TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Xxxx XXXX, Pages 000-000
S 0002-9947(XX)0000-0
HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES
OF COMPACT LIE GROUPS
ALEXEI STROUNINE
Abstract.Let p be a prime number and G be a compact Lie group. A
homology decomposition for the classifying space BG is a way of building
BG up to mod p homology as a homotopy colimit of classifying spaces of
subgroups of G. In this paper we develop techniques for constructing such
homology decompositions. In [6] Jackowski, McClure and Oliver construct a
homology decomposition of BG by classifying spaces of p-stubborn subgrou*
*ps
of G. Their decomposition is based on the existence of a finite-dimensio*
*nal
mod p acyclic G - CW-complex with restricted set of orbit types. We apply
our techniques to give a parallel proof of the p-stubborn decomposition *
*of BG
which does not use this geometric construction.
1.Introduction
Let p be a fixed prime number and G be a compact Lie group. A homology
decomposition for the classifying space BG is a mod p homology isomorphism
hocolimDF "!pBG
where D is a small category, F is functor from D to the category of spaces, and
for each object d of D, F (d) has the homotopy type of BH for some subgroup H
of G: Jackowski, McClure and Oliver [6] construct a homology decomposition of
BG by classifying spaces of p-stubborn subgroups of G. Their decomposition is
based on the existence of a finite-dimensional mod p acyclic G - CW -complex wi*
*th
restricted set of orbit types [6, 2.14]. In this paper we give a parallel proof*
* of the
p-stubborn decomposition of BG which does not use [6, 2.14] and has a potential
to be extended to p-compact groups.
We will call a set C of closed subgroups of G a collection if it is closed un*
*der
the process of taking conjugates in G: Let C be a collection. We define the C-o*
*rbit
category O(C) whose objects are the G-sets G=H, H 2 C, and whose morphisms
are G-maps. We regard O(C) as a topological category, where the set of objects
is discrete and where the morphism sets have compact-open topology. By JO(C),
or simply J, we will denote the inclusion functor from O(C) to the category of
G-spaces. Composing J with the Borel construction EG xG - gives a functor
EG xG J : O(C) ! Spaces
____________
1991 Mathematics Subject Classification. Primary 55R35; Secondary 55R40.
.
cO1997 American Mathematical Socie*
*ty
1
2 ALEXEI STROUNINE
whose value EG xG G=H at an object G=H has the homotopy type of BH. Let
* denote the one-point space with the trivial action of G. The natural maps
EG xG G=H ! EG xG * = BG induce a map
(1.1) hocolimO(C)EG xG J ! BG
Here the operator hocolimis the homotopy colimit defined as a special case of t*
*he
bar construction 2.2.
Definition 1.1.Call a collection C of subgroups of G ample if the map (1.1) in-
duces an isomorphism on mod p homology.
Since the Borel construction commutes with taking homotopy colimits
EG xG hocolimO(C)J ~=hocolimO(C)EG xG J
to prove that a collection C is ample, it suffice to show that in the fibration
hocolimO(C)J ! EG xG hocolimO(C)J ! BG
the fiber hocolimO(C)J is Fp-acyclic (i.e. the map hocolimO(C)J ! * induces
an isomorphism on mod p homology). For a collection of subgroups C, we write
EO(C) = hocolimO(C)J. The action of G on the values of the functor J induces an
obvious G-action on this space. In section 3 we consider sufficient conditions *
*that
imply the Fp-acyclicity of EO(C).
Suppose that the category O(C), for some collection C, has finite morphism se*
*ts.
Then the standard Bousfield-Kan cohomology spectral sequence of a homotopy
colimit [1, XII.5.8] associated to hocolimO(C)EG xG J has the form:
(1.2) Ei;j2= limiO(C)Hj(EG xG J; Fp) ) Hi+j(hocolimO(C)EG xG J; Fp)
Definition 1.2.An ample collection C of subgroups of G is said to be sharp if O*
*(C)
is a discrete category (in the sense that the morphism sets of O(C) have discre*
*te
topology) and the spectral sequence (1.2) collapses, that is Ei;j2= 0 for i > 0*
* and
E0;j2 Hj(BG; Fp).
The E2-term of the above spectral sequence can be identified with the ordi-
nary equivariant cohomology groups of the G-space EO(C) with certain coefficient
systems. In fact a collection C is sharp if and only if EO(C) has equivariant c*
*oho-
mology of a point. In section 4 we obtain a refinement of the equivariant trans*
*fer
map constructed by Oliver [9] which is used to show that, for suitable collecti*
*ons
C, the E2-term of the spectral sequence described above has appropriate vanishi*
*ng
properties.
Recall that a compact Lie group P is called p-toral if its identity component
P0 is a torus and P=P0 is a finite p-group. A p-toral subgroup P of G is called
p-stubborn if NG (P )=P is finite and has no nontrivial normal p-subgroups.
The main result of this work is the following
Theorem 1.3. Suppose that G is a compact Lie group that contains an element of
order p. Then the collection of nontrivial p-toral subgroups of G is ample and *
*the
collection of nontrivial p-stubborn subgroups of G is sharp.
Throughout the paper p is a fixed prime number. The symbol Fp denotes the
field with p elements. A map f : X ! Y between two spaces is said to be an Fp-
equivalence or mod p equivalence and is denoted X!"pY if it induces an isomorph*
*ism
HOMOLOGY DECOMPOSITIONS FOR COMPACT LIE GROUPS 3
f* : H*(X; Fp) ! H*(Y; Fp). A space X is said to be Fp-acyclic or mod p acyclic
if the projection into a one point space X ! * is an Fp-equivalence.
2. Bar constructions, homotopy colimits and G-spaces
Throughout this paper a topological category is a category D with topologized
morphism sets such that composition is continuous and for each object d of D the
inclusion idd ! Mor(d; d) is a closed cofibration.
Definition 2.1.Let D be a small topological category, : Dop ! T op and
: D ! T op be continuous functors. The bar construction B(; D; ) is the
topological realization |Bo|of the simplicial space Bo, which in dimension k is*
* the
following disjoint union of spaces indexed by ordered sets of k + 1 objects of *
*D:
a
Bk = (dk) x MorD (dk;dk-1) x : :x:MorD (d1;d0) x (d0)
(d0;:::;dk)
The face operators ffii are induced by composite maps
MorD (di+1;di) x MorD (di;di-1) ! MorD (di+1;di-1)
when 0 < i < k, and by evaluation maps
MorD (d1;d0) x (d0) ! (d1);
(dk) x MorD (dk;dk-1) ! (dk-1)
when i is respectively 0 and k.
The degeneracies are given by appropriate inclusions of identities.
Definition 2.2.Let * denote the constant one point valued functor on D, consid-
ered as necessary to be either covariant or contravariant. With the above notat*
*ion
the homotopy colimits of functors and are
hocolim = B(; D; *); hocolim = B(*; D; )
Recall that the n-skeleton Bnoof a simplicial space is the smallest subobject*
* that
contains Bk for all k n. The mod p homology spectral sequence associated to
the filtration
fififi fi fi
B0ofi fiB1ofi : : :|Bno| : : :
fi fi fi fi
of |Bo|has the E1-term E1i;j= Hj+i(fiBiofi; fiBi-1ofi; Fp) Hj(Bi=sBi; Fp) and *
*is
first quadrant strongly convergent. Here sBi denotes the subspace of Bi which is
the union of images of Bi-1under the degeneracy operators. A simple application
of this spectral sequence is the following:
Lemma 2.3. Suppose that D is a small topological category, , 0 : Dop ! T op
and , 0 : D ! T op are continuous functors. If u : ! 0 and v : !
are natural transformations such that for each object d of D u(d) and v(d) are *
*mod
p-equivalences, then the map B(u; D; v) : B(; D; ) ! B(0; D; 0) is a mod p
equivalence.
Proof.Let Bo and B0obe the simplicial spaces used to define the bar constructio*
*ns
B(; D; ) and B(0; D; 0) respectively. The transformations u and v induce
a map (u; v) : Bo ! B0oof simplicial spaces such that the corresponding maps
Bi ! B0iand sBi ! sB0iare mod p homology isomorphisms in each dimension.
Since the inclusions sBi ! Bi and sB0i! B0iare cofibrations, (u; v) induces an_
isomorphism of the spectral sequences described above. |__|
4 ALEXEI STROUNINE
Suppose that C is a collection of subgroups of G and X is a G-space. Define X*
*C,
the C resolution of X, to be the bar construction B(MapG (J; X); O(C); J). There
is an obvious G-map 'C(X) : XC ! X induced by the evaluation maps in each
dimension of the simplicial space
a
Bk = G=Hk x MapG (G=Hk;G=Hk-1) x : :x:MapG (G=H0; X) ! X
(G=H0;:::;G=Hk)
Essentially the space XC contains all the equivariant homotopy information of*
* X
which is detectable by the subgroups of G in C. It is clear form the constructi*
*on that
the isotropy subgroups of XC are those subgroups H in C for which the fixed-poi*
*nt
space XH = MapG (G=H; X) is nonempty.
A map f : X ! Y of G-spaces is said to be a weak G-equivalence (respectively
a G-mod-p-equivalence) if for every subgroup H of G the induced map on the fixed
point sets fH : XH ! Y H is a weak equivalence (respectively mod p equivalence).
We will be interested in finding out to what extent the weak G-homotopy (ho-
mology) type of a G space X is determined by the fixed-point sets XH for some
restricted type of subgroups H of G. The following facts are simple consequences
of general properties of the bar construction:
Theorem 2.4. Let G be a compact Lie group, C a collection of subgroups of G and
X a G-space. Then
(a) for every H in C, the map on the fixed-point sets (XC)H ! XH is a homotopy
equivalence;
(b) if H is in C then 'C(G=H) : (G=H)C ! G=H is a G-equivalence;
(c) if X is a G-CW -complex such that for each orbit type G=H of X, 'C(G=H)
is a G-equivalence then so is 'C(X) : XC ! X.
Proof.Note that for a topological category D, functors : Dop ! T op and :
D ! T op, and an object d of D, the evaluation maps induce natural maps
(2.1) B(; D; MorD (d; -)) "! (d)
(2.2) B(MorD (-; d); D; ) "! (d)
which are strong deformation retractions [4, 3.1(5)].
(a) For H in C
(XC)H = B(MapG (J; X); O(C); J)H =
= B(MapG (J; X); O(C); MorO(C)(G=H; -))!"MapG (G=H; X) XH
(b) For H in C and Q any subgroup of G
(G=HC)Q = B(MapG (J; G=H); O(C); J)Q =
= B(MorO(C)(-; G=H); O(C); MapG (G=Q; J))!"
"! MapG (G=Q; G=H) (G=H)Q
(c) A G-CW -complex X is constructed from a collection of "G-cells" {G=HxDn}
by taking disjoint unions, pushouts in which one of the maps is a cofibration,
and sequential colimits. Clearly for a subgroup Q of G the fixed point functor
MapG (G=Q; -) preserves this three operations and the bar construction commutes
with colimits. Hence 'C(X) is a weak G-equivalence if it is one for each_G-cell*
* of
X. |__|
HOMOLOGY DECOMPOSITIONS FOR COMPACT LIE GROUPS 5
Corollary 2.5.If X is a G-space and C contains all isotropy subgroups of X then
'C(X) is a weak G-equivalence.
Corollary 2.6.If f : X ! Y is a G-map such that fH : XH ! Y H is a mod p
equivalence whenever H is an isotropy subgroup of X or Y , then f is a G-mod-p-
equivalence.
Proof.Let C be a collection of isotropy subgroups of X and Y . By 2.4 'C(X) :
XC ! X and 'C(Y ) : YC ! Y are weak G-equivalences, and by 2.3 fC : XC ! YC_
is a G-mod-p-equivalence. |__|
Remark 2.7.Note that the G-space EO(C) associated to a collection C is the C
resolution of a point *C, so its isotropy subgroups are those in C and for any *
*H in
C the fixed point set (EO(C))H is contractible. We will study the space EO(C) in
more detail in the next section.
We conclude this section by a lemma that considers C resolutions of transitive
G-spaces.
Lemma 2.8. Suppose that H < G is a pair of compact Lie groups and C is a
collection of subgroups of G. Let C|H be the collection of those subgroups of H*
* which
are in C and : G xH EO(C|H ) ! G=H be the H-bundle induced by projection
EO(C|H ) ! *. Then there is a weak G-equivalence f : G xH EO(C|H ) ! G=HC
such that = 'C O f.
Proof.The map f is defined on bar constructions in the following way: First we
identify the constant functor * : O(C|H ) ! T op with MapH (J; *) and note that*
* the
functor G xH - commutes with the bar construction, then we obtain a map of bar
constructions induced by the functor G xH - : O(C|H ) ! O(C) :
G xH B(*; O(C|H ); J)= B(MapH (J; *); O(C|H ); G xH J) !
! B(MapG (J; G xH *); O(C); J)
Note that the G-spaces G xH EO(C|H ) and G=HC have the same set of isotropy
subgroups, namely those subgroups Q in C which are subconjugates of H. For any
such Q consider the maps on the fixed point spaces
Q 'QC
Q : (G xH EO(C|H ))Q f! (G=HC)Q ! (G=H)Q :
The map 'QCis a weak equivalence by 2.4. The composite Q is a fiber bundle
with the fiber over a point gH in (G=H)Q being equal to (EO(C|H ))g-1Qg which
is contractible since g-1Qg 2 C|H . Thus f is a weak equivalence on the fixed p*
*oint
sets for isotropy subgroups of G xH EO(C|H ) and G=HC, and hence it is a_weak_
G-equivalence by 2.6. |__|
3. The G-space EO(C)
As mentioned before a collection of subgroups C is ample if the space EO(C) h*
*as
mod p homology of a point. In this section we state some conditions that imply
the mod p-acyclicity of this space. We first note that this is a necessary cond*
*ition
for ampleness of C if ss0(G) is a p-group:
Lemma 3.1. Let G be a compact Lie group such that ss0(G) is a finite p-group.
Then C is an ample collection of subgroups of G if and only if EO(C)!"p*.
6 ALEXEI STROUNINE
Proof.If EO(C)!"p* the conclusion is obvious. Suppose now that EGxG EO(C)!"pBG.
Let ss = ss1(G) G=G0. Since ss is a p-group it acts nilpotently on Fp[ss] and *
*the
homology isomorphism EGxG EO(C) ! EGxG * with coefficients in Fp implies the
isomorphism with twisted coefficients H*(EG xG EO(C); Fp[G=G0]) ! H*(EG xG
*; Fp[G=G0]). By Shapiro's lemma H*(EGxG X; Fp[G=G0]) H*(EGxG0 X; Fp),
thus we have a fibration EO(C) ! EG xG0 EO(C) ! BG0 in which the base space
is simply-connected and the projection map is a homology isomorphism. Serre lon*
*g __
exact sequence on homology of a fibration implies the acyclicity of the fiber. *
* |__|
Definition 3.2.Call a subgroup H in C a core subgroup of the collection C if H *
*is
normal in G and HP is in C for every P in C.
The following is a theorem of Quillen in a slightly more general form:
Lemma 3.3. If a collection of subgroups C has a core subgroup then EO(C) is
contractible.
Proof.Let P be a core subgroup of C. Consider the following subcollection of C
CP = {H 2 C : P H}:
Note that P acts trivially on every object of the associated orbit category O*
*(CP ),
therefore we can identify the functor JO(CP)with MorO(CP)(G=P; -). By (2.1) the
space EO(CP ) = B(*; O(CP ); MorO(CP)(G=P; -)) is contractible.
The map EO(CP ) ! EO(C) induced by inclusion of categories is a homotopy
equivalence: The inverse is induced by the functor (-)P : O(C) ! O(CP ) that
sends every object of G=H of O(C) its P -orbit space (G=H)P . Since P is a norm*
*al
subgroup of G, (G=H)P is a transitive G space of type G=HP . The projection
G=H ! (G=H)P defines a natural transformation between the identity functor on
O(C) and the composite O(C) (-)P!O(CP ) ! O(C). By [4, 3.1(7)] this transforma-_
tion gives a homotopy between the maps on EO(C) induced by these functors. |__|
We denote by Op(G) (respectively Rp(G)) the orbit category associated to the
collection of all nontrivial p-toral subgroups (respectively all nontrivial p-s*
*tubborn
subgroups).
Remark 3.4.If G has a nontrivial normal p-subgroup P , then P is a core subgroup
of the collection of nontrivial p-toral subgroups, hence EOp(G) ' *. This is the
original result of Quillen.
The following theorem will allow us to reduce the proof of the homology decom-
position theorem to the case of compact Lie groups with simpler structure (e.g.
groups of smaller dimension or groups with ss0 a p-group).
Theorem 3.5. Let G be a compact Lie group and p be a prime number. Suppose
that there exists a normal subgroup K of G such that for any p-toral subgroup Q*
* of
G (including the trivial subgroup) EOp(QK)!"p*. Then EOp(G)!"p*.
Proof.Denote the collection of nontrivial p-toral subgroups of G by C. Let D =
C [ {KQ : Q 2 C} [ {K}. Clearly D is a collection of subgroups. Consider the
following diagram of functors
HOMOLOGY DECOMPOSITIONS FOR COMPACT LIE GROUPS 7
Op(G) ______-JG-spaces
F | Fh*Jaaeae>eae>
|?aeaaeJe
O(D) ae
Here F is the inclusion functor and Fh*J is the Kan extension of J along F as
defined in [4, 5]. Then
EOp(G) ~=hocolimO(D)Fh*J!"phocolimO(D)J ~=*
The first map is an equivalence by the homotopy pushdown theorem [4, 5.5], the
third one is the equivalence of lemma 3.3 (Note that K is a core subgroup of D).
The map in the middle is constructed in the following way:
For each object G=H of D, by definition the Kan extension is the bar construc*
*tion
Fh*J(G=H) = B(MorD (F; G=H); Op(G); J), which is in fact the C resolution of
G=H. There is a natural transformation Fh*J ! J defined on each object G=H of
D to be the map 'C(G=H) : G=HC ! G=H. We will show that this map is mod
p equivalence for each object G=H of D which will imply the mod p equivalence of
the map on the homotopy colimits by 2.3.
Note that the projection of the fiber bundle G xH EOp(H) ! G=H is a mod p
equivalence since its fiber EOp(H) is mod p acyclic by assumption when H is in
the form KQ for a p-toral Q or it is contractible when H is nontrivial p-toral.*
* By
2.8 this projection factors through a weak G-equivalence:
G xH EOp(H)!"G=HC 'C!G=H
Therefore 'C is a mod p equivalence and this finishes the proof. |*
*___|
Remark 3.6.Let G be an infinite compact Lie group, we let K = G0, the identity
component of G. For any p-toral Q, ss0(QG0) is a p-group, thus Theorem 3.5 com-
bined with lemma 3.1 allows us to reduce the proof of the homology decomposition
theorem to the case of a compact Lie group ss0 of which is a p-group.
4.The transfer on equivariant cohomology
Let G be a compact Lie group. We will denote by O(G) the orbit category O(C)
when C is the collection of all subgroups of G.
Definition 4.1.A contravariant coefficient system M for G is a contravariant
functor from O(G) to Ab, the category of abelian groups, such that if f and g a*
*re
G-homotopic maps in O(G), M(f) = M(g).
Let X be a G-space and M a coefficient system. By H*G(X; M) we will denote
the ordinary equivariant cohomology of X with coefficients in M as defined in [*
*5]
and [12].
Suppose F is a functor from a small discrete (in the sense that the set of mo*
*r-
phisms has the discrete topology) category D to the category of G-spaces such t*
*hat
F (d) is a transitive G-space for each object d of D . There is an equivariant *
*co-
homology spectral sequence of hocolimDF associated to the skeleton filtration of
hocolimDF :
(4.1) Ei;j2= limiDHjG(F; M) ) Hi+jG(hocolimDF; M)
8 ALEXEI STROUNINE
Note that by the dimension axiom of the ordinary equivariant cohomology
HjG(F; M) = 0 when j > 0 and H0G(F; M) = M(F ). Thus the spectral sequence
(4.1) collapses and
(4.2) HiG(hocolimDF; M) limiDM(F ):
The mod p cohomology Bousfield-Kan spectral sequence for hocolimDEG xG F
has the form
Ei;j2= limiDHj(EG xG F; Fp) ) Hi+j(hocolimDEG xG F; Fp)
Clearly its E2 term can be identified with (4.2): Mj = Hj(EG xG -; Fp) de-
fines a contravariant coefficient system and Ei;j2 HiG(hocolimDF; Mj). The next
proposition is obvious.
Proposition 4.2.Let G be a compact Lie group and Mj = Hj(EG xG -; Fp) a
contravariant coefficient system. A collection C of subgroups of G; for which O*
*(C)
is discrete, is sharp if and only if EO(C) has Mj equivariant cohomology of a p*
*oint:
HiG(EO(C); Mj) "HiG(*; Mj):
We now describe sufficient conditions that imply the equivariant acyclicity o*
*f a
G-space X. First we make a few definitions. Let P be a maximal p-toral subgroup
of G, M a coefficient system for G. Define M|P , the restriction of M to P , to*
* be
the composite O(P ) GxP--!O(G) M! Ab. Following the terminology of [7] we will
call a subgroup Q of G sub-p-toral if it is contained in a p-toral subgroup of *
*G.
We call a sub-p-toral subgroup Q of G p-relevant if it contains an element of o*
*rder
p (equivalently, it is not a finite group of order prime to p). For a G-space X*
* we
let Xs = {x 2 X : Gx \ P is p-relevant_}, a P -invariant subspace of X. Recall
that a G-space X is said to have finitely many orbit types if there are finitel*
*y many
conjugacy classes of isotropy subgroups of X. The following is a result of Slom*
*inska:
Proposition 4.3.([10, 1.1]) Let G be a Lie group and P be a maximal p-toral
subgroup of G. Suppose that X is a G-CW complex with finitely many orbit types.
With the notation above, if Xs is P -mod-p-equivalent to a point, then, for j >*
* 0
H*G(X; Mj) "H*G(*; Mj)
Proof.The main idea behind the proof, which can be found in [10], is that the
functors Mj are restrictions of Mackey functors defined on stable orbit category
of G as described in [8], for which a natural equivariant cohomology transfer m*
*ap
is constructed. The transfer exhibits the map H*G(X; Mj) "H*G(*; Mj) as a re-
tract of H*P(X; Mj|P ) "H*P(*; Mj|P ). Since the coefficients Mj|P (j > 0) vani*
*sh
on the P -orbits of X outside Xs, the inclusion map Xs ! X induces an iso-
morphism H*P(X; Mj|P )!"H*P(Xs; Mj|P ): Finally H*P(Xs; Mj|P ) is isomorphic_to
H*P(*; Mj|P ) by [6, A.13]. |__|
Remark 4.4.The problem one faces trying to get a result analogous to 4.3 when
j = 0 is that the functor M0|P does not vanish on outside of Xs and the map
H*P(X; Mj|P ) ! H*P(Xs; Mj|P ) induced by inclusion is not necessarily an isomo*
*r-
phism. Another result of Slominska [10, 1.2] covers the case j = 0, however it *
*does
depend on Jackowski-McClure-Oliver construction. We will give an independent
proof, which is based on ideas we are planning to exploit later.
HOMOLOGY DECOMPOSITIONS FOR COMPACT LIE GROUPS 9
Proposition 4.5.Let G be a compact Lie group, P be a maximal p-toral subgroup
of G, and X be a G-CW -complex with finitely many orbit types such that each po*
*int
in X is fixed by an element in G of order p. If Xs=P is mod-p-acyclic then so is
the orbit space X=G.
Note that the coefficient system M0 is a constant system,
M0(G=H) H0(BH; Fp) = Fp
and the equivariant cohomology H*G(X; Fp) is the ordinary cohomology of the orb*
*it
space H*(X=G; Fp). Our proof of 4.5 is analogous to the proof of 4.3 and is bas*
*ed
on the construction of the transfer map given by Oliver [9]. For the rest of t*
*his
section H* denotes the Cech cohomology.
Theorem 4.6. ([9]) Let G be a compact Lie group, P a subgroup of G and R any
coefficient group then for any paracompact and perfectly normal G-space X, there
is a transfer homomorphism
(4.3) trf : H*(X=P ; R) ! H*(X=G; R);
natural in X, and such that the composite
trf O ss* : H*(X=G; R) ! H*(X=G; R);
where ss denotes the projection X=P ! X=G, is multiplication by the Euler char-
acteristic O(G=P ).
Note that any G - CW -complex is paracompact and perfectly normal. We will
show that
Lemma 4.7. If P is a maximal p-toral subgroup of G, X is a G-space such that
each point in X is fixed by an element in G of order p, and R = Fp, then the
*
transfer map (4.3) factors as H*(X=P ; Fp) i! H*(Xs=P ; Fp) ! H*(X=G; Fp),
where i* is induced by inclusion Xs ! X.
First we recall basic properties of the Dold fixed point index [2] and the key
constructions form [9]:
The fixed point index assigns an integer Ind(f) to any map f : U ! M, where
M is an ENR (Euclidean neighborhood retract), U an open subset of M, and
F ix(f) = {u 2 U : f(u) = u} is compact. We will use the following properties of
the index:
Lemma 4.8. ([2, 1.3-1.8]) Let f : U ! M be as above.
(a) If V U is open and F ix(f) V , then Ind(f|V ) = Ind(f).
`n Pn
(b) If U = Ui, a finite disjoint union of open sets, then Ind(f) = Ind(*
*f|Ui).
i=0 i=0
(c) If ' : U x I ! M is a homotopy such that F ix(') = {u 2 U : '(u; t) = u
for some t 2 I} is compact, then Ind('0) = Ind('1) ('t= '(-; t), t 2 I).
(d) If M and M0 are ENR's, U M and U0 M0 are open sets, and f :
U ! M0, g : U0 ! M are maps, then the two composites gf : f-1 (U0) ! M
and fg : g-1(U) ! M0 have homeomorphic fixed point sets and, if these sets are
compact, Ind(gf) = Ind(fg).
As a simple consequence of 4.8 we have the following:
10 ALEXEI STROUNINE
Lemma 4.9. Suppose that M is an ENR, V an open subset of M, f : V ! M is
a compactly fixed map and q : M ! M is an automorphism of M. Then the map
-1 f q
qfq-1 : q(V ) q! V ! M ! M has the same index as f.
Proof.Note that qf : V ! M and q-1|q(V ): q(V ) ! M are maps whose two __
composites have the same index by 4.8 (d). |__|
Definition 4.10.Let X be a space and Y be a G-space. A homotopy
' : X x I ! Y=G
is called stratum decreasing if for any x in X and t s in I the isotropy subgr*
*oups
of the orbit '(x; t) in Y are subconjugates to the ones of '(x; s).
Lemma 4.11. ([9, Theorem 2]) Let G be a compact Lie group acting on spaces X
and Y , where X is paracompact and perfectly normal and Y is completely regular,
then, for any G-map f : X ! Y , any stratum decreasing homotopy of f=G can be
lifted to a G-homotopy of f.
Lemma 4.12. ([9, Theorem 3]) For any pair P G of compact Lie groups and any
paracompact perfectly normal G-space X there is a stratum decreasing homotopy
' : X=P x I ! X=P of the identity on X=P , commuting with the projection
ss : X=P ! X=G; and such that the restriction of ss :
^ss: F ix('1) ! X=G ('1 = '(-; 1))
has finite fibers.
The "stratum decreasing" property of ' implies that it restricts to a homotopy
's : Xs=P x I ! Xs=P with analogous properties. We will denote by ^sssthe
restriction of ^ssto F ixs('1) = F ix('1) \ Xs=P .
Let H (respectively Hs) denote the sheaf over X=G induced by the presheaf U !
H0(^ss-1(U); Fp) (respectively U ! H0(^ss-1s(U); Fp)) (U open in X=G). Since ^s*
*sand
^sssare closed maps, H (respectively Hs) has stalks Hx = H0(^ss-1(x); Fp) (resp*
*ec-
tively (Hs)x = H0(^ss-1s(x); Fp)) (x 2 X=G). Since each of the sets ^ss-1(x), ^*
*ss-1s(x)
is zero dimensional, H*(F ix('1); Fp) ~= H*(X=G; H) and H*(F ixs('1); Fp) ~=
H*(X=G; Hs).
For any x 2 X=G, choose "xin the orbit x. Let K denote the isotropy subgroup
G"xof "x. The fiber ss-1(x) can be identified with the orbit space (G=P )K of t*
*he K-
action on G=P through the homeomorphism (G=P )K ! ss-1(x) X=P that sends
the equivalence class [gP ] in (G=P )K to [g-1x"] in X=P . Under this identific*
*ation
' restricts to a homotopy '|(G=P)K xI: (G=P )K x I ! (G=P )K , which is stratum
decreasing with respect to the action of K by [9, Lemma 5 (iii)]. By 4.11 it ca*
*n be
lifted to a K-equivariant homotopy of the identity map on G=P . Note that the
fixed point set of the resulting map 1 = (-; 1) projects into ^ss-1(x) (G=P *
*)K .
For any point y 2 ^ss-1(x) (G=P )K the corresponding K-orbit y G=P has
a K-equivariant tubular neighborhood U that does not contain any fixed point
sets of 1 other than those in y: Let I(y) denote the Dold fixed point index of
1|U : U ! G=P . This index is independent of the choice of "x, and U. Oliver
defines a map J(') from the sheaf H to the constant sheaf X=G x Fp, which is a
homomorphism
M P I(y)
J(')x : H0(^ss-1(x); Fp) ~= H0(y; Fp) -! Fp
y2^ss-1(x)
HOMOLOGY DECOMPOSITIONS FOR COMPACT LIE GROUPS 11
on stalks. The map J(') is continuous [9, 14], and hence a sheaf homomorphism.
Our proof of 4.7 is based on the following fact:
Lemma 4.13. With the notation above, I(y) is divisible by p for every y 2 ^ss-*
*1(x),
which is not in ^ss-1s(x).
Proof.Let y 2 ^ss-1(x) - ^ss-1s(x) (G=P )K . It follows from the definition of*
* Xs
that the K-isotropy subgroups of any point of the corresponding orbit yin G=P a*
*re
finite of order prime to p. Since 1 is a K-equivariant map it either fixes the*
* entire
K-orbit yor has no fixed points in y. In the latter case I(y) = 0 and the concl*
*usion
is obvious. For the rest of the proof we assume that y F ix( 1). Let Q be a cyc*
*lic
subgroup of K of order p, and U be a K equivariant tubular neighborhood of y in
G=P that does not contain fixed point sets of 1 others than those in y. Clearl*
*y Q
acts freely on U. Consider projections U ff!U=Q fi!U=K. Since any subspace of a
paracompact and perfectly normal space is paracompact and perfectly normal ([9,
Lemma 1 (2)]), by 4.12 there exist a stratum decreasing homotopy of the identity
! : U=Q x I ! U=Q which commutes with the projection U=Q ! U=K, and such
that the resulting map !1 has finite fixed-point set on each fiber of U=Q ! U=K.
Composing ! with 1=Q we get a homotopy
U=Q x I !!U=Q 1=Q!(G=P )Q
which, by 4.11, can be lifted to a Q-equivariant homotopy ! : U x I ! G=P
of 1|U . Note that the fixed point set of ! projects to the fixed point set of
'1|U=K : U=K ! (G=P )K , which consists of one point y, therefore F ix(! ) is
compact. By 4.8 (c) Ind( 1|U ) = Ind(!1). Now the p-fold covering ff projects
F ix(!1) into a finite set F ix(!1=Q) = F ix(!1) \ fi-1(y). For all z 2 F ix(!1*
*=Q)
chose disjoint neighborhoods Vz 3 z in U=Q which are evenly covered by ff. Note
that {ff-1Vz}z2Fix(!1=Q)isPan open covering of F ix(!1) by disjoint sets, by 4.8
(b) Ind(!1) = Ind(!1|ff-1Vz). Each of the open sets ff-1Vz is a disjoi*
*nt
z2Fix(!1=Q)
union of p open`sets each of which is homeomorphicPto Vz. Let V denote one of t*
*hem,
then ff-1Vz = qV and Ind(!1|ff-1Vz) = Ind(!1|qV_). By 4.9 Ind(!1|qV) =
q2Q q2Q
Ind(!1|V ) for each q 2 Q (note that !1 is a Q-equivariant map and !1|qV_=
(q!1q-1|qV). |__|
Proof of 4.7.Note that the sheaf map i* : H ! Hs induced by inclusions ^ss-1s(U*
*) !
^ss-1(U) (U open in X=G) is a quotient map, and 4.13 implies that J(')|Ker(i*)=*
* 0.
* J(')s
Therefore J(') can be factored as H i!Hs ! X=G x Fp. Consider the following
commutative diagram
H*(Xs=H; Fp) ! H*(F ixs('1); Fp) ~=H*(X=G; Hs)
|6 |6 |6 @ Js
| | | @@R
H*(X=H; Fp) ! H*(F ix('1); Fp) ~=H*(X=G; H) J!H*(X=G; Fp)
Here J and Js are the maps induced by J(') and J(')s respectively and the
unlabeled maps are those induced by inclusions. The bottom row composite is
Oliver's transfer trf : H*(X=P ; Fp) ! H*(X=G; Fp) which factors through_the top
row. |__|
12 ALEXEI STROUNINE
5. Relation between two decompositions
In this section we establish a relation between the p-toral and p-stubborn de-
compositions. The following properties of p-toral subgroups will be used:
Lemma 5.1. ([7, 3.1, 3.2]) Let G be a compact Lie group and Q be a sub-p-toral
subgroup of G.
(a) For any p-toral subgroup P Q in G, NP (Q)=Q is p-toral and nontrivial.
6=
(b) If G acts smoothly on a compact manifold M, then O(MQ ) O(M) mod p.
(c) If NG (Q)=Q is finite and has no nontrivial normal p-subgroup, then Q is
p-toral.
Lemma 5.2. ([6, 1.6]) The category Rp(G) is equivalent to a finite category.
Lemma 5.3. Let G be a compact Lie group which is not a finite group of order
prime to p. Suppose that for any p-relevant subgroup Q of G such that NG (Q)=Q
is infinite, EOp(NG (Q)=Q)!"p*. Then for any p-relevant subgroup Q the fixed po*
*int
space ERp(G)Q is mod p acyclic.
Proof.To simplify the notation denote ERp(G) by X. Note that X has finitely
many orbit types by 5.2. Let S denote the set of p-relevant subgroups Q such th*
*at
XQ is not mod p acyclic. We will prove that S is empty. Assuming the contrary,
we order the elements of S by inclusion and use Zorn's lemma to prove that S has
a maximal element:
Let Q1 Q2 : : :Qi : :b:e a chain in S and Q be the closure of the union
of Qi's. Note that a subgroup H of G is sub-p-toral if and only if (G=P )H 6= ;*
*, where
P is a maximal p-toral subgroup of G. For i sufficiently large, (G=P )Qi = (G=P*
* )Q
and XQi = XQ [11, Proposition IV.3.4], therefore Q is in S.
Now let Q be a maximal element of S. We prove that XQ is mod p acyclic and
get a contradiction.
If NG (Q)=Q is finite and has no nontrivial normal p subgroups then Q is p-
toral and therefore XQ ~=*. Otherwise consider XQ and (G=P )Q as a W =
NG (Q)=Q spaces. The isotropy subgroups of the action of W on XQ are of the
form (H \ NG (Q))=Q = NP (Q)=Q for H a p-stubborn subgroup containing Q. By
5.1 (a) these subgroups of W are p-toral and nontrivial. Let H be a nontrivial *
*p-
toral subgroup of W and H NG (Q) its preimage. Note that H properly contains
Q and is a sub-p-toral subgroup of G since (G=P )H is nonempty:
O((G=P )H ) = O(((G=P )Q )H ) O((G=P )Q ) O(G=P ) 6 0 (mod p)
By the maximality of Q the fixed point set (XQ )H = XH is mod p acyclic.
Therefore XQ is mod p equivalent to EOp(W ) which is either mod p acyclic by
the hypothesis if W is infinite or contractible if W has a nontrivial normal_p-
subgroup. |__|
Theorem 5.4. Under the hypothesis of 5.3, the map ERp(G) ! EOp(G) induced
by the inclusion of categories Rp(G) ! Op(G) is a G-mod-p-equivalence, which
implies that one of the collections is ample if and only if the other is.
Proof.The map induces mod p homology isomorphism on the fixed point sets
ERp(G)H ! EOp(G)H for every isotropy subgroup H of ERp(G) or EOp(G). __
The result follows from 2.6. |__|
HOMOLOGY DECOMPOSITIONS FOR COMPACT LIE GROUPS 13
Theorem 5.5. Suppose that G is a compact Lie group that contains an element
of order p. Under the hypothesis of 5.3, the collection of nontrivial p-stubbo*
*rn
subgroups of G is sharp.
Proof.Let P be a maximal p-toral subgroup of G. Consider the P space (ERp(G))s.
By definition, isotropy subgroups of (ERp(G))s are p-relevant and for every iso*
*tropy
subgroup Q of (ERp(G))s, including P , (ERp(G))Qs= ERp(G)Q is mod p acyclic
by 5.3. Therefore the projection map ERp(G) ! * is an equivalence by 2.6. The_
result follows from 4.3 and 4.5. |__|
6. Proof of the Homology Decomposition Theorem
In this section we complete our proof of 1.3. First we observe the following
elementary facts:
Lemma 6.1. Suppose that G is a compact Lie group and ss0(G) is a p-group. If
N is a proper subgroup of G that contains G0, then N is a proper subgroup of its
normalizer in G.
Proof.The p-group N=G0 acts on G=N by left translation. The fixed point set of
this action contains a nontrivial coset gN. The element g is in the normalizer_*
*of N
and not in N. |__|
Lemma 6.2. Let G be a compact Lie group and N be a normal subgroup of G that
contains G0. If P is a maximal p-toral subgroup of G, then P \ N is a maximal
p-toral subgroup of N.
Proof.Clearly P0 is the identity component of P \N and (P \N)=P0 is a subgroup
of the p-group P=P0, hence P \N is p-toral. Since N is normal in G the coset sp*
*ace
N=(N \ P ) is homeomorphic to NP=P . Therefore O(N=(N \ P )) = O(NP=P ) 6 0_
mod p which implies the maximality of N \ P in N. |__|
Lemma 6.3. Suppose that G is a compact Lie group, ss0(G) is a p-group, N is a
subgroup of G that contains G0. If P is a maximal p-toral subgroup of G, then
P \ N is a maximal p-toral subgroup of N.
Proof.By 6.1 there exists a normal series N = N1 / N2 / : :/:Nm = G. The proof
is by downward induction on the subscript of Ni. The inductive step is provided_
by 6.2. |__|
Now we are ready to prove the main result. Note that under the hypothesis of
5.3 the conclusion of 1.3 has been proven in the previous section (5.5 and 5.4)*
*. The
following theorem implies this hypothesis.
Theorem 6.4. For any infinite compact Lie group G, the G-space EOp(G) is mod
p acyclic.
Proof.The proof is by induction on the dimension of G.
When dimG = 0 the statement of the theorem is vacuously true.
Let n be a positive integer. Suppose the theorem holds for any infinite G who*
*se
dimension is less than n. First we prove that EOp(G)!"p* for compact Lie group
G for which dim G = n and ss0(G) is a p-group and then proceed to an arbitrary
compact Lie group of dimension n.
Suppose ss0(G) is a finite p-group. We consider two cases:
14 ALEXEI STROUNINE
Case 1. For any p-relevant subgroup Q of G, dim NG (Q)=Q < dim G. Then
by the inductive assumption the hypothesis of lemma 5.3 is satisfied and hence
EG xG EOp(G)!"pBG. Now an application of 3.1 shows that EOp(G)!"p*
Case 2. There is a p-relevant subgroup Q of G with dim NG (Q)=Q = dim G.
Then obviously Q is a finite group and its normalizer N = NG (Q) contains the
identity component of G. Let K denote the normal subgroup of G which is the
intersection of all maximal p-toral subgroups of G. We will show that K contains
an element of order p.
Let P be a maximal p-toral subgroup of G that contains Q. The subgroup Q is
contained in P \ N, which is a maximal p-toral subgroup of N by 6.3. Since Q is
normal in N, it is contained in the intersection of maximal p-toral subgroups o*
*f N,
which is, again by 6.3, K \ N. Therefore K contains Q and hence an element of
order p.
If G is itself a p-toral group then EOp(G) ' *:
If G is not p-toral then for any p-toral subgroup H of G dimH < n and, since
KH is contained in some maximal p-toral subgroup of G, dim KH < n. If K is
infinite then by the inductive assumption EOp(KH)!"p* and EOp(G)!"p* by 3.5.
If K is finite, then the group of its inner automorphisms A = G=CG (K) is also
finite and is a p-group. The action of A on K fixes a subgroup KA of K and
|KA | |K| 0. The Sylow p-subgroup of KA is a nontrivial normal p-subgroup
of G and EOp(G) ' * by 3.3.
This finishes the proof of the case when ss0(G) is a p-group.
The case of an arbitrary compact Lie group of dimension n now follows by_an
application of 3.5. |__|
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Mathematics Department, University of Notre Dame, Notre Dame, IN 46556
E-mail address: alexei.strounine.1@nd.edu