot*
*ropy
subgroups_of_X r X have prime power order, and extensions _' : X ! Y of ' a*
*nd
f : X ! BG_O of f such that _' is (nonequivariantly) a homotopy equivalence*
* and
fY O_'' LG Of. If, furthermore,
(3) X and Y are finite Gcomplexes with XG 6= ; and Y connected,
(4) O(XH ) = O(Y H) for all H G not of prime power order, and
fi
(5) O ('1 YiP)H = O (YiP)H for each prime pfiG, each psubgroup 1 6= P G,*
* each
connected component YiP of Y P, and each cyclic subgroup 1 6= H=P N(P )=P of
order prime to p,
__
then X can be chosen to be a finite Gcomplex.
Proof. We concentrate on the proof in the case where X and Y are finite complex*
*es and
conditions (3)(5) hold. The proof in the finite dimensional case is simpler, a*
*nd some
remarks will be made afterwards as to how it differs from that for finite compl*
*exes.
Define Z to be the (Gequivariant) homotopy pullback in the following square:
Z ________wflBG O

fi 
 LG (*
*6)
u u
Y ________wfYB*GO:
Lemma 2.1(b) and Condition (2) imply that for any P G of prime power order,
P ) ss1(fiP )
ss0(ZP ) ss0(fi!~ss0(Y P) and ss1(ZP ; x)  i ss1(Y P; fi(x))(*
*8x72)ZP :
= onto
By the homotopy commutativity of (1), there is a Gmap ff : X ! Z such that fi*
* Off ' '
and fl O ff ' f.
Finite case: Step 1: Let P1; : :;:Pk be conjugacy class representatives f*
*or all
subgroups 1 6= P G of prime power order, ordered from largest to smallest (i.*
*e.,
i j if Pi contains a subgroup conjugate to Pj). We first construct finite Gco*
*mplexes
14
X = X0 X1 . . .Xk, together with maps ffi : Xi !Z (where ff0 = ff), satisfyi*
*ng
the following conditions for all 1 i k:
(a) Xir Xi1 contains only orbits of type G=Pi,
(b) ffiXi1 = ffi1, and
(c) (fiOffi)Pi : (Xi)Pi ! Y Piis an Fpihomology equivalence, where pi is t*
*he prime
such that Pi is a pigroup.
Fix some 1 i k, and assume that Xi1 and ffi1 have been constructed. Cons*
*ider
the maps
Pi fiPi
(Xi1)Pi (ffi1)!ZPi ! Y Pi:
After restricting to any connected component of Y Pi(and to those connected com*
*po
nents of ZPi and (Xi1)Pi which map into it), these maps satisfy hypotheses (2)*
*(6)
of Lemma 2.2, applied with the action of the group N(Pi)=Pi. Note in particular*
* that
conditions (2) and (5) in Lemma 2.2 follow from (7) and Lemma 2.1 (and the pull*
*back
square (6)), that condition (3) follows from the assumptions on Xi1, and that *
*condition
(6) follows from condition (5) here. So by Lemma 2.2, there is a finite N(Pi)=P*
*icomplex
W (Xi1)Pi, and an equivariant map bff: W ! ZPi which extends (ffi1)Pi, such*
* that
fiPiObffis an Fpihomology equivalence. And if we set Xi = Xi1[ Gx (incl)(Gx N*
*(Pi)W )
and ffi = ffi1[ (Gx bff), then the pair (Xi; ffi) satisfy conditions (a), (b),*
* and (c) above.
Step 2: It remains to deal with the free orbits. Note that fi : Z ! Y is (*
*nonequiv
ariantly) a homotopy equivalence since LG is (by Lemma 2.1). Since Xk and Y are*
* both
finite, we can attach free orbits of cells to Xk, eliminating all relative homo*
*topy groups
ssi(Y; Xk) ~=ssi(Z; Xk) for small i, until we get a new finite Gcomplex X0 Xk*
* and a
map ff0: X0 ! Z extending ffk, such that Hi(Y; X0) = 0 for all i dim(X0) dim*
*(Y ).
Set d = dim(X0) + 1.
By Lemma A.11 below, Hd(Y; X0) ~= Hd(Z; X0) is a projective Z[G]module; and
there exists a finite Gcomplex T such that T H = pt for all H G not of prime *
*power
order, and such that for some m, T is (m  1)connected and He*(T ) = Hm (T ) *
*~=
Hd(Y; X0) as Z[G]modules. Upon replacing T by an appropriate suspension, we c*
*an
assume that m d, and that m  d is even. Set X00= X0_ T (recall (X0)G = XG 6= *
*;),
and extend ff0 to ff00: X00! Z by sending T to a point. Then H*(Y; X00) ~=H*(Z*
*; X00)
vanishes except in dimensions d and m + 1, and Hd(Y; X00) ~= Hm+1 (Y; X00) as Z*
*[G]
modules. Since Y and Z are connected, we can now attach (finitely many)_free or*
*bits
of cells Gx Di to X00, for d + 1 i m + 1, to obtain a Gmap _ff: X ! Z which
is_a (nonequivariant) homotopy equivalence. By construction, all isotropy subgr*
*oups of
X r X have prime power order.
Countable case: The main difference in the proof when X and Y are countable
and finite dimensional is that since we are working with countably generated mo*
*dules,
the group Hd(Y; X0) is stably free by Lemma A.11. So the last part of Step 2, a*
*nd in
particular the replacement of X0 by a wedge product, are not needed.
Note that the condition XG 6= ; in Proposition 2.3 is needed only in the la*
*st step
of the construction, when removing the projective obstruction to making X0 ! Y*
* a
homotopy equivalence. If XG is empty, the calculations of projective obstruct*
*ions in
[OP, x4] provide other conditions under which the lemma still holds.
15
The main technical theorem on the construction of Gvector bundles can now be
shown.
Theorem 2.4. Assume that G is a finite group not of prime power order. Let ' : *
*X !
Y be a Gmap, where X is a finite Gcomplex and Y is countable and finite dimen*
*sional.
Fix a Gbundle j#X, and a P vector bundle P #Y for each subgroup P G of prime
power order, such that the following conditions hold:
(1) For each prime p, the P (for psubgroups P G) are plocally "consistent up*
* to
isomorphism" with respect to morphisms in Op(G); i.e., they define an elemen*
*t in
the inverse limit
lim KOP (Y )(p)= lim KOG (G=P xY )(p):
G=P2Op(G) G=P2Op(G)
(2) If 1 6= P G is a psubgroup, then ['*(P )] = [jP ] in KOP (X)(p).
(3) [1] 2 KO(Y )G , and '*(1) ~=j (as nonequivariant vector bundles over X).
__
Then there is_a_countable finite dimensional Gcomplex_X_ X such that all isot*
*ropy
subgroups of X rX have prime power order, a Gmap__': X ! Y which extends ' and
is a homotopy equivalence, a Gvector bundle _j#X, and a (real) Grepresentatio*
*n_V , so
that__jX ~=j (V xX) as Gvector_bundles; and such that [_j] = [_'*(P ) (V xX*
* )] in
KO(X ) (if P = 1) or in KOP (X )(p)(if P G is a psubgroup). If, furthermore,
(4) X and Y are finite Gcomplexes with XG 6= ; and Y connected,
(5) O(XH ) = O(Y H) for all H G not of prime power order, and
fi
(6) O ('1 YiP)H = O (YiP)H for each prime pfiG, each psubgroup 1 6= P G,*
* each
connected component YiP of Y P, and each cyclic subgroup 1 6= H=P N(P )=P of
order prime to p,
__
then X can be chosen to be a finite Gcomplex.
Proof. Let f : X ! BG O be the classifying map for j (Lemma A.3). Write Y = [1*
*i=1Yi,
where Y1 Y2 . . .Y are finite Gsubcomplexes. By Proposition 1.3 (and Lemma
A.3 again), for each i, the P combine to define a Gmap f0i: Yi ! B*GO which *
*is
unique up to Ghomotopy. In particular, f0iYi1 ' f0i1for all i, and hence t*
*he f0i
combine to give a map fY : Y ! B*GO. Since X is a finite Gcomplex, conditions*
* (2)
and (3) (and Proposition 1.3 again) show that fY O' ' LG Of. These maps satisfy*
* all of
the appropriate hypotheses of Proposition 2.3. Note in particular that conditio*
*n (2) in
Proposition 2.3 is satisfied since the P are actual bundles (see Lemma 2.1(b)).
Proposition_2.3 now applies_to give a_countable_finite dimensional (or finit*
*e) G
complex X X, and Gmaps _': X ! Y and f : X !_BG O extending ' and f, such
that _'is a homotopy_equivalence, such that LG Of ' fY O_', and such that_all i*
*sotropy
subgroups of X r X have prime power order. It remains to check_that f is induc*
*ed
by a Gbundle. For all H G not of prime power order, Im (ss0(f H)) = Im (ss0(*
*fH ))
is finite since f is induced by an actual bundle. If P G has prime power ord*
*er,
_ __ _f
then Im(ss0(f P)) is finite since the composite X ! BG O ! BP O(p)is P homot*
*opic to
16
_
the classifying map for the P bundle (' )*P (and ss0((BG O)P_) ~=IRO (P ) inje*
*cts into
ss0((BP O(p))P ) by Proposition A.2). Thus, by Lemma A.3, f factors_through BG *
*O(n)
for some n, and hence is the classifying map of some Gbundle _j#X, whose restr*
*iction_to
X is stably isomorphicftoij, and which is P equivariantly stably isomorphic to*
* '*(P )
for each prime pfiG and each psubgroup P G.
Theorem 2.4 can easily be combined with Theorem A.12 below, to allow the con*
*struc
tion of smooth Gmanifolds with various properties. But since it seems quite di*
*fficult to
formulate such a theorem in the greatest possible generality, we limit the appl*
*ications
to the case of actions on disks and euclidean spaces described in the next sect*
*ion.
3. Smooth actions on disks and euclidean spaces
We are now ready to describe the fixed point sets, and the tangent bundles o*
*ver fixed
point sets, for actions of a finite group not of prime power order on a disk or*
* euclidean
space. We first recall the definition of the number nG which determines which h*
*omotopy
types can occur among fixed point sets of actions of G on disks.
Consider the set {O(XG ) 1  X a finite contractible Gcomplex} Z. This*
* is a
subgroup of Z (as seen by taking wedge products and suspensions of Gcomplexes),
and hence has the form nG .Z for some unique nG 0. Thus, by definition, for *
*any
k 2 Z, there is a finite contractible Gcomplex X such that O(XG ) = k if and o*
*nly if
k 1 (mod nG ); and the main theorem in [O1] says that any finite complex F wi*
*th an
"allowable" Euler characteristic can be realized as a fixed point set in this w*
*ay. This is
also a special case of Theorem 2.4 above: if Y is a finite contractible Gcomp*
*lex, and
if O(F ) =_O(Y_G),_then_that theorem describes how to construct a finite contra*
*ctible
Gcomplex X with X G = F (while taking all maps to BG O and B*GO to be trivial*
*).
Proof of Theorem 0.1. By assumption, F is a smooth manifold, and j#F is a G
bundle such that (1) j is nonequivariantly a product bundle, (2) [jP ] 2 gKO P*
*(F ) is
infinitely pdivisible for all primes p and all psubgroups P G, and (3) jG ~*
*=o(F ).
Let V be the fiber over any point of F (regarded as a Grepresentation).
Finite case: Assume that F is compact and O(F ) 1 (mod nG ). If F = ;, then
nG = 1, and G has a fixed point free action on a disk by [O1, corollary to Theo*
*rem 3].
So we can assume F 6= ;. By the above definition of nG , there is a finite cont*
*ractible
Gcomplex Y with O(Y G) = O(F ) (and Y G 6= ;). Set X = F _ (Y=Y G), let ' : X *
*! Y
be any Gmap, extend j to a Gbundle j#X by letting it be trivial over Y=Y G, a*
*nd set
P = (V P )x Y for each P . Then O(XH ) = O(Y H) for all H G, and Y P is acyc*
*lic
and hence connected for each P G_of_prime power order. _By_Theorem 2.4, there
is a finite contractible complex X X and a Gbundle _j#X_such that _jX is st*
*ably
isomorphic to j. In particular, by condition (1) above, (jF )G is stably isomo*
*rphic to
o(F ); and so by Theorem A.12 there is a smooth action of G on a compact contra*
*ctible
manifold M with fixed point set F and with o(M)F stably isomorphic to j. By t*
*he
hcobordism theorem (cf. [Mi]), M is a disk if its boundary is simply connecte*
*d. So
if M is not itself a disk, then we can replace it by Mx D(V ) for any Greprese*
*ntation
V 6= 0 with V G = 0.
Countable case: Let f : F ! BG O be the classifying map for [j]  [F xV ]. *
* By
17
Proposition 1.3 (and the assumptions on j), (LG Of)F 0' * for any finite subco*
*mplex
F 0 F . In particular, the image of LG Of is contained in the identity connecte*
*d compo
nent of (B*GO)G . By Corollary 1.4, (B*GO)G is an Hspace, and (Z=n) [X; (B*GO*
*)G ]
is finite for any finite complex X and any n > 0. Hence Lemma A.9 applies to s*
*how
that there is a countable finite dimensional Z=Gacyclic complex Y F and a *
*map
fY : Y ! (B*GO)G which extends LG Of.
Recall that LG : BG O ! B*GO and the forgetful map BG O ! BO are both none*
*quiv
ariantly homotopy equivalences: the first by Lemma 2.1(a) and the second by Pro*
*po
sition A.2(b). Let #Y be any bundle which is classified by fY : Y ! B*GO '*
* BO.
Then F is a (stably) product bundle, since j#F is by assumption a product bun
dle. Let 0#(Y=F ) be an inverse bundle to (Y is finite dimensional), conside*
*r it as
a Gbundle over Y with trivial Gaction, and let : Y ! Y=F ! (BG O)G be*
* its
classifying map. We can now replace [fY ] by [fY ] + [LG O ] (LG is a homomorph*
*ism of
Hspaces by Corollary 1.4), and thus arrange that fY : Y ! B*GO be nonequivari*
*antly
nullhomotopic.
By Proposition 2.3, applied with X =_F_and ' : X ! Y the inclusion, there_*
*is a
countable_finite_dimensional Gcomplex X F_, together with extensions _': X *
*! Y
and f : X ! BG O of ' and f, such that_X G = F and _' is nonequivariantly*
* a
homotopy equivalence. In_particular,_X is Z=Gacyclic; andfsoiby Smith theo*
*ry (cf.
[Br, Theorem III.7.11]),_X P is Fpacyclic for each prime pfiG_and each psu*
*bgroup
P G. Set d = dim(X ), and consider the dskeleton of the join X *EG. By Lemma
__
A.11, Hd (X * EG)(d) is Z[G]stably free, and so free orbits of d and (d + 1)*
*cells can
*
* __
be attached to produce a countable_finite_dimensional_contractible_Gcomplex Z *
* X .
By construction, all orbits in Zr X are free. Since X *EG is contractible, the*
* inclusion
__
of the dskeleton extends to a map : Z ! X *EG(d+1).
__
We next show, inductively on n, that there are Gmaps fn : X *EG(n)! BG O f*
*or
_ __
all n 0 which extend f : X ! BG O. Since fY is_(nonequivariantly) nullhomot*
*opic
and LG : BG O !_B*GO is a homotopy equivalence, f is also nullhomotopic. So we*
* can
construct f0 : X *EG(0)! BG O.
__
Now assume, for some n 1, that fn1 : X *EG(n1)! BG O has been constructe*
*d.
__
The obstruction to extending fn1 to X *EG(n)is an element
n __ def n __
ffln 2 Cn EG; gKO( X ) = Hom ZG Cn(EG); gKO( X ) :
__
We can regard gKO (nX ) as the set of homotopy classes of maps of pairs
i __ j
(Dn; Sn1 )  ! map (X ; BG O) ; (constant maps) ;
and under this identification addition is given by juxtaposition of disks. Thi*
*s view_
point makes it clear that ffln_is a cocycle, and hence is a coboundary since gK*
*O *(X )
is uniquely Gdivisible (X is Z=Gacyclic). And if ffln is the coboundary *
*of ffn1 in
18
__
Cn1 EG; gKO(nX ) , then ffn1 provides the "recipe" for_changing_fn1 on (n *
*1)
simplices in EG to obtain a map which can be extended to X *EG(n).
By Lemma A.3, fd+1O : Z ! BG O induces a Gbundle _j#Z, and _j(F =ZG ) is
stably isomorphic to j. Theorem A.12 can now be applied to construct a smooth *
*G
action on a contractible manifold M with fixed point set F , such that o(M)F i*
*s stably
isomorphic to j. If @F = ;, then we may assume that @M = ; (otherwise just repl*
*ace
M by its interior). If dim(M) 5, then by [St, Theorem 5.1], M is a euclidean s*
*pace if
it is simply connected at infinity; in particular, if there is a sequence K1 K*
*2 . .o.f
compact subspaces whose union is M and such that each Mr Ki is simply connected.
And if M does not satisfy this property, then Mx R does; and so we can get a eu*
*clidean
space by replacing M by Mx V for any Grepresentation V 6= 0 with V G = 0.
It remains to prove Theorem 0.2; and in particular to characterize which smo*
*oth
manifolds have Gvector bundles which satisfy the hypotheses of Theorem 0.1.
In the proofs of the next three lemmas, a pair of (real or complex) Grepres*
*entations
(V; W ) will be called a "Pmatched pair" of type n if V P ~=W P for all P G*
* of prime
power order, and dim (V G)  dim(W G) = n. In these terms, MC MC+ MR are
the classes of finite groups for which there exist a Pmatched pair (V; W ) of *
*complex,
selfconjugate, or real Grepresentations, respectively, of type 1. If we only *
*assume that
V P ~=W P for psubgroups P G (for some given prime p), then (V; W ) will be*
* called
a pmatched pair.
Lemma 3.1. The following hold for any finite group G not of prime power order.
(a) G 2 MC if and only if G contains an element not of prime power order; and
G 2 MC+ if and only if G contains an element not of prime power order which is
conjugate to its inverse.
(b) G 2 MR if and only if there are subgroups K C H G such that H=K is
dihedral of order 2n for some n not a prime power.
Proof. (a) Assume g 2 G has order n = km, where k; m > 1 and (k; m) = 1. For a*
*ny
nth root of unity , we let C be the 1dimensional representation where g*
* acts
via multiplication by . Set ff = exp(2ssi=k) and fi = exp(2ssi=m), and define
G
V = IndGC1 Cfffiand W = IndCff Cfi:
Then (V; W ) is a Pmatched pair of type 1, and so G 2 MC. If g is conjugate to*
* g1 ,
then V and W are selfconjugate _ the character of any Grepresentation induced*
* from
is real valued by the formula for the character of an induced representatio*
*n (cf. [Se,
x7.2, Proposition 20]) _ and so G 2 MC+ .
If G 2 MC, let (V; W ) be a Pmatched pair of complex representations of typ*
*e 1.
Then V 6~=W , but the characters of V and W agree on elements of prime power or*
*der.
So G must contain an element not of prime power order.
Now assume that G 2 MC+ . Recall that a subgroup H G is pelementary (for
a prime p) if it is a product of a pgroup with a cyclic group; and is 2Relem*
*entary
if it is a semidirect product H = Cn o P , where Cn is cyclic of order n, P is *
*a 2
group, and each element of P centralizes Cn or acts on it via (g 7! g1 ). For*
* each
19
H G, let IP (H)+ R (H) be the subgroup of selfconjugate elements which vanish
upon restriction to prime power order subgroups of H; or equivalently the subgr*
*oup of
elements whose characters are real valued and vanish on elements of prime power*
* order.
Then IP (H)+ is a module over RO (G), where multiplication by [V ] is induced b*
*y tensor
product with C RV (or by multiplication with its character). Also, the IP (H*
*)+ are
preserved under induction and restriction maps (by the formula for the characte*
*r of an
induced representation again). Since RO (G) is generated by induction from subg*
*roups
which are pelementary or 2Relementary [Se, x12.6, Theorem 27], it now follow*
*s by
Frobenius reciprocity that IP (G)+ is also generated by induction from such sub*
*groups
(cf. [Lam, Theorem 3.4(III)]). Hence, since induction leaves unchanged the dime*
*nsions
of fixed point sets, there is some pelementary or 2Relementary subgroup G0 G
and a Pmatched pair (V; W ) of selfconjugate G0representations of type a wit*
*h 2a.
In particular, G0 contains elements not of prime power order, and so G0 2 MC. T*
*hus,
there exist Pmatched pairs of selfconjugate G0representations of type 2; hen*
*ce of type
a for any a 2 Z; and so G02 MC+ .
Now note that Syl2(G0) 6C G0: since otherwise
0 Syl(G0) Syl(G0) G0
dim (V G ) dim(V 2 ) = dim(W 2 ) dim(W ) (mod 2):
(The representations V Syl2(G0)=V_G0_and W Syl2(G0)=W G0 of the odd order*
* group
G0= Syl2(G0) both split as sums U U , and hence are even dimensional, by Propo
sition A.1(c)). Hence G0 is 2Relementary but not 2elementary. Write G0= Cn o*
* P ,
where Cn = is cyclic of odd order n and P is a 2group. We must show that *
*G0
contains an element not of prime power order conjugate to its inverse, and g is*
* such
an element if n is not a prime power. If n > 1 is an odd prime power, and if P*
*  4,
then write g0 = gx for any element x 2 Z(P ) of order 2 which centralizes g; an*
*d g0 has
order 2n and is conjugate to its inverse. And the remaining possibilities _ P *
* = 2 and
n a prime power, or n = 1 _ both contradict the above observation that G0 conta*
*ins
elements not of prime power order.
(b) Assume first that G is dihedral of order 2n, where n is not a prime power. *
*Then
G 2 MC by (a). Also, every CGrepresentation has the form C RV for some RG
representation V (this holds for any dihedral group). Thus, there is a Pmatche*
*d pair
(C RV; C RW ) of complex Grepresentations of type 1, so (V; W ) is a Pmatch*
*ed
pair of real Grepresentations of type 1, and G 2 MR.
Clearly, G 2 MR if any quotient group of G lies in MR. And if H G and (V; W*
* )
is a Pmatched pair of real Hrepresentations of type 1, then (IndGH(V ); IndGH*
*(W )) is a
Pmatched pair of real Grepresentations of type 1.
Conversely, assume that G 2 MR. We must show that G contains a dihedral subq*
*uo
tient of order 2n for some n not a prime power. The same argument as that used *
*in (a)
shows that G contains a 2Relementary subgroup which is not 2elementary and w*
*hich
lies in MR. Upon replacing G by this subgroup, we may assume that G ~= Cn o P ,
where n is odd and P is a 2group, where P0 = P \ CG (Cn) has index 2 in P , an*
*d where
elements in P rP0 act on Cn via (a 7! a1 ). Then G=P0 is dihedral of order 2n,*
* and we
are done if n is not a prime power. If P is not cyclic, then let Fr(P ) denote *
*its Frattini
subgroup (generated by squares and commutators in P ); P= Fr(P ) is elementary *
*abelian
of order at least 4 (cf. [Go, Corollary 5.1.2 & Theorem 5.1.3]), and so G= Fr(P*
* ) contains
a subgroup which is dihedral of order 4n.
20
It remains to consider the case where P is cyclic; i.e., where
fi pk 2m 1 1 ff
G = a; b fia = 1 = b ; bab = a :
We must show that G =2MR. Assume otherwise: let (V; W ) be a Pmatched pair of
RGrepresentations of type 1. Decompose V and W as sums
V = V11 V1x Vx1 Vxx and W = W11 W1x Wx1 Wxx;
where V11 V1x = V and V11 Vx1 = V (and similarly for W ). Thus, for ex*
*ample,
Vxx and Wxx are the sums of those irreducible components where neither a nor b2*
* acts
trivially. Then 4 dim(Vxx) and 4 dim(Wxx): each irreducible real Grepresenta*
*tion on
which neither a nor b2 acts trivially is 4dimensional (and of complex or quate*
*rnion
type). Since dim(V ) = dim(W ) and dim(V ) = dim(W ) by assumption, this *
*shows
that dim(Vx1) dim(Wx1) (mod 4). And since dim(Vx1) = 1_2dim(Vx1) (b acts o*
*n each
irreducible representation in Vx1 with equally many eigenvalues +1 and 1), we *
*now
see that
dim (V G) = dim(V )  1_2dim(Vx1) dim(W )  1_2dim(Wx1) = dim(W G) (mod*
* 2):
And this contradicts the assumption that dim(V G)  dim(W G) = 1.
The condition in Lemma 3.1(b) for G to lie in MR was pointed out to me by Er*
*kki
Laitinen.
Recall that Fix(G) denotes the class of smooth manifolds F for which there i*
*s a G
vector bundle j such that j isfnonequivariantlyia product, [jP ] 2 gKOP (F ) i*
*s infinitely
pdivisible for all primes pfiG and all psubgroups P G, and jG ~=o(F ). W*
*e are
now ready to start proving necessary and sufficient conditions for a manifold F*
* to lie
in Fix(G), for a given group G.
The standard induction and forgetful maps between the groups of real, comple*
*x, and
quaternion vector bundles over F are denoted here as follows:
0
gKO(F ) ________wcu________eK(F])K________u________cSp(F *
*):
r q w
As usual, F is called stably complex if [o(F )] 2 r(Ke(F )); or equivalently if*
* o(F ) Rk
has a complex structure for some k. Note that this requires that the dimensions*
* of the
connected components of F all have the same parity.
Recall that for any abelian group A, qdiv(A) denotes the intersection of th*
*e kernels
of all homomorphisms from A to free abelian groups. In particular, qdiv(A) = to*
*rs(A)
if A is finitely generated.
Lemma 3.2. Fix a finite group G not of prime power order, and a smooth manifold
F . Then the following hold.
(a) F 2 Fix(G) if G 2 MR, or if G 2 MC and F is stably complex.
(b) F 2 Fix(G) if G 2 MC+ and c([o(F )]) 2 c0(]KSp (F )) + qdiv(Ke(F )).
21
(c) F 2 Fix(G) if [o(F )] 2 r qdiv(Ke(F )) , or if Syl2(G) 6C G and o*
*(F ) 2
qdiv(KgO (F )).
Proof. For any F 2 Fix(G), we let TgG (F ) denote the class of Gbundles over F*
* satis
fying conditions (1)(3) in Theorem 0.1.
(a) Assume first that G 2 MR, and let (V; W ) be a Pmatched pair of real G
representations such that dim(V G) = 1 and W G = 0. If F is any compact manifol*
*d, let
o(F ) and (F ) denote the tangent and normal bundles, and set
j = o(F ) V (F ) W (*
*1)
(as a real Gbundle over F ). Then jP is a product P vector bundle for any P *
* G of
prime power order, and jG ~= o(F ). Thus j 2 TgG (F ), and so F 2 Fix(G). If G *
*2 MC
and F is stably complex, then the same construction as in (1), but with complex*
* bundles
and representations, again produces a bundle j 2 TgG(F ).
(b) Assume G 2 MC+ and c([o(F )]) 2 c0(]KSp (F )) + qdiv(Ke(F )). By (a), the*
*re are
elements g; x 2 G such that g is not a prime power and xgx1 = g1 . Set G0= *
*.
We can choose a subgroup K C G0such that G0=K is either dihedral of order 2n wh*
*ere
n is not a prime power, or quaternion of order 4p for some odd prime p. In the*
* first
case, G 2 MR by Lemma 3.1(b), and so F 2 Fix(G) by (a). So we are reduced to the
case where G0=K is quaternion of order 4p.
Fix a 2 G0 which generates the cyclic subgroup of order 2p in G0=K, and set *
*H =
C G0. Set i = exp (2ssi=p). Then there are RGrepresentations V 0; W 0*
*and
HGrepresentations V 00; W 00such that
C RV 0~=IndGH(C1); C RW 0~=IndGH(Ci); V 00C ~=IndGH(Ci );
W 00C ~=IndGH(C1 ):
Here C denotes the 1dimensional H=Krepresentation where a acts via multiplic*
*ation
by . Set
V = (C RV 0) (V 00C) and W = (C RW 0) (W 00C):
Then (V; W ) is a selfconjugate Pmatched pair of Grepresentations of type 1;*
* and
(V 0; W 0) and (V 00; W 00) are 2matched pairs of Grepresentations of types 1*
* and 0,
respectively.
Set o = o(F ), and let be a normal bundle for F . By assumption on F , ther*
*e are
Hbundles o00and 00such that o00 00is a product Hbundle, and such that [o00C]*
* =
[C Ro] 2 eK(F )=(qdiv). Since all infinitely 2divisible elements in eK(F )+ (*
*the elements
invariant under complex conjugation) are in the image of infinitely 2divisible*
* elements
in ]KSp(F ) (see Lemma A.5(a)), we can assumefthatithe difference [o00C]  [C *
* Ro] is
infinitely pdivisible for all odd primes pfiG (Lemma A.5(b)). Set
j = (o RV 0) (o00 HV 00) ( RW 0) (00 HW 00):
By construction, [jP ] = 0 2 gKOP (F ) for any 2subgroup P G. Also,
C Rj = ((C Ro) C(C RV 0)) (o00 CV 00) ((C R) C(C RW 0)) (00 CW 00)
(C Ro) CV (C R) CW
22
fi
modulo elements which are infinitely pdivisible for all odd primes pfiG. So*
* for each
such p and each psubgroup P G, 2.[jP ] = r([C RjP ]) is infinitely pdivi*
*sible,
and hence [jP ] is infinitely pdivisible by Lemma A.5(a). Thus, j 2 TgG (F )*
*, and so
F 2 Fix(G).
(c) Assume first that G has the property that for each prime p, there is a pma*
*tched
pair (Vp; Wp) of real Grepresentations of type 1. We can assume that dim((Vp)G*
* ) = 1
and (Wp)G = 0. LetPF be such that [o(F )] 2 qdiv(KgO (F )). By Lemma A.5(b), we*
* can fi
write [o(F )] = pG[op], where each [op] is infinitely qdivisible for all *
*primes qfiG
different from p. Choose p such that each op p is a product bundle. Set
M i j
j = (op Vp) (p Wp) :
pG
Then j 2 TgG(F ), and so F 2 Fix(G).
If o(F ) 2 r qdiv(Ke(F )) , and if there is for each prime p a pmatched pai*
*r of complex
representations of type 1, then the same construction (taken with complex bundl*
*es) gives
a Gbundle j 2 TgG(F ).
It remains to show that for each prime p, G has a pmatched pair (Vp; Wp) of*
* complex
representations of type 1, and a pmatched pair of real representations of type*
* 1 if
Syl2(G) 6C G. The complex representations are easily constructed: let g 2 Gr 1 *
*be any
element of order m prime to p, let C1 and Ci be the 1dimensional represent*
*ations
where g acts via multiplication by 1 or i = exp(2ssi=m), and set
Vp = IndG(C1) and Wp = IndG(Ci): (*
*2)
fi G *
* G
Also, if 2fiG and g is any element of order 2, then Vp = Ind(R+ ) and Wp =*
* Ind(R )
form for any odd prime p a pmatched pair of real representations of type 1.
If G is dihedral of order 2m, where m > 1 is odd, and if g generates the sub*
*group
of index 2, then the representations in (2) are induced from a 2matched pair o*
*f real
Grepresentations. So to finish the proof, we need only show that any group G s*
*uch that
Syl2(G) 6C G contains a subquotient of that form. Upon dividing out by the inte*
*rsection
O2(G) of the Sylow 2subgroups, we can assume that this intersection is trivial*
*. Let S
be any conjugacy class of elements of order 2 in G. By [Go, Theorem 3.8.2], eit*
*her S
generates a normal 2subgroup of G (which is clearly not the case), or some pai*
*r x; y of
elements in S generates a subgroup not of 2power order. And then is dih*
*edral,
and contains a dihedral subgroup of order 2m for some odd m > 1.
Lemma 3.2 gave sufficient conditions for a manifold to be contained in Fix(G*
*). It
remains to show that these conditions are also necessary.
Lemma 3.3. Fix a finite group G not of prime power order, and assume that F 2
Fix(G).
(a) If Syl2(G) C G, then F is stably complex.
(b) If G 62 MC, then o(F ) 2 qdiv(KgO (F )), and o(F ) 2 r qdiv(Ke(F )) if *
*Syl2(G) C
G.
23
(c) If G 62 MR, then c([o(F )]) 2 c0(]KSp (F )) + qdiv(Ke(F )). If G 62 MC+*
* , then
[o(F )] 2 r(Ke(F ))+ qdiv(KgO (F )).
Proof. Fix some F 2 Fix(G). Let j#F be a Gvector bundle which satisfies condit*
*ions
(1)(3) in Theorem 0.1: jG ~=fo(Fi), [j] = 0 in gKO (F ), and jP is infinitely*
* pdivisible
in gKO P(F ) for each prime pfiG and each psubgroup P G.
(a,b) If all elements in G have prime power order, then R (G) is detected by *
*re
striction to the Sylow subgroups, and so c([j]) = [C Rj] lies in qdiv(KeG(F *
*)) ~=
qdiv(Ke(F )) R (G). Since rOc([j]) = 2.[j], it now follows that [j] 2 qdiv(Kg*
*O G(F )),
and in particular that [jG ] = [o(F )] lies in qdiv(KgO (F )).
Now assume that Syl2(G) C G, and write G2 = Syl2(G) for short. Then [jG2 ]
is infinitely 2divisible in gKO (F ), and hence is the image of the infinitely*
* 2divisible
element c 1_2.[jG2 ] 2 eK(F ). In particular, [jG2 ] 2 r qdiv(Ke(F )) . Let V0*
* = R; V1; : :V:k
be the distinct irreducible real representations of G=G2. Write
jG2 = j0 j1 . . .jk;
where each fiber in ji is a sum of copies of Vi (in particular, j0 = jG ). Sinc*
*e G=G2
is odd, each representation V1; : :;:Vk can be given a complex structure by Pro*
*position
A.1(c), and hence (by Proposition A.1(a)) each ji has the form ji ~=i CVi for *
*some
complex bundle i#F . Thus [ji] is a complex bundle for each i 1, and so o(F ) *
*~=j0
is a stably complex bundle. If, in addition, all elements of G have prime power*
* order,
then we have seen that [j] 2 qdiv(KgO G(F )), so [i] 2 qdiv(Ke(F )) for all 1 *
*i k,
and [ji] 2 r qdiv(Ke(F )) for each i 1. Also, [jG2 ] 2 r qdiv(Ke(F )) as see*
*n above;
and hence [o(F )] = [j0] lies in r qdiv(Ke(F )) .
(c) Note first that G 2 MR (G 2 MC+ ) if there is a Pmatched pair (V; W ) of*
* real
(self conjugate complex) representations of type a for any odd a. To see this, *
*note first
that if there is such a Pmatched pair, then G has an element not of prime powe*
*r order,
and hence G 2 MC by Lemma 3.1(a). This implies that there is a Pmatched pair of
real Grepresentations of type 2; and hence a Pmatched pair of real (or self c*
*onjugate)
Grepresentations of type 1.
*
* _
Let T : eK(F ) ! Z be any conjugation invariant homomorphism (i.e., T () = *
*T ( ));
and let TG denote the induced homomorphism
TG = T Id: eKG(F ) ~=Ke(F ) R (G)  ! R (G):
We first analyze TG (C Rj). Let V0; V1; : :;:Vn be the distinct irre*
*ducible RG
representations, where V0 ~=R is the trivial representation, and where Vi has r*
*eal type
for 0 i k, has complex type (with given complex structure) for k + 1 i m, a*
*nd
has quaternion type (with given structure) for m + 1 i n. Then
iM k j i Mm j i Mn j
j ~= i RVi i CVi i HVi ;
i=0 i=k+1 i=m+1
24
(Proposition A.1(a)), where the i are real, complex, or quaternion vector bundl*
*es,
respectively, and where 0 ~= o(F ). For convenience, we write T (i) = T ([C R*
*i]) if
i k, T (i) = T ([i]) if k+ 1 i m, and T (i) = T ([iC]) if m + 1 i n. Then
Xk mX __
TG ([C Rj]) = T (i).c([Vi]) + (iT). [Vi] + [V i]
i=0 i=k+1
Xn (*
*1)
+ T (i).c0([Vi]) 2 R(G):
i=m+1
Since (for given T ) the TG commute with restriction of subgroups, w*
*e see that
TG ([C Rj])P = 0 for any P G of prime power order. Thus TG ([C Rj]) = [V ] *
* [W ],
where (V; W ) is a Pmatched pair of selfconjugate Grepresentations of type T*
* (0).
Assume that [o(F )] =_[0] =2r(Ke(F ))+ qdiv(KgO (F )). Then c([0]) is not a *
*multiple
of 2 in Ke(F )=<[]  [ ] 2 (qdiv)>. So by Lemma A.5(c), we can choose T such*
* that
T (0) is odd. Then TG ([C Rj]) = [V ]  [W ] where (V; W ) is a Pmatched pai*
*r of
selfconjugate Grepresentations of type T (0), and G 2 MC+ by the above remark*
*s.
If c([o(F )]) = c([0]) =2c0(]KSp (F )) + qdiv(Ke(F )), then qOc([0]) is not *
*a multiple of
2 in ]KSp(F )=(qdiv). By Lemma A.5(c) again, we can choose T to be a composite *
*of
the form eK(F ) q!]KSp(F ) ! Z, and such that T (0) is odd. In particular, T *
*(i) 2 2Z
for m + 1 i n (the quaternion case). By (1), TG ([C Rj]) = c([V ]  [W ]), w*
*here
Xk mX Xn T ( )
[V ]  [W ] = T (i).[Vi] + T (i).r([Vi]) + ___i_.rOc0([Vi]*
*);
i=0 i=k+1 i=m+1 2
and (V; W ) is a Pmatched pair of real Grepresentations of type T (0). It fol*
*lows that
G 2 MR.
We now get immediately:
Proof of Theorem 0.2. If F = MG for any contractible manifold M with smooth
Gaction, then the Gvector bundle j = o(M)F satisfies conditions (1)(3) in T*
*heorem
0.1, and so F 2 Fix(G). Also, O(F ) 1 (mod nG ) if M is a disk. Conversely*
*, by
Theorem 0.1, if F 2 Fix(G), then F is the fixed point set of a smooth Gaction *
*on a
euclidean space if @F = ;, and F is the fixed point set of a smooth Gaction on*
* a disk
if F is compact and O(F ) 1 (mod nG ).
The necessary and sufficient conditions for F to be in Fix(G) were shown in *
*Lemmas
3.2 and 3.3. Note in particular case (C) in Theorem 0.2. If G 2 MCr MC+ and
Syl2(G) 6C G, then F 2 Fix(G) if [o(F )] 2 r(Ke(F )) (Lemma 3.2(a)) or if [o(F*
* )] 2
qdiv(KgO (F )) (Lemma 3.2(c)). So from the definition of Fix(G), it follows th*
*at F 2
Fix(G) if [o(F )] 2 r(Ke(F )) + qdiv(KgO (F )).
The following example shows how Theorem 0.2 applies in the case of a dihedra*
*l or
quaternion group acting on a disk.
25
Example 3.4. If G is dihedral of order 2n or quaternion of order 4n, where n is*
* not a
prime power, then a compact manifold F is the fixed point set of a Gaction on *
*some
disk if and only if O(F ) is odd. If G is dihedral of order 2pa for some odd pr*
*ime p, then
a compact manifold is the fixed point set of a G action on a disk if and only i*
*f O(F ) = 1
and [o(F )] is torsion in gKO (F ). If G is quaternion of order 4pa for some o*
*dd prime
p, then a compact manifold is the fixed point set of a G action on a disk if an*
*d only
if O(F ) = 1, and there is an Hvector bundle #F such that c([o(F )]) c0([]) m*
*odulo
torsion in eK(F ).
Proof. If G is dihedral of order 2n or quaternion of order 4n, then by Theorem *
*0.3,
nG = 2 if n is not a prime power, and nG = 0 if n is a power of an odd prime. T*
*he rest
of the corollary follows from Lemma 3.1 and Theorem 0.2.
As another example, note that for G = A4x 3, any compact smooth manifold F c*
*an
be the fixed point set of a smooth Gaction on a disk. This is in fact the smal*
*lest group
with that property (see [O1, Theorem 8]).
Appendix
We collect here some results which are well known, but which either are hard*
* to find
in the literature, or which have been used often enough to state here explicitl*
*y.
Real Gvector bundles and their classifying spaces
We start with the following proposition, which describes some of the basic s*
*tructure
of real Gvector bundles and real Grepresentations.
Proposition A.1. Fix a finite group G. Let V0; V1; : :;:Vk be the distinct irre*
*ducible
RGrepresentations, where V0 ~= R with the trivial Gaction. For each i, set D*
*i =
End RG (Vi) (~=R, C, or H).
(a) LetLX be space with trivial Gaction, and let #X be a real Gvector bund*
*le.
Then ~= ki=0(Vi Dii), where each i is a (nonequivariant) Divector bundle o*
*ver
X.
(b) Let V be any orthogonal Grepresentation, and let OG (V ) be the group *
*of G
equivariant orthogonal self maps of V . Then
Mk Yk
V ~= (Vi)ni implies OG (V ) ~= O(ni; Di);
i=0 i=0
where we write O(n; R) = O(n), O(n; C) = U(n), and O(n; H) = Sp(n).
(c) If G is odd, then Di ~=C for all i 6= 0.
Proof. For each i, End RG(Vi) is a division algebra over R by Schur's lemma (cf*
*. [Ad,
Lemma 3.22]), and hence is isomorphic to R, C, or H. Part (b) also follows from*
* Schur's
lemma, and part (c) from [Se, Exercise 13.12].
26
To see part (a), set i = Hom RG (Vi; ) (defined fiberwise) for each i. Then*
* i is a
Dibundle, and the evaluation maps define an isomorphism
Mk ~
Vi DiHom RG (Vi; ) =! :
i=0
An irreducible RGrepresentation V will be said to have real, complex, or qu*
*aternion
type, depending on whether End RG(V ) is isomorphic to R, C, or H.
For each n 0, BG O(n) will denote the classifying space for ndimensional G*
*vector
bundles: constructed using either infinite joins (cf. [tD, xI.8]), or Grassma*
*nnians of
ndimensional subspaces in an appropriate infinite dimensional Grepresentation*
*. It
has a universal Gvector bundle EnG#BG O(n) with respect to which pullback defi*
*nes a
bijection between [X; BG O(n)]G and the set of locally trivial ndimensional or*
*thogonal
Gbundles over X, for any countable Gcomplex X (cf. [tD, Theorem I.8.12], whe*
*re
the classifying space is denoted B(G; O(n))). Note that BG O(n) is connected fo*
*r all n,
since ss0(BG O(n)) contains just one element: the class of the product bundle G*
*x Rn#G.
For each orthogonal Grepresentation V , and each m 0, direct sum with V de*
*fines
a Gmap
V : BG O(m) ! BG O(m+ dim (V ));
which is well defined up to Ghomotopy. We define BG O to be the homotopy dire*
*ct
limit (i.e., infinite mapping cylinder, or mapping telescope)
i j
BG O = hocolim!BG O(0) RG!BG O(d) RG!BG O(2d) RG!: :;:
where RG denotes the regular representation and d = dim(RG) = G. For each n, *
*we
let
n : BG O(nd) ! BG O
denote the inclusion of the nth stage into this telescope.
If X is any finite Gcomplex, then any map X ! BG O factors through some fi*
*nite
stage BG O(nd) in the mapping telescope, and similarly for homotopies between m*
*aps.
Hence
i *
* j
[X;BG O] ~=lim![X; BG O(0)] RG![X; BG O(d)] RG![X; BG O(2d)] RG!:*
* : :
i j
~=lim VectR;G(X) RG!VectR;G(X) RG!VectR;G(X) RG!: : :
! 0 d 2d
(d = G*
*)
~=Ker KOG (X) dim!Z ;
Here, VectR;Gm(X) denotes the set of isomorphism classes of mdimensional ortho*
*gonal
Gvector bundles over X; and the last step holds since any Gvector bundle over*
* X
is a summand of a product bundle RGkx X for some k (since any Grepresentation *
*is
contained in some multiple of the regular representation RG). In particular, th*
*is shows
that Zx BG O is the classifying space for the equivariant Ktheory functor KOG *
*().
We next look more closely at the fixed point sets (BG O)H and (BG O(n))H .
27
Proposition A.2. Fix a finite group G and a subgroup H G.
(a) For each n 0,
a
(BG O(n))H ' BOH (V )
[V ]2RepRn(H)
where Rep Rn(H) is the set of isomorphism classes of ndimensional orthogonal H
representations.
(b) Let V0; V1; : :;:Vk be the distinct irreducible orthogonal Hrepresentat*
*ions. Then
Yk
(BG O)H ' IRO (H) x Bi;
i=0
where IRO (H) = Ker[RO (H) dim!Z] is the augmentation ideal, and where Bi ~=B*
*O,
BU, or BSp depending on whether End RG(Vi) ~=R, C, or H.
H
(c) For any n > 0, Hn : BG O(nd) ! (BG O)H (d = G) sends the compone*
*nt
H
BG O(nd) V corresponding to the representation V to the component of (BG O)H *
*cor
responding to [V ]  [RGn] 2 IRO (H); and the map between this pair of componen*
*ts is
mconnected if each irreducible Hrepresentation occurs in V with multiplicity *
*at least
m.
(d) For any finite Hcomplex X,
( i
KOH (X) if i > 0
ssi map H (X; BG O) ~= dim
Ker KOH (X) ! Z if i = 0.
Proof. Consider the Hequivariant maps BG O(n) f1!BH O(n) f2!BG O(n), where f1
classifies the universal bundle EnG#BG O(n) regarded as an Hbundle, and where *
*f2
classifies the Gbundle (Gx HEnH)#(Gx HBH O(n)). These are easily checked to be*
* H
homotopy inverses; and show that BG O(n) is Hequivariantly homotopy equivalent*
* to
BH O(n). So BG O is Hhomotopy equivalent to BH O. In particular, it suffices t*
*o prove
the proposition when H = G.
(a) For any n,
ss0((BG O(n))G ) ~=VectR;Gn(pt) ~=Rep Rn(G):
Let BG O(n) GVdenote the component corresponding to the representation V . For*
* any
X (without group action),
[X; (BG O(n))G ] ~=[X; BG O(n)]G ~=Vect R;Gn(X);
and so [X; (BG O(n))GV] is the set of isomorphism classes of Gvector bundles o*
*ver X
with fiber V . The structure group for such bundles is OG (V ), and hence (BG O*
*(n))GV'
BOG (V ).
28
(b,c) The descriptions of the components of (BG O)G , and of Gn : (BG O(nd))*
*G !
(BG O)G , follow from part (a) and Proposition A.1(b), upon taking limits with *
*respect
to direct sum with the regular representation RG. Note in particular that
i j
IRO(G) ~=lim!RepR0(G) RG!RepRd(G) RG!RepR2d(G) RG!: :::
And the last statement in (c) follows since the inclusions BO(m) ! BO, BU(m) *
*! BU,
and BSp(m) ! BSp are mconnected for all m.
(d) We have already seen that [X; BG O]G ~=Ker [KOG (X) dim!Z] when X is a*
* finite
Gcomplex. And for any i > 0,
i G i
ssi map G(X; BG O) ~=[ (X+ ); BG O]* ~=KOG (X):
As was noted above, any map from a finite Gcomplex X to BG O factors through
some n : BG O(nd) ! BG O, and hence induces (stably) a Gbundle over X. This d*
*oes
not hold in general for finite dimensional Gcomplexes, but the next lemma desc*
*ribes
conditions under which maps X ! BG O do induce Gbundles.
Lemma A.3. Fix a countable finite dimensional Gcomplex X. Then for each n 0,
pullback of the universal bundle EnG#BG O(n) defines a bijection between [X; BG*
* O(n)]G
and the set of isomorphism classes of ndimensional Gbundles over X. Also, a G*
*map
f : X ! BG O factors through m : BG O(md) ! BG O for some m (where d = G)
if and only if Im (ss0(fH )) ss0((BG O)H ) is finite for all H G. And any two*
* liftings
fm ; f0m : X ! BG O(md) of f are homotopic after some finite stabilization; i*
*.e., the
induced Gbundles over X are stably isomorphic.
Proof. The bijection between ndimensional Gbundles over X and [X; BG O(n)]G i*
*s a
special case of [tD, Theorem I.8.12].
If f : X ! BG O factors through some BG O(md), then Im(ss0(fH )) must be fi*
*nite for
all H since (BG O(md))H has only finitely many connected components (correspon*
*ding
to the finite set of mdimensional Hrepresentations). Conversely, set n = dim*
* (X),
and assume that Im (ss0(fH )) is finite for all H G. For each H, we can choose*
* some
mH 0 large enough so that the image of fH is contained in components of (BG O*
*)H
corresponding to some family of virtual Hrepresentations vi = [Vi][RGmH ] 2 I*
*RO (H)
(1 i k), and such that each irreducible Hrepresentation occurs in each Vi wi*
*th
multiplicity at least n. Thus, the image of any connected component of XH is l*
*ies in
one of the components (BG O)Hvi, which is in the image of (BG O(mH .d))HVi; and*
* the
inclusion of those components is nconnected by Proposition A.2(c). Hence, if *
*we set
m = max {mH  H G}, then f : X ! BG O factors through BG O(md).
Divisible and quasidivisible subgroups
The purpose of the following two lemmas is to set up some notation and resul*
*ts to
work with cohomology and Ktheory groups of countably infinite CW complexes. He*
*nce,
we concentrate on the class of what we call PFG groups ("profinitely generat*
*ed"):
abelian groups which are products of the form lim(Mi) x lim1(M0i), where Mi a*
*nd M0i
are two inverse systems of finitely generated abelian groups. We first note tha*
*t the lim1
factor is divisible (i.e., ndivisible for all n > 0).
29
i j
Lemma A.4. Fix a sequence . . .!M2 ! M1 ! M0 of abelian groups. Then
for any n > 0, lim1(Mi) is ndivisible if Mi=nMi is finite for all i. In parti*
*cular, if the
Mi are all finitely generated, then lim1(Mi) is divisible, and hence injective.
Proof. Since lim1is right exact, the sequence
lim1(Mi) .n!lim1(Mi) ! lim1(Mi=nMi) ! 0
i i i
is exact for all n > 0, and lim1(Mi=nMi) = 0 if the Mi=nMi are finite. So lim*
*1(Mi) is
ndivisible in this case.
For any abelian group A, we define qdiv(A) to be the smallest possible kerne*
*l of a
homomorphism from A to a product of copies of Z. Clearly, all elements in A wh*
*ich
are infinitely pdivisible for any prime p are contained in qdiv(A). So by Lemm*
*a A.4,
if A = lim(Mi) x lim1(M0i) where Mi and M0iare inverse systems of finitely ge*
*nerated
abelian groups, then qdiv(A) = lim(tors(Mi)) x lim1(M0i).
Lemma A.5. If X is a countable CW complex, and if h* is any (representable) coh*
*o
mology theory such that hi(pt) is finitely generated for all i, then hi(X) is a*
* PFG group
for all i. In particular, eK(X), gKO(X), and ]KSp(X) are all PFG groups. Furt*
*hermore,
the following hold for any PFG group A:
(a) If x 2 A is divisible, i.e., if x 2 nA for all n > 0, then x is "sequent*
*ially divisible" in
that for any sequence n1; n2; : :o:f positive integers there is a sequence x = *
*x0; x1; x2; : : :
in A such that nixi = xi1 for all i. Similarly, if x 2 A is infinitely pdivis*
*ible for any
prime p, then there is a sequence x = x0; x1; x2; : :s:uch that pxi = xi1 for *
*all i. And
if nx is infinitely pdivisible for any prime pn, then x is also infinitely p*
*divisible.
P
(b) For any n, and any x 2 qdiv(A), we can write x = pnxp, where each xp *
*is
infinitely qdivisible for all primes q 6= p dividing n.
(c) If x 2 A, and x =22A+ qdiv(A), then there is a homomorphism ' : A ! Z s*
*uch
that '(x) is odd.
Proof. Fix a countable CW complex X, and write X = [ 1i=1Xi, where X1 X2
X3 . . .are finite subcomplexes. Then for any representable cohomology theory *
*h*,
there is for each j a short exact sequence
j1 j j
0 ! lim1eh (Xi) ! h (X) ! limh (Xi) ! 0;
i i
where the hj(Xi) and ehj1(Xi) are all finitely generated. The extension split*
*s, since
the first term is injective by Lemma A.4, and so hj(X) is a PFG group.
Now assume A is a PFG group, and write A = lim(Mi)xlim1(M0i), where the *
*Miand
M0iare all finitely generated. Point (a) follows upon noting that the divisible*
* elements
in A are precisely those in lim1(M0i), and that the pdivisible elements (for *
*any prime
p) are those in limp0tors(Mi) x lim1(M0i). Point (b) is immediate. Point (c*
*) follows
upon noting that if x =22A + qdiv(A), then the image of x in some Mi=(tors) is *
*not
a multiple of 2. And then there is a homomorphism Mi=(tors) ! Z which sends t*
*he
image of x to an odd integer.
30
Homotopy and homology groups
We collect here some miscellaneous lemmas on homotopy and homology groups and
the Hurewicz map.
Lemma A.6. All homotopy groups of a countable CW complex are countable.
Proof. The homotopy groups of a finite simply connected complex are finitely ge*
*nerated
(cf. [Hu, Corollary X.8.3]). The fundamental group of a countable complex is co*
*untably
generated and hence countable. So the homotopy groups of the universal cover o*
*f a
countable complex are countable direct limits of finitely generated groups, and*
* hence
are countable.
The following version of the relative Hurewicz theorem is needed in Section *
*2 when
constructing spaces and maps. For convenience, when a map f : X ! Y is underst*
*ood,
we write ss*(Y; X) for ss*(Zf; X) (where Zf denotes the mapping cylinder), and *
*similarly
for H*(Y; X).
Lemma A.7. Fix a prime p and n 2. Assume that f : X ! Y is a map between
connected complexes such that ss1(f) is onto, such that Ker(ss1(f)) is abelian *
*and torsion
prime to p, and such that ssi(Ye; eX) Z(p)= 0 for all i < n. Here, Xe and eY *
*denote
the universal covers. Then Hi(Y; X) Z(p)= 0 for all i < n, and the Hurewicz m*
*ap
ssn(Ye; eX) Z(p)i Hn(Y; X; Z(p)) is onto. If, furthermore, X and Y are finite *
*complexes
and Ker (ss1(f)) is finite, then ssn(Ye; eX) Z(p) is finitely generated as a Z*
*(p)[ss1(X)]
module.
(Note that ss2(Ye; eX) ~=Im [ss2(Y ) ! ss2(Y; X)], and ssi(Ye; eX) = ssi(Y; X)*
* for i > 2. The
lemma is formulated using ssi(Ye; eX) rather than ssi(Y; X) to allow the possib*
*ility that
ss2(Y; X) is not abelian.)
Proof. Let F be the homotopy fiber of f : X ! Y , and let eFbe its universal c*
*over (F
is connected by assumption). Then
ssi(Fe) Z(p)~=ssi+1(Y; X) Z(p)= 0 for all 2 i < n 1.
So by the generalized Hurewicz theorem (cf. [Hu, Theorem X.8.1]), applied to th*
*e class
of torsion abelian groups of order prime to p,
Hi(Fe; Z(p)) = 0 for all i < n 1and Hn1 (Fe; Z(p)) ~=ssn1 (Fe) Z(*
*p):(1)
Set
h i
= ss2(Ye; eX) ~=Im ss2(Y ) ! ss2(Y; X)~=ss1(F ) = Ker ss1(F ) ! ss*
*1(X) ;
__
and let F = Fe= be the covering of F with fundamental group . By Lemma A.8
below, acts trivially on H*(Fe). If n 3, then is abelian and_torsion prime t*
*o p, by
assumption, and the spectral sequence for the fibration eF! F ! B gives
__
H*(F ; Z(p)) ~=H0(; H*(Fe; Z(p))) ~=H*(Fe; Z(p)):
31
Together with (1), this shows that
__ __
ssi(F ) Z(p)~=Hi(F ; Z(p)) for all i n  1 (*
*2)
whenever i 2, and this clearly also holds for i = 1.
__
Since Hi(F ; Z(p))_= 0 = Hi(Fe; Z(p)) for i < n1 by (1) and (2), the spectr*
*al sequence
for the fibration F ! Xe ! eYshows that
__
Hi(Ye; eX; Z(p)) = 0 for i < n and Hn(Ye; eX; Z(p)) ~=Hn1 (F ; Z(*
*p)):(3)
And this together with (2) shows that the plocal Hurewicz homomomorphism for t*
*he
pair (Ye; eX) is a composite of isomorphisms:
__ __
ssn(Ye; eX) Z(p)~=ssn1 (F ) Z(p)~=Hn1 (F ; Z(p)) ~=Hn(Ye; eX; Z(p)*
*):(4)
Set K = Ker(ss1(f)). By assumption, K is abelian and torsion prime to p. H*
*ence
H*(Ye; eX=K; Z(p)) ~=H0(K; H*(Ye; eX; Z(p))). Since Hi(Ye; eX; Z(p)) = 0 for i *
*< n by (3),
the spectral sequence for the fibration (Ye; eX=K) ! (Y; X) ! ss1(Y ) shows t*
*hat
Hi(Y; X; Z(p)) ~=H0 ss1(Y ); Hi(Ye; eX=K; Z(p)) ~=H0(ss1(X); Hi(Ye; eX; Z*
*(p)))
for all i n. Together with (3) and (4), this shows that Hi(Y; X; Z(p)) = 0 for*
* i < n,
and that the Hurewicz homomorphism sends ssn(Ye; eX) Z(p)onto Hn(Y; X; Z(p)).
Now assume that X and Y are finite complexes, and that Ker[ss1(X) i ss1(Y )]*
* is fi
nite. The kernel has order prime to p, by assumption, and so any projective Z(p*
*)[ss1(Y )]
module is also projective as a Z(p)[ss1(X)]module. Each term in the relative *
*cellular
chain complex C* = C*(Ye; eX; Z(p)) is thus a finitely generated projective Z(p*
*)[ss1(X)]
module. Since C* has no homology below dimension n, Zn def=Ker[Cn @!Cn1 ] is*
* a
direct summand of Cn and hence finitely generated. So Hn(Ye; eX; Z(p)) = Zn=@(C*
*n+1 )
is also finitely generated over Z(p)[ss1(X)]; and is isomorphic to ssn(Ye; eX) *
* Z(p)by (4).
It remains to prove the following lemma, which says in particular that the h*
*omotopy
fiber of a map between connected and 1connected spaces is simple.
Lemma A.8. Let F ! X f!Y be a fibration of path connected spaces such that F
has a universal cover eF; and set = Ker[ss1(F ) ! ss1(X)]. Then the translati*
*on action
of any element of on eFis homotopic to the identity. In particular, acts triv*
*ially on
ss*(Fe) and on H*(Fe).
Proof. Fix a basepoint x0 2 F X, and set y0 = f(x0) 2 Y . Let fl : I ! F be
any loop (fl(0) = fl(1) = x0) which represents an element of , and choose a hom*
*otopy
G : Ix I ! X such that G(t; 0) = fl(t), and G(t; s) = x0 if s = 1 or t 2 {0; 1*
*}. Then
[f O G] 2 ss2(Y ), and @([f O G]) = [fl].
32
Define ff = f O G O proj: F xIx I ! B. By the homotopy lifting property fo*
*r the
fibration, there exists a map A : F xIx I ! X such that A(x; t; s) = x if s =*
* 1 or
t 2 {0; 1}. Let fi : F xI ! F be the map fi(x; t) = A(x; t; 0). Then fi(; 0) *
*= fi(; 1) =
IdF , and so fi can be lifted to a unique homotopy efi: eFxI ! eFsuch that efi*
*(; 0) = Id.
Also, the loop fi(x0; ) is homotopic to fl by construction, so efi(; 1) is th*
*e covering
transformation induced by fl, and is thus homotopic to the identity.
The next lemma is much more technical. It is needed in the proof of Theorem *
*0.1,
to handle fixed point sets not of finite homotopy type.
Lemma A.9. Fix n > 0, and let B be a connected Hspace with the property that
(Z=n) [K; B] is finite for any finite CW complex K. Let X be a countable fini*
*te
dimensional complex, and let f : X ! B be a map which is nullhomotopic on all *
*finite
__ _f
subcomplexes of X. Then f factors as a composite_X ,! X ! B, for some countab*
*le
finite dimensional Z=nacyclic complex X X.
Proof. Write X = [1i=1Xi, where X1 X2 X3 . .a.re all finite subcomplexes. By
assumption, [f] 2 [X; B] lies in the image of the first term in the short exact*
* sequence
0 ! lim1[(Xi); B] ! [X; B] ! lim[Xi; B] ! 0:
By Lemma A.4, the group lim1[(Xi); B] is ndivisible, and so there is a sequen*
*ce of
maps f = f0; f1; f2; : :::X ! B such that n.[fi] = [fi1] for each i. Hence f *
*factors as
a composite
bf
X ! bB! B;
i .n .n .n j
where bBis the homotopy inverse limit of the sequence : ::! B ! B ! B .
We next claim that Bb is Z=nacyclic. To see this, note that each map B .n*
*!B
induces multiplication by n in homotopy groups, and so all homotopy groups of b*
*Bare
uniquely ndivisible. Hence, via spectral sequences for the fibrations, it will*
* suffice to
show that He*(K(M; i); Z=n) = 0 for all i 1 and all Z[ 1_n]modules M. It suff*
*ices (by
taking direct limits) to show this for finitely generated M; and hence for M = *
*Z[ 1_n]
and M finite of order prime to n. The latter case is clear. When M = Z[ 1_n],*
* then
K(M; 1) is a Z[ 1_n]Moore space (and hence Z=nacyclic); and its deloopings ar*
*e seen to
be Z=nacyclic using the usual spectral sequences.
It_remains_to show that bfextends to a countable finite dimensional Z=nacyc*
*lic com
plex X X. To see this, first replace bBby a CW complex (of the same weak homot*
*opy
type) which contains X as a subcomplex. Since homology is supported by finite c*
*om
plexes, there is a sequence X = X0 X1 X2 . .o.f countable subcomplexes of bB
such that each inclusion Xi1 Xiis trivial in Z=nhomology. Set X1 = [1i=1Xi.*
* Then
X1 is countable and Z=nacyclic, but need not be finite dimensional. Set d = d*
*im(X),
and consider the free abelian group Bd(X1 ) def=Ker[Hd((X1 )(d)) ! Hd(X1 )]. E*
*very
element in Bd(X1 ) is in the image of the_Hurewicz homomorphism for (X1 )(d). *
*So
there is a (d + 1)dimensional complex X with the same dskeleton as X1 , such*
* that
33
__ __ __
the_inclusion_X (d) X1 extends_to_X , and such that Hi(X ) ~=Hi(X1_) for i d *
*and
Hi(X ) = 0 for i > d. Then eH*(X ) is uniquely ndivisible, so X is Z=nacycli*
*c, and we
are done.
Projective and stably free homology of Gcomplexes
The following lemma is basically taken from [O1], although not stated there *
*explicitly.
Lemma A.10. Let G be any finite group, and let f : X ! Y be a map between
countable finite dimensional Gcomplexes. Set n = max {dim (X); dim(Y )}. Ass*
*ume,
for some prime p, that Hei(Zf; X; Fp) = 0 for all i n, and that fP : XP ! Y*
* P is
an Fphomology equivalence for all psubgroups 1 6= P G. Then Hn+1 (Zf; X; Fp)
is projective as an Fp[G]module, and hence is stably free as a countably gener*
*ated
Fp[G]module. If in addition, X and Y are finite complexes, and O(XH ) = O(Y H)*
* for
any cyclic subgroup 1 6= H G of order prime to p, then Hn+1 (Zf; X; Fp) is a f*
*ree
Fp[G]module.
Proof. By replacing Y with the mapping cone of f, we can assume that X is a po*
*int
(and dim (Y ) n+ 1). Throughout the proof, for any subcomplex Y 0 Y , we let
C*(Y; Y 0; Fp), denote the cellular chain complex of (Y; Y 0): the complex who*
*se nth
degree term is the free Fpmodule with one generator for each ncell in Y not i*
*n Y 0.
Fix a Sylow psubgroup S G, and let Ys be the union of the fixed point sets
Y P taken over all nontrivial subgroups 1 6= P S. Then Ys is a union of Fpac*
*yclic
subcomplexes such that all intersections are also Fpacyclic. Hence Ys is itsel*
*f Fpacyclic
(seen using MayerVietoris sequences). We thus get an exact sequence
0 ! Hn+1 (Y ; Fp) ! Cn+1 (Y; Ys; Fp) ! Cn(Y; Ys; Fp) ! : ::!C0(Y; Ys; F*
*p) ! 0;
and each term Ci(Y; Ys; Fp) is free as an Fp[S]module since S acts freely on Y*
* rYs.
Thus, Hn+1 (Y ; Fp) is stably free as an Fp[S]module; and hence is projective *
*as an
Fp[G]module (cf. [Rim, Corollary 2.4 & Proposition 4.8]). And by the "Eilenb*
*erg
swindle", any countably generated projective module M is stably free in the cat*
*egory of
countably generated modules: we can write M N ~=F for some countably generated
free module F , and then
M F 1 ~=M (N M) (N M) . .~.=(M N) (M N) . .~.=F 1:
Assume now that X and Y are finite complexes, and that O(Y H) = 1 for any cy*
*clic
subgroup 1 6= H G of order prime to p. We want to show that Hn+1 (Y ; Fp) is F*
*p[G]
free. In general, by [Se, x16.1], two projective Fp[G]modules M1 and M2 are is*
*omorphic
if and only if [M1] = [M2] 2 RFp(G) (the representation ring of all finitely ge*
*nerated
Fp[G]modules modulo short exact sequences). By [Se, x18.2], this is the case *
*if and
only if M1 and M2 have the same modular character, where the modular character *
*of an
Fp[G]module is a complex valued function defined on the set of elements of G o*
*f order
prime to p. Thus, a projective Fp[G]module M is free if and only if M ~=(Fp[G]*
*)s for
s = rkFp(MG ), if and only if MG~= (Fp[G])r for r = rkFp(M), if and only if M*
* is free
as a Fp[H]module for each cyclic subgroup H G of order prime to p. And for any
34
such H, since O(Y K) = 1 for all 1 6= K H by assumption, a count of the number*
*s of
Horbits of cells in Y shows that in RFp(H),
Xn
(1)n[Hn+1 (Y ; Fp)] = (1)i[Ci(Y; pt; Fp)] 0 (mod ):
i=0
The next lemma provides an analogous result for integral homology.
Lemma A.11. Let G be any finite group, and let f : X ! Y be a map between
countable finite dimensional Gcomplexes. Set n = max {dim (X); dim(Y )}. Ass*
*ume
that Hei(Zf; X; Z) = 0 for allfii n, and that fP : XP ! Y P is an Fphomology
equivalence for all primes pfiG and all psubgroups 1 6= P G. Then Hn+1 (Zf;*
* X; Z)
is Z[G]projective, and hence is stably free as a countably generated Z[G]modu*
*le. If in
addition X and Y are finite complexes, and O(XH ) = O(Y H) for all 1 6= H G, t*
*hen
there is a finite Gcomplex T such that T H = pt for all H G not of prime power
order, and such that for some d, T is (d 1)connected and He*(T ; Z) = Hd(T ; *
*Z) ~=
Hn+1 (Zf; X; Z) as Z[G]modules.
Proof. By LemmafA.10,iHn+1 (Zf; X; Fp) = Z=p Hn+1 (Zf; X) is Fp[G]projective *
*for
all primes pfiG. Also, Hn+1 (Zf; X; Z) is Zfree, since dim (Zf) = n+ 1. In p*
*articular,
Hn+1 (Zf; X; Z) is Gcohomologically trivial; and hence is Z[G]projective by R*
*im's
theorem [Rim, Theorem 4.11]. And by the "Eilenberg swindle" again, this implies*
* that
Hn+1 (Zf; X; Z) is Z[G]stably free in the category of countably generated Z[G]*
*modules.
Now assume that X and Y are finite complexes. Let Cf denote the mapping cone
of f : X ! Y , so Hn+1 (Cf) ~=Hn+1 (Zf; X) is Z[G]projective. Let CNPf be the*
* set of
elements x 2 Cf whose isotropy subgroup Gx does not have prime power order. Then
O((CNPf)H ) = 1 for all H G (since O(XH ) = O(Y H) when H 6= 1, and all free
orbits have been removed). Hence, by [O2, Proposition 5], there is a finite con*
*tractible
Gcomplex Z CNPf such that all isotropy subgroups of Zr CNPf have prime power
order. By [O2, Lemma 11], applied to the inclusion Z Z[ CNPfCf, there exists a*
* finite
contractible Gcomplex Z0 Cf such that all isotropy subgroups of Z0r Cf have p*
*rime
power order. If we now set T = Z0=Cf, then T ' (Cf) (nonequivariantly); and so T
is (n+ 1)connected and eH*(T ) = Hn+2 (T ) ~=Hn+1 (Zf; X).
Since this last argument is rather indirect, we now outline a more direct ar*
*gument
to help explain what is really going on. The details are similar to those used *
*in [O1, x3]
to study fixed point sets. For any Gcomplex X, XNP X denotes the union of fi*
*xed
point sets of subgroups not of prime power order. A finite Gcomplex X will be *
*called
simple if O(XH ) = 1 for all H G. A finite Gcomplex X will be called a Greso*
*lution
if X is ndimensional and (n 1)connected for some n, and Hn(X) is Z[G]projec*
*tive.
For any Gresolution X, set flG (X) = (1)n[Hn(X)] 2 Ke0(Z[G]) (n = dim (X)). *
*Let
B0(G) eK0(Z[G]) be the subset of all flG (X) for Gresolutions X such that XNP*
* is a
point. Using direct geometric constructions, one now shows that B0(G) is a subg*
*roup,
and that there is a well defined function
G : {finite simple Gcomplexes}  ! eK0(Z[G])=B0(G)
35
__ __ __
which sends X to flG (X ) for any Gresolution X such that X NP ~= XNP . Also, *
*G (X)
depends only on the Ghomotopy type of X, and G is additive in the sense that
G (Y=X) = G (Y )  G (X) for any pair X Y . It is now straightforward to show
that any such function from finite simple Gcomplexes to an abelian group is tr*
*ivial.
And when applied to the mapping cone Cf defined above (more precisely, to CNPf)*
*, this
shows that [Hn+1 (Zf; X)] 2 B0(G).
An equivariant thickening theorem
The procedure for constructing manifolds with smooth Gaction, starting with*
* a G
vector bundle over a countable finite dimensional Gcomplex, is based on the fo*
*llowing
equivariant thickening theorem. As has been seen, it provides a good tool when*
* con
structing smooth Gactions on disks and euclidean spaces. In contrast, it cann*
*ot be
used (at least not directly) to construct smooth actions on closed manifolds.
Theorem A.12. [Pawalowski] Fix a finite group G, a countable finite dimensional*
* G
complex X, and a Gvector bundle #X. Assume that F = XG is given the structure
of a smooth manifold, and that (F )G is stably isomorphic to the tangent bund*
*le
o(F ). Then there is a smooth manifold M with smooth Gaction, containing X as*
* a
Gdeformation retract, such that MG is diffeomorphic to F , and such that o(M)*
*X is
stably Gisomorphic to (i.e., (o(M)X) (V xX) ~= (W xX) for some pair of G
representations V and W ). If X is a finite Gcomplex, then M can be chosen t*
*o be
compact.
Proof. See [Pa2, Theorems 2.4 & 3.1] (where the result is stated more precisely*
*). The
idea of the proof is the following. After adding a product bundle to , we can a*
*ssume
that G #F ~=o(F ) (Rkx F ) for some k; and that
dim ((x)H ) > 2.dim (XH ) + k and dim ((x)H )  dim((x)> H) > dim(XH )
for all H $ G and all x 2 XH . Here, (x)> H denotes the union of the fixed poin*
*t sets
of subgroups of Gx strictly containing H.
Choose Ginvariant subcomplexes F = X0 X1 X2 . .X., such that X =
[Ni=0Xi (where N 1), and such that each Xi is obtained from Xi1 by attaching
one orbit of cells G=H x Dj for some H and j. Manifolds M0 M1 M2 . . .
are now constructed such that for each i, (Mi)G = F , Xi Mi and (@Mir @F )
(Mir Xi) are Gdeformation retracts, and o(Mi)Xi (Rkx Xi) ~= Xi. To start t*
*he
procedure, let M0 be the disk bundle of (F )=(G ). The induction step is carri*
*ed out
using standard (nonequivariant) embedding theorems, and theorems about destabil*
*izing
vector bundles and isomorphisms between them. The manifold M = [Ni=0Mi now
satisfies the conclusions of the theorem.
References
[Ad] J. F. Adams, Lectures on Lie groups, Benjamin (1969)
[Br] G. Bredon, Introduction to compact transformation groups, Academic Press (*
*1972)
[tD] T. tom Dieck, Transformation groups, de Gruyter (1987)
36
[EL] A. Edmonds & R. Lee, Topology 14_(1975), 339345
[Go] D. Gorenstein, Finite groups, Harper & Row (1968)
[Hu] S.T. Hu, Homotopy theory, Academic Press (1959)
[JM] S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces via
elementary abelian subgroups, Topology 31_(1992), 113132
[JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of selfma*
*ps of
BG via Gactions, Annals of Math. 135_(1992), 183270
[JO] S. Jackowski & B. Oliver, Vector bundles over classifying spaces of compac*
*t Lie
groups, preprint
[Jo] L. Jones, The converse to the fixed point theorem of P. A. Smith: I, Anna*
*ls of
Math. 94_(1971), 5268
[Lam] T.Y. Lam, Induction theorems for Grothendieck groups and Whitehead groups
of finite groups, Ann. Scient. Ec. Norm. Sup. 1_(1968), 91148
[Mi] J. Milnor, Lectures on the hcobordism theorem, Princeton Univ. Press (196*
*5)
[O1] R. Oliver, Fixedpoint sets of group actions on finite acyclic complexes, *
*Comment.
Math. Helv. 50_(1975), 155177
[O2] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149_(1976),*
* 7996
[O3] R. Oliver, Gactions on disks and permutation representations II, Math. Z*
*. 157_
(1977), 237263
[O4] B. Oliver, Higher limits via Steinberg representations, Comm. in Algebra 2*
*2_(1994),
13811393
[OP] R. Oliver & T. Petrie, GCWsurgery and K0(ZG), Math. Z. 179_(1982), 1142
[Pa1] K. Pawalowski, Group actions with inequivalent representations at fixed p*
*oints,
Math. Z. 187_(1984), 2947
[Pa2] K. Pawalowski, Fixed point sets of smooth group actions on disks and eucl*
*idean
spaces, Topology 28_(1989), 273289
[Rim] D. S. Rim, Modules over finite groups, Ann. of Math. 69_(1959), 700712
[Ro] D. Robinson, Finiteness conditions and generalized soluble groups, Part 1,*
* Springer
Verlag (1972)
[Se] J.P. Serre, Linear representations of finite groups, SpringerVerlag (197*
*7)
[Sm] P. A. Smith, Fixedpoints of periodic transformations, Amer. Math. Soc. *
* Coll.
Pub. XXVII (1942), 350373
[St] J. Stallings, The piecewise linear structure of euclidean space, Proc. Ca*
*mbridge
Phil. Soc. 58_(1962), 481488
[Wo] Z. Wojtkowiak, On maps from holim F to Z, Algebraic topology, Barcelona, 1*
*986,
Lecture Notes in Math. 1298, SpringerVerlag (1987), 227236
37