On the homotopy groups of the self equivalences of linear spheres

0.23 MB·English

by  Assaf Libman

ON THE HOMOTOPY GROUPS OF THE SELF
EQUIVALENCES OF LINEAR SPHERES
3
1 ASSAFLIBMAN
0
2
Abstract. LetS(V)beacomplexlinearsphereofafinitegroupG. Let

n S(V)∗n denote the n-fold join of S(V) with itself and let autG(S(V)∗)
a denotethespaceofG-equivariantselfhomotopyequivalencesofS(V)∗n.
J
We show that for any k≥1 there exists M >0 which depends only on
1 V such that |πkautG(S(V)∗n)|≤M is for all n≫0.
1

]
T
A 1. Introduction
.
h Let aut(X) denote the space of self homotopy equivalences of a space X
t
a with the identity as a basepoint. If X is a G-space we write autG(X) for
m the equivariant self equivalences. It is well known that for any k ≥ 1 the
[ sequence of groups

1 π aut(S0)→ π aut(S1) → ··· → π aut(Sn)−π−k−(ϕ−7→−−Σ−ϕ→) π aut(Sn+1) → ...
v k k k k
1 stabilizes on a finite group, namely for all n ≫ 0 all the arrows become iso-
1
morphismsoffinitegroups. Thestablegroupisthestablek-homotopygroup
5
2 of S0. This can be deduced from the fibration ΩnSn → map(Sn,Sn) → Sn
. together with [12, 19.1.2]) and the classical fact that the groups π Sn
1 n+k
0 stabilize on πS.
k
3 Equivariantly, one replaces S0 with a sphere X on which a finite group
1
G acts. Let X∗n denote the n-fold join of X with itself equipped with the
:
v natural action of G. We obtain a sequence of spaces
i
X
aut (X) → ··· → aut (X∗n) −ϕ−7→−−ϕ−∗1−→X aut (X∗(n+1))→ ...
r G G G
a If the G-sphere X has a fixed point, the stabilization of {π aut (X∗n)}
k G n
where k ≥ 1 can be deduced from the results of Hauschild [5, Satz 2.4]. In
the absence of fixed points, the problem is much harder.
A complex representation V of a finite group G admits a G-invariant
scalar product, unique up to equivalence, and the subspace S(V) of V con-
sisting of the unit vectors is called a linear sphere. The stabilization of
{π aut (S(V)∗n)} was established if G is cyclic by Schultz [14, Proposi-
k G n
tion6.5], orifGactsfreelyonS(V)byBeckerandSchultz[1]. Thefiniteness
ofthegroupsπ aut (S(V)∗n)forallsufficientlylargenwasprovenbyKlaus
k G
[8, Proposition 2.5] and U¨nlu¨-Yalcin [17, Theorem 3.1].
Definition 1.1. A sequence of (abelian) groups E ,E ,... is called essen-
1 2
tially bounded if there exists some M > 0 such that |E | ≤ M for all n ≫ 0.
n
1
2 ASSAFLIBMAN
The main result of this paper is the following theorem. It has its origin
and motivation in the problem of constructing free actions of finite groups
on products of spheres, see [17], [8]. We will prove it in Section 7.
Theorem 1.2. Let V be a complex representation of a finite group G. Then
for any k ≥ 1 the sequence of groups {π aut (S(V)∗n)} is essentially
k G n≥1
bounded.
Corollary 1.3. For any k ≥ 1 the group lim π aut (S(V)∗n) is finite.
−→n k G
2. Polytopes and isotropy groups
Let G bea finite group. Let X bea G-space For any x ∈ X let G denote
x
the isotropy group of x.
Definition 2.1. For any K ≤ G let XK denote the subspace of X fixed by
K. Set
X>K := {x ∈ X : G > K} = XH
x
K[<H
When K is the trivial subgroup, X>e is the subspace of X consisting of
the non-free orbits of G.
Remark 2.2. Clearly X>K ⊆XK and both are invariant under the action
of N K. Hence, they both admit an action of W = N K/K. Clearly, if we
G G
regard XK as a W-space, then X>K ⊇ (XK)>e. If K < H ≤ N (K) and
G
X>K ⊆ A ⊆ XK is W-invariant then AH = XH because XH ⊆ X>K ⊆
A⊆ XK.
Recall from [15] that an abstract simplicial complex X is a collection of
non-empty subsets, called simplices, of an underlying set V of “vertices”,
which contains all the singletons in V and if σ ∈ X is a simplex then any
nonempty τ ⊆ σ isalso asimplex. Apolytope is thegeometric realization of
a simplicial complex X. By abuse of terminology we will not distinguish be-
tween a simplicial complex X and the associated polytope. A sub-polytope
of X is the geometric realization of a sub-complex.
Throughout, whenever a finite group G acts on a polytope X, it is un-
derstood that it acts simplicially. By possibly passing to the barycentric
subdivision sdX we may always assume that for any K ≤ G, XK is a sub-
polytope of X on which N K acts simplicially. In particular, also X>K is
G
a sub-polytope. See [6, Sec. 1] for more details.
Lemma 2.3. Let W be a finite group acting simplicially on a finite polytope
Y. Let A be a W-invariant sub-polytope of Y which contains Y>e. Then,
by possibly passing to the second barycentric subdivision sd2(Y), there is a
W-invariant open subset U ⊆ Y which contains A and
(1) U and B := Y\U are W-invariant sub-polytopes of Y.
(2) The inclusions A ⊆ U ⊆ U and Y\U ⊆ B ⊆ Y\A are W-homotopy
equivalences.
SELF EQUIVALENCES OF LINEAR SPHERES 3
Proof. By [11, Lemma 72.2] and the remarks in the beginning of [11, §70],
the open subset U := St(A,sd2(Y)), namely the star of A in the second
barycentric subdivision of Y, has the properties in points (1) and (2) non-
equivariantly. SinceAisW-invariant,itisclearthatU mustbeW-invariant,
and hence so are U and B. Thus, point (1) follows.
To prove point (2) we must show that for any H ≤ W the inclusions
AH ⊆ UH ⊆ UH and(Y\U)H ⊆BH ⊆(Y\A)H arehomotopy equivalences.
If H 6= 1 then AH = UH = UH and (Y\A)H = ∅ by Remark 2.2 (with
K = 1). If H = 1 then these inclusions are homotopy equivalences by the
choice of U. (cid:3)
3. Bredon homology and cohomology
In this section we will recall the definitions and some of the basic prop-
erties of Bredon homology and cohomology groups. Most of the results are
contained in Bredon’s book [3] in the cohomological setting.
Let G be a finite group. Let O denote the category of transitive left
G
G-sets. It is equivalent to its full subcategory whose objects are the left
op
cosets G/H. A cohomological coefficient functor is a functor M: O →
G
Ab. The cohomological coefficient functors form an abelian category with
op
natural transformations as morphisms which will be denoted O -mod. It
G
has enough injectives and projectives. Similarly, a homological coefficient
functor is a functor M: O → Ab. The abelian category of homological
G
coefficient functors will be denoted O -mod; It has enough injectives and
G
projectives.
A G-module M gives rise to the following coefficient functors described
in [3, I.4]. For any G-set Ω let Z[Ω] denote the permutation G-module
whose underlying set is the free abelian group with Ω as a basis. If M is
a left (resp. right) G-module, there is a cohomological (resp. homological)
coefficient functor
M: Ω 7→ Hom (Z[Ω],M), M: Ω 7→ M ⊗ Z[Ω].
ZG ZG
Associated to a cohomological coefficient functor M there is a unique equi-
variant cohomology theory, called Bredon cohomology, defined on the cat-
egory of G-CW complexes with the property that H∗(Ω;M) = M(Ω) for
G
any G-set Ω viewed as a discrete G-space. See [3, I.6] for details. When the
coefficient functor M is associated with a G-module M these cohomology
groups have a particularly nice description as the homology groups of the
cochain complex
C∗(X;M) := Hom (C (X;Z),M)
G ZG ∗
where C (X;Z) is the ordinary (cellular) chain complex of X. See [3, I.9, p.
∗
I-22]. If X is a polytope on which G acts simplicially, then H∗(X;M) can
G
be calculated by applying Hom (−,M) to the simplicial chain complex of
ZG
X. See [15, §4.3].
4 ASSAFLIBMAN
In a similar way, one defines the Bredon homology groups HG(X;M)
∗
with respect to a homological coefficient functor. If M is associated with a
right G-module M then these groups are the homology groups of the chain
complex
CG(X;M) := M ⊗ C (X;Z).
∗ ZG ∗
IfX isapolytopeonwhichGactssimpliciallythenC (X;Z)canbereplaced
∗
with the simplicial chain complex of X.
For any G-space X and n ≥ 0 there is an associated coefficient functor
H (X;Z) in Oop-mod, see [3, I.9],
n G
H (X;Z): G/H 7→ H (XH;Z).
n n
In other words this functor has the effect Ω 7→ H (map (Ω,X);Z). Given
n G
any cohomological coefficient functor M there is a first quadrant cohomo-
logical spectral sequence, see [3, Section I.10, (10.4)],
Ep,q = Extp (H (X;Z),M) ⇒ Hp+q(X;M).
2 Oop-mod q G
G
Similarly, for any homological coefficient functor M thereis a firstquadrant
homological spectral sequence
E2 = TorOG-mod(H (X;Z),M) ⇒ HG (X;M).
p,q p q p+q
Recall thata map f: X → Y of G spaces is a weak G-homotopy equivalence
if it induces weak homotopy equivalences on the fixed points XH ≃ YH for
all H ≤ G. The spectral sequences above imply the homotopy invariance of
Bredon (co)homology.
4. An equivariant Lefschetz duality
The main result of this section is Proposition 4.2 which is an equivariant
formofLefschetzdualityforBredon(co)homology withrespecttocoefficient
functorsassociated withtrivialmodules. Itshypothesesshouldbecompared
with Lemma 2.3.
For any set X let Z[X] denote the free abelian group with X as a basis.
The assignment X 7→ Z[X] is clearly functorial. If X is a (left) G-set then
Z[X] is naturally a (left) ZG-module.
Lemma 4.1. Let M be a trivial ZG-module. Then in the category of finite
G-sets there are isomorphisms, natural in Ω
∼=
(1) Ψ: Hom(Z[Ω],Z)⊗ M −→ Hom (Z[Ω],M)
ZG ZG
∼=
(2) Θ: Hom (Hom(Z[Ω],Z),M) −→ M ⊗ Z[Ω].
ZG ZG
Proof. The G-module Z[Ω] has a canonical basis {ω} . For any abelian
ω∈Ω
group R, any ω ∈ Ω and any r ∈ R, let χR(r) ∈ Hom(Z[Ω],R) denote
ω
the homomorphism determined by the assignment ω′ 7→ 0 if ω′ 6= ω and
ω 7→ r. Since Ω is finite, the(right) G-module Hom(Z[Ω],Z) has a canonical
G-invariant basis χ := χZ(1) where ω runs through Ω. For any ω ∈ Ω let
ω ω
Gω denote the orbit of ω in Ω. Given Ω we will also choose a set {ω } of
i
SELF EQUIVALENCES OF LINEAR SPHERES 5
representatives to the orbits of G on Ω; the index i runs through Ω/G. The
homomorphisms Ψ and Θ are defined by
Ψ : χω ⊗ZGm 7→ χMω′(m)
ωX′∈Gω
Θ : ϕ 7→ ϕ(χ )⊗ ω .
ωi ZG i
X
i∈Ω/G
To see that Ψ is well defined one uses the fact that G acts trivially on M
and χR(r)◦g−1 = χR (r). For the same reasons Θ is independent of the
ω gω
choice of the representatives ω . By inspection Ψ and Θ are natural with
i
respect to G-maps f: Ω → Γ. The details are left to the reader. (cid:3)
For any polytope X we will write C (X) for the simplicial chain complex
∗
of X, see [15, §4.3]. In the presence of a simplicial action of the group G,
this becomes a chain complex of permutation G-modules, namely C (X) is
n
the ZG-module Z[X ] where X is the G-set of the n-simplices of X. If G
n n
acts freely on X then C (X) is a chain complex of free ZG-modules. The
∗
cochain complex C∗(X) is by definition Hom(C (X),Z). If M is an abelian
∗
group then by definition
C (X;M) = C (X)⊗M and C∗(X;M) = Hom(C (X),M).
∗ ∗ ∗
Proposition 4.2(Lefschetzduality). Suppose that Gisafinite groupacting
simplicially on a finite n-dimensional polytope Y which is a compact con-
nected and orientable homology n-manifold. Let U be an open G-subspace
which contains Y>e. Assume that U and B := Y\U are sub-polytopes of Y
and that the inclusion Y\U ⊆ Y\U is a homotopy equivalence. Also assume
that G acts trivially on H (Y) = Z and that H (Y) → H (Y,U) is an iso-
n n n
morphism. Set D = B ∩U. Then for any abelian group M and any p ≥ 0
there are isomorphisms
excision
Hp(B;M) ∼= HG (B,D;M) ∼= HG (Y,U;M) and
G n−p n−p
excision
HG(B;M) ∼= Hn−p(B,D;M) ∼= Hn−p(Y,U;M).
p G G
Proof. ThroughoutweletC (Y),C (B),C (B,D)etc. denotethesimplicial
∗ ∗ ∗
chain complexes of these finite polytopes. For every g ∈ G we obtain an
automorphism g of C (B) which in every degree p permutes the basis
# ∗
elements of C (B), namely permutes the set of p-simplices of B. Similarly
p
there is an automorphism g# of C∗(B,D) which has the following effect on
a p-cochain cp
g#(cp): σ 7→ cp(g (σ)), σ is a p-simplex of B.
#
In this way C (B) and C∗(B,D) become (co)chain complexes of left ZG-
∗
modules where
g·c = g (c ), and g·cp = (g−1)#(cp).
p # p
6 ASSAFLIBMAN
Since U ⊇ Y>e, the action of G on B and D is free and therefore C (B) and
∗
C∗(B,D) are (co)chain complexes of finitely generated free ZG-modules.
Let Γ be an orientation cycle for Y, namely Γ is an n-cycle of Y whose
image [Γ] ∈ H (Y) = Z is a generator. Let [Γ ] ∈ H (B,D) be the image
n U n
of [Γ] under the isomorphism
∼= ∼=
H (Y)−−→ H (Y,U) −−−−−→ H (B,D)
n n n
excision
For dimensional reasons, Z (Y)= H (Y). Since G acts trivially on H (Y),
n n n
it follows that Γ is G-invariant. Hence, [Γ ] is G-invariant and therefore
U
its preimage Γ ∈ Z (B,D) is also a G-invariant orientation cycle since
U n
Z (B,D)= H (B,D). By [11, Theorem 66.1] the assignments
n n
Φp: Cp(B,D) −−c−p−7→−(−−−1−)n−−−p−·(−cp−∩−Γ−U−)→ C (B)
n−p
form a morphism of cochain complexes (we view C (B) as a cochain com-
n−∗
plex). It follows from the G-invariance of Γ and from the naturality state-
U
ment in [11, Theorem 66.1] that Φ is a morphism of cochain complexes of
ZG-modules because for any p, set ǫ = (−1)n−p and then
g·Φ(g−1·cp) = ǫ·g (g#(cp)∩Γ ) = ǫ·cp∩g (Γ )= ǫ·cp∩Γ = Φ(cp).
# U # U U
We now apply [11, Theorem 70.6] to the inclusion D ⊆ B and the orienta-
tion class Γ , and use excision together with the fact that j : H (Y\U) →
U ∗ ∗
H (Y\U) is an isomorphism by hypothesis on Y\U ⊆ Y\U, to deduce that
∗
Φ induces an isomorphism in homology.
Set R (B,D) := Cn−∗(B,D). Thus, R (B,D) is a chain complex which
∗ ∗
is obtained from the cochain complex C∗(B,D) by simply re-indexing the
modules. So
Φ: R (B,D)→ C (B)
∗ ∗
is a morphism of chain complexes of finitely generated free ZG-modules. It
follows from Ku¨nneth’s spectral sequence [16, Theorem 5.6.4] that the maps
below induce isomorphism in (co)homology
M ⊗ R (B,D)−−M−⊗−Z−G−Φ→ M ⊗ C (B)
ZG ∗ ZG ∗
Hom (C (B),M) −−H−o−m−Z−G−(Φ−,−M−→) Hom (R (B,D),M).
ZG ∗ ZG ∗
By applying Lemma 4.1 to C (B,D) and C∗(B,D) and using the notation
∗
and results in Section 3, we obtain isomorphisms of chain complexes
M ⊗ R (B,D) = M ⊗ Cn−∗(B,D) ∼= Cn−∗(B,D;M)
ZG ∗ ZG G
Hom (R (B,D),M) = Hom (Cn−∗(B,D),M) ∼= CG (B,D;M)
ZG ∗ ZG n−∗
Therefore Hn−p(B,D;M) ∼= HG(B;M) and Hp(B;M) ∼= HG (B,D;M).
G p G n−p
(cid:3)
SELF EQUIVALENCES OF LINEAR SPHERES 7
5. The key lemma
Throughout this section we will fix a finite group G and a sequence
{X } of compact polytopes on which G acts simplicially. By possibly
n n≥1
passingtothebarycentricsubdivisions,wemay assumethatfor any H ≤ G,
the subspaces (X )H are sub-polytope of X . We will make the following
n n
assumptions on {X } . Important examples are given by X = S(V)∗n
n n n
where S(V) is a linear sphere; See Propositions 7.2 and 7.4.
(I) For any H ≤ G either XH are empty for all n ≫ 0, or for any n ≫ 0
n
these are connected compact and orientable homology N-manifolds
for some N (which depends on n) such that N (H) acts trivially on
G
H (XH;Z) ∼= Z.
N n
(II) For any H ≤ G, if i≥ 1 then H (XH;Z)= 0 for all n≫ 0.
i n
(III) If H′ ≤ H then either
(i) lim (dimXH′ −dimXH) = ∞, or
n→∞ n n
(ii) XH = XH′ for all n ≫ 0.
n n
Let us now fix a subgroup K ≤ G and set W = N K/K. For every
G
n ≥ 1 set Y = XK and A = X>K (Definition 2.1). These are polytopes
n n n n
on which W acts simplicially. By Remark 2.2, A ⊇ (Y )>e.
n n
By Proposition 2.3, for every n ≥ 1 we can choose a W-invariant neigh-
bourhood U ⊆ Y of A such that U and B := Y \U are sub-polytopes
n n n n n n n
of Y , and the inclusions A ⊆ U ⊆ U and Y \U ⊆ B ⊆ Y \A are
n n n n n n n n n
W-homotopy equivalences. Set D = B ∩U .
n n n
Lemma 5.1. Let G be a finite group. Let {X } be a sequence of compact
n n
G-polytopes satisfying (I)–(III) above. Fix K ≤ G, set W = N K/K, and
G
let A ,U ,B ,D ⊆ X be the subspaces defined above. Assume further that
n n n n n
A ( Y for all n≫ 0. Write N for the dimension of Y . Then
n n n
(A) For any abelian group T and any k ≥ 0 there are isomorphisms
HW(B ;T) ∼= H (W;T) for all n ≫ 0. The right hand side is group
k n k
homology with the trivial W–module T.
(B) For anyfiniteabeliangroup R andanyk ≥ 0, thesequencesof groups
{HW(Y ;R)} and {HW(A ;R)} are essentially bounded.
k n n k n n
(C) If k ≥ 1 then {HN−k(Y ;Z)} and {HN−k(Y ,A ;Z)} are essen-
W n n W n n n
tially bounded. If R is a finite abelian group then {HN−k(Y ;R)}
W n n
and {HN−k(B ;R)} and {HN−k(D ;R)} are essentially bounded
W n n W n n
for any k ≥ 0.
(D) Consider the homomorphisms Hi (D ;Z) −λ→i Hi (B ;Z)inducedby
W n W n
D ⊆ B . Then {kerλ } is essentially bounded for any k ≥ 1
n n N−k n
and {cokerλ } is essentially bounded for any k ≥ 2.
N−k n
Proof. For all n≫ 0, Y 6= ∅since A ( Y . Sincedim(A ) is themaximum
n n n n
of dim(XH) where H > K, hypothesis (III) implies
n
(1) lim (dimY −dimA )= ∞.
n n
n→∞
8 ASSAFLIBMAN
Proof of (A): Recall that N denotes dimY . Given 0 ≤ i ≤ k, we obtain
n
the isomorphisms below for all n ≫ 0. The first isomorphism follows from
the equivalence Y \U ≃ Y \A and from Lefschetz duality [11, Theorem
n n n n
70.2] for A ⊆ Y which is applicable by hypothesis (I) since Y = XK. The
n n n n
second isomorphism follows from (1), the third from Poincar´e duality, and
the fourth from hypotheses (II) and (I) since Y 6= ∅.
n
H (B ;Z)∼= HN−i(Y ,A ;Z) ∼= HN−i(Y ;Z)
i n n n n
Z if i= 0
∼= H (Y ;Z) ∼=
i n (cid:26) 0 if 1≤ i ≤ k
Byconstruction,W actsfreelyonB ,soC (B )isachaincomplexoffinitely
n ∗ n
generated free W-modules. We obtain a Ku¨nneth’s spectral sequence [16,
Theorem 5.6.4] for XW(B ;T)= C (B )⊗ T
∗ n ∗ n ZW
E2 (n) = TorZW(H (B );T) ⇒ HW (B ;T)
p,q p q n p+q n
We have seen above that if n ≫ 0 then H (B ) = 0 for any 1 ≤ q ≤ k and
q n
therefore we obtain isomorphisms, for all 0 ≤ i≤ k
HW(B ;T)∼= TorZW(H (B ),T) = TorZW(Z,T) = H (W;T).
i n i 0 n i i
Proof of (B): Consider the coefficient functors H (Y ) defined in Section 3.
j n
Given i > 0, hypothesis (II) implies that H (Y ) = 0 for all n ≫ 0 and,
i n
that the sequence of functors {H (Y )} stabilizes on a coefficient functor
0 n n
F: Oop → Ab whose values are the groups 0 or Z (depending on whether
W
XH, where K ≤ H ≤ N (H), are connected or empty for all n ≫ 0). As a
n G
consequence, for all n ≫ 0, the spectral sequence
E2 (n)= TorOW-mod(H (Y ),R) ⇒ HW (Y ;R)
i,j i j n i+j n
vanishesfor1 ≤ j ≤ kandE2 (n) = TorOW-mod(F,R). HenceHW(Y ;R) ∼=
i,0 i k n
TorOW-mod(F,R) whoseorder is boundedby |R|α for some α which depends
k
only on O by Lemma 5.2. Thus, {HW(Y ;R)} is essentially bounded.
W k n n
We now prove the second assertion of point (B). The first step is to show
that there is some number β ≥ 1 such that for any 0 ≤ i ≤ k we have
rkH (A ) ≤ β for all n ≫ 0. Let I denote the poset of all the subgroup
i n
H ≤ G such that H > K. For every n there is a functor F : Iop → Spaces
n
given by F (H) = XH. By hypothesis (II), for all sufficiently large n, if
n n
0 ≤ j ≤ k then H (F ) = 0, and the sequence of functors H (F ) stabilize
j n 0 n
of a functor F′: Iop → Ab whose values are the groups 0 or Z.
Note that F (H)∩F (H′) = F (hH,H′i) and since A = ∪ F (H),
n n n n H∈I n
it follows that A = colimF . In fact since XH are polytopes, this also
n n n
Iop
shows that the functors F are Reedy cofibrant in the sense of [4, Section
n
22]. Therefore the natural maps
hocolimF → colimF = A
n n n
Iop Iop
SELF EQUIVALENCES OF LINEAR SPHERES 9
are homotopy equivalences. We obtain a Bousfield-Kan spectral sequence
E2 (n) = colimIop-modH (F ;Z) ⇒ H (A ;Z).
i,j i j n i+j n
We have seen that if n ≫ 0 then E2 (n)= 0 for all 1 ≤ j ≤ k and therefore
i,j
H (A ) ∼= colimIop-modF′ = TorIop-mod(Z,F′) for all 0 ≤ i ≤ k. It follows
i n i i
from Lemma 5.2(b) that there is β ≥ 1 such that rkH (A ) ≤ β for any
i n
0≤ i ≤ k provided n ≫ 0,
For every j ≥ 0 consider the coefficient functors H (A ): H 7→ H (AH)
j n j n
definedinsection 3(hereH ≤ W). IfH 6= 1thenbyRemark2.2,AH = XH˜
n n
where H˜ is the preimage of H ≤ W in G. By hypothesis (II), H (AH) = 0
j n
for all 1 ≤ j ≤ k provided n ≫ 0. Also, the sequence {H (AH)} stabilizes
0 n n
on a either Z or 0. If H = 1 then we have seen that rkH (AH) ≤ β for
j n
all 0 ≤ j ≤ k provided n ≫ 0. Thus, for any n ≫ 0, if 0 ≤ j ≤ k then
rk(H (A )(−)) ≤ β. Applying Lemma 5.2(a) to the spectral sequence (see
j n
section 3)
E2 (A ) = TorOW-mod(H (A );R) ⇒ HW (A ;R)
i,j n i j n i+j n
we deduce that there is α > 0 such that for any 0 ≤ i,j ≤ k we have
|E2 (A )| ≤ |R|αβ provided n ≫ 0. As a result |HW(A ;R)| ≤ |R|(k+1)αβ
i,j n k n
for all n ≫ 0, namely {HW(A ;R)} is essentially bounded.
k n n
Proof of (C) and (D): Let M be an abelian group. Since Y 6= ∅, it is, by
n
hypothesis (I) a compact connected orientable homology N-manifold. For
any0 ≤ i ≤ k, equation (1), thechoice of U ⊇ A , Proposition4.2, excision
n n
and point (A), yield the following isomorphisms for all n ≫ 0
HN−i(Y ;M) ∼= HN−i(Y ,A ;M) ∼= HN−i(Y ,U ;M) ∼=
W n W n n W n n
HN−i(B ,D ;M) ∼= HW(B ;M) ∼= H (W;M).
W n n i n i
If k ≥ 1 and M = Z then |H (W;Z)| ≤ |W||W|k by Remark 5.3, and
k
therefore {HN−k(Y ;Z)} is essentially bounded. This is the first part of
W n n
point (C). Similarly, if k ≥ 0 and M is a finite group then |H (W;M)| ≤
k
|M||W|k, and therefore {HN−k(Y ;M)} is essentially bounded.
W n n
The inclusions A ⊆ U are W-equivalences and, from (1) it follows that
n n
HN−k(U ;M) = 0 for all n≫ 0. For every n≥ 1 we have Y = B ∪ U ,
W n n n Dn n
so provided n ≫ 0, the Mayer–Vietoris sequence yield the exact sequence
··· → HN−k(Y ;M) → HN−k(B ;M) −−λ−N−−−k→ HN−k(D ;M) →
W n W n W n
→ HN−k+1(Y ;M) → ...
W n
If M = Z then point (D) now follows from the first part of point (C) which
we have proven above. It remains to prove the second part of point (C).
We have shown above that if M is finite then {HN−i(Y ;M) is essentially
W n
bounded for any i≥ 0. Proposition 4.2 gives the isomorphisms
HN−k(B ;M) ∼= HW(B ,D ;M) ∼= HW(Y ,U ;M) ∼= HW(Y ,A ;M)
W n k n n k n n k n n
10 ASSAFLIBMAN
and {HW(Y ,A ;M)} is essentially bounded by point (B) and the long
k n n n
exactsequenceinhomology. Hence{HN−k(B ;M)} isessentiallybounded.
W n n
The Mayer–Vietoris sequence above shows that also {HN−k(D ;M)} is
W n n
essentially bounded. (cid:3)
Lemma 5.2. Let C be a finite category and k ≥ 0 an integer. Let α be the
number of sequences of k composable morphisms in C. Let F: Cop → Ab
be a functor such that rk(F(C)) ≤ r for any C ∈ C. Let G: C → Ab be a
functor.
(a) If there is some M > 0 such that |G(C)| ≤ M for all C ∈ C then
|TorC-mod(F,G)| ≤ Mrα.
(b) If rk(G(C)) ≤ m for all C ∈ C, then rkTorC-mod(F,G) ≤αmr.
Proof. The groups TorC-mod(F,G) are the homology groups of a chain com-
plex (the cobar construction) whose nth group has the form
G(C )⊗F(C ).
0 n
C0→M···→Cn
Point (a) follows since |G(C ) ⊗ F(C )| ≤ Mr and (b) since rk(G(C ) ⊗
0 k 0
F(C )) ≤ rm. (cid:3)
k
Remark 5.3. As a consequence we see that if G is a finite group and M
is an abelian group of rank r then rkH (G;M) ≤ r|G|k. It follows that
k
|H (G;Z)| ≤ |G|(|G|k) for any k ≥ 1 since |G| annihilates H (G;Z). Also, if
k k
M is a finite group then |H (G;M)| ≤ |M|(|G|k).
k
6. The Barrat-Federer spectral sequence
The purpose of this section is to prove Theorem 6.1 below. It is a special
case of the Barrat-Federer spectral sequence tailored to our purposes.
Recall that a space Y is called simple if it is path connected and for any
choice of basepoint, π Y acts trivially on π Y. In this case basepoints can
1 ∗
be ignored in the sense that the basepoint-change isomorphisms for π Y are
q
canonical. Put differently, if q ≥ 1 then π Y can be identified with the set
q
of (unpointed)homotopy classes of (unpointed)mapsSq → Y, which inthis
case has a natural group structure. Hence, if a group G acts on Y then π Y
q
has a natural structure of a G-module.
ForG-spacesX andY letmap (X,Y)denotethespaceofG-maps. Write
G
map (X,Y) for the path component of f: X → Y with f as a basepoint.
G f
Theorem 6.1. Let X be a finite dimensional polytope on which a finite
group G acts freely and simplicially. Let Y be a simple G-space and fix a
G-map f: X → Y. Then there exists a second quadrant homological spectral
sequence
E2 = Hp(X;π Y) ⇒ π map (X,Y) , (0 ≤p ≤ q).
−p,q G q q−p G f
The differential in the Er-page has degree (−r,r −1) and the E∞ -terms,
−p,q
where q−p = k, are the quotients of a finite filtration of π map (X,Y) .
k G f