PARTITION COMPLEXES, TITS BUILDINGS AND
SYMMETRIC PRODUCTS
G. Z. ARONE AND W. G. DWYER
Abstract. We construct a homological approximation to the par-
tition complex, and identify it as the Tits building. This gives
a homological relationship between the symmetric group and the
affine group, leads to a geometric tie between symmetric powers of
spheres and the Steinberg idempotent, and allows us to use the self-
duality of the Steinberg module to study layers in the Goodwillie
tower of the identity functor.
1. Introduction
A partition complex is a geometric object associated to the poset
of equivalence relations on a finite set; a Tits building is a geometric
object associated to the poset of subspaces of a vector space. The two
objects are formally alike, but aside from that they don't seem to have
much in common. For instance, they have very different symmetries;
permutation groups act on partition complexes, while affine groups or
general linear groups act on Tits buildings. Nevertheless, in this pa-
per we describe a close relationship between the two, based upon the
observation that when we feed the partition complex to a particular
homological approximation machine, it is essentially a Tits building
that comes back out (x4-5) . This gives some surprising connections
between the homology of symmetric groups and the homology of affine
groups or general linear groups (1.2, 1.5). In view of a tie we establish
between the partition complex and symmetric powers of spheres (1.11),
it also explains, in a geometric way, why the homology of these sym-
metric powers is related to the Steinberg idempotent. Finally, although
the partition complex is not stably self-dual, the Tits building is. The
relationship between the two leads to a kind of shifted self-duality for
the partition complex, and this allows us to prove some conjectures of
Arone and Mahowald about the layers in the Goodwillie tower of the
identity functor (1.16).
____________
Date: January 28, 1999.
Both authors were partially supported by the NSF.
1
2 G. Z. ARONE AND W. G. DWYER
In the course of the paper, we clarify some aspects of [1] and weave to-
gether many threads from earlier work of Kuhn [11 ], Kuhn and Priddy
[12 ], Mitchell [16 ], Mitchell and Priddy [17 ], and others. Part of the
appeal of our approach is that it amounts to setting up a general ap-
proximation technique and then just working out its consequences in
one particular case.
We will now describe the results of the paper in more detail.
Partitions and subspaces. Let n denote the finite set {1; : :;:n}.
An equivalence relation on n is said to be nontrivial if there is more
than one -equivalence class and at least two distinct elements of n are
-equivalent to one another. The collection of nontrivial equivalence
relations on n is a poset (with the convention that 1 2 if 1 is a
refinement of 2) and the associated space P n is called the partition
complex of n (see 1.18). The group n of permutations of n acts on
P nin a natural way.
Suppose that n = pk (p a prime) and identify n in some way with
the vector space = (Fp)k over the field Fp. Say that an equiva-
lence relation on is linear if it is stable under translations (i.e.,
v ~ w ) v + x ~ w + x). Giving a linear equivalence relation
amounts to specifying the subspace of all elements equivalent to 0, and
so the poset of nontrivial linear equivalence relations is isomorphic to
the poset (under inclusion) of nontrivial proper subspaces of . The
associated space T k is called the Tits building of . The affine group
Affk;p= GL k(Fp) n acts on and on T k. The identification of
with n gives a subgroup inclusion Affk;p! n, and treating linear
equivalence relations as equivalence relations gives a map T k ! P n
which is equivariant with respect to this inclusion.
An approximation theorem. Our first result states that the map
T k! P nis in a certain sense an equivariant homology approximation.
We need some notation to describe this. If G is a subgroup of a sym-
metric group and M is a G-module, let M denote the twist of M by
the sign representation of G; in particular, Fp denotes the sign repre-
sentation of G on Fp. For a pair (X; Y ) of G-spaces, let H G*(X; Y ; M)
denote the relative Borel construction homology
H *( (E G x X)=G; (E G x Y )=G; M ) :
This is relative homology with local coefficients. If X has a G-basepoint
x0, the reduced homology H"G*(X; M) is defined to be H G*(X; x0; M).
For any G-space X, the unreduced suspension X} of X is the pointed
G-space obtained by collapsing the base of the cone on X to a point
and using this as the basepoint.
PARTITION COMPLEXES AND TITS BUILDINGS 3
1.1. Theorem. Suppose that n > 1 is an integer. If n is not a power
of p, then "Hn*(P }n; Fp ) vanishes. If n = pk, then the map T k ! P n
induces an isomorphism
H"Affk;p*( T}k; Fp ) ~=H"n*( P}n; Fp ) :
Group homology interpretations. Theorem 1.1 leads to a surpris-
ing group homology calculation. The spaces T k and P n above are
spherical, in the sense that each is homotopy equivalent to a wedge
of spheres which all have the same dimension. For T k this dimension
is k - 2, while for P n it is n - 3. (See [20 ] [2] or [19 , Th. 2] for the
Tits building, and [18 , 4.109] for the partition complex.) The sus-
pended spaces T }kand P }nare also spherical, with the dimensions of
the spheres involved increased by one. The top group "Hk-1(T }k; Z) is
a module over Affk;pwhich is called the Steinberg module Stk. The top
group L = "Hn-2(P }n; Z) is a module over n without a standard name,
but since the Z-dual of the twisted module L is sometimes denoted
Lien, we will denote L itself by Lie*n.
1.2. Theorem. Suppose that n > 1 is an integer. If n is not a power
of p, then H *(n; Fp Lie*n) vanishes. If n = pk, then for i 0 there
are isomorphisms
H i(Affk;p; Fp Stk) ~=H i-n+k+1(n; Fp Lie*n) :
1.3. Remark. In the above statement and in similar situations involv-
ing group homology later on in the paper, H i = 0 for i < 0. There
is also a cohomological version of this theorem, which appears in 6.2.
The calculations of 6.2 show that Lien and Lie*n are not in general
isomorphic as n-modules.
Since the subgroup (Fp)k of translations in Affk;p= GL k(Fp) n (Fp)k
acts trivially on the complex T k of linear equivalence relations, Stk is
actually a module over GL k(Fp) = GL k;p. It turns out that Fp Stk is
projective as a module over Fp[GL k;p], and that there is an idempotent
fflStk2 Fp[GL k;p], called the Steinberg idempotent, such that for any
Fp[GL k;p]-module M there is an isomorphism
(1.4) fflStk. M ~= H0(GL k;p; M Stk) :
See [21 ], [17 ], or [10 ]. A little bit of calculation with a collapsing
spectral sequence leads to the following consequence of 1.2.
1.5. Theorem. Suppose that n = pk > 1. Let = (Fp)k and let
GL k;p= Aut () n act on the group homology H *(; Fp ) in the
natural way. Then for i 0 there are isomorphisms
fflStk. Hi(; Fp ) ~= Hi-n+k+1 (n; Fp Lie*n) :
4 G. Z. ARONE AND W. G. DWYER
1.6. Remark. In the above statement, the twist in Fp affects the action
of GL k;pon H *(; Fp ) but does not affect the group homology itself.
A geometric interpretation. Theorem 1.1 also has a more geomet-
ric interpretation. If X is a G-space, let XhG denote the Borel construc-
tion (E G x X)=G; if X has basepoint x0, let X"hGdenote the reduced
Borel construction XhG =(x0)hG = XhG = BG.
1.7. Theorem. Suppose that n > 1, that ` is odd, and that n acts
on the sphere S`n by permuting the factors of S`n = (S`)^n. If n is
not a power of p, then the reduced mod p homology of (S`n ^ P }n)"hn
vanishes. If n = pk, then the inclusion T k ! P n induces a mod p
homology isomorphism
(S`n ^ T }k)"hAffk;p! (S`n ^ P }n)"hn :
In the course of proving 1.7 we also prove a dual result, which it
turns out is needed in studying the Goodwillie tower. If X is a space
or spectrum, let X# denote the Spanier-Whitehead dual of X (1.18).
1.8. Theorem. Suppose that n > 1, that ` is odd, and that n acts
on the sphere S`n by permuting the factors of S`n = (S`)^n. If n is not
a power of p, then the reduced mod p homology of (S`n ^ (P }n)# )"hn
vanishes. If n = pk, then the inclusion T k! P n gives rise to a mod p
homology isomorphism
(S`n ^ (P }n)# )"hn ! (S`n ^ (T }k)# )"hAffk;p:
1.9. Remark. Say that an equivalence relation on n is completely reg-
ular if all of the equivalence classes have the same size, and let R n
denote the space associated to the poset of nontrivial completely reg-
ular equivalence relations on n. If n = pk, the inclusion T k ! P n
factors as T k! R n ! P n, and for odd ` Theorem 1.7 guarantees that
the induced composite
(S`n ^ T }k)"hAffk;p! (S`n ^ R }n)"hn ! (S`n ^ P }n)"hn
gives an isomorphism on mod p homology. It follows that the p-completion
of (S`n^ P}n)"hn is a retract of the p-completion of (S`n^ R}n)"hn. This
answers a question of Kuhn [9]; the space (Sn ^ R }n)"hn is related to
the space Yk of [8, x5].
1.10. Remark. Theorems 1.7 and 1.8 also hold (in the appropriate sta-
ble sense) for negative odd values of `. If p = 2, the assumption that `
is odd can be removed from both theorems.
PARTITION COMPLEXES AND TITS BUILDINGS 5
Relationship to symmetric powers. If X is a space, let SP n(X)
denote the n'th symmetric power of X, i.e., the quotient space Xn =n.
Choice of a basepoint gives an inclusion SP n-1(X) ! SP n(X).
1.11. Theorem. Suppose that n 1 and that S` is the `-sphere. Then
in the stable range with respect to ` there is an equivalence
SP n(S`)= SPn-1 (S`) ' S` ^ (Sn ^ P }n)"hn :
1.12. Remark. A map is an equivalence "in the stable range with re-
spect to `" if it is an m-equivalence for m = 2` - ffl, where ffl is a
small constant. We leave it to the reader to determine whether in any
particular case that comes up in this paper ffl should be 1, 2, or 3.
Passing to the limit and making a homology calculation leads to the
following result.
1.13. Theorem. Suppose that n > 1 and that S0 is the stable zero
sphere. Then there is an equivalence
SP n(S0)= SPn-1 (S0) ' 1 (Sn ^ P }n)"hn :
The i'th mod p homology group of this spectrum is isomorphic to
Hi-2n+2(n; Fp Lie*n) :
1.14. Remark. The statement that SP n(S0)= SPn-1 (S0) is the suspen-
sion spectrum of a space appears in a paper of Lesh [13 ]; in the course
of our arguments we recover her identification of the space involved as
the classifying space of a particular family of subgroups of n (7.4).
Combining the above result with Theorem 1.5 gives a connection be-
tween the homology of symmetric power spectra and certain images of
the Steinberg idempotent. Relationships like this first appeared in a
paper of Mitchell and Priddy [17 ]. Theorems 1.7 and 1.13 give a direct
geometric explanation for such a relationship.
1.15. Goodwillie layers. As described in [1], evaluating the n'th
layer in the Goodwillie tower for the identity functor at a pointed space
X yields the spectrum
Dn(X) = Map *(S1 ^ P }n; 1 X^n )"hn :
The following theorem was conjectured by Arone and Mahowald; it
is a type of duality statement. The proof depends upon combining
the relationship described above between the Tits building and the
partition complex with the fact that the Tits building is stably self-
dual.
6 G. Z. ARONE AND W. G. DWYER
1.16. Theorem. Suppose that X is an odd sphere and that n = pk.
Then after p-completion there is an equivalence
S2(k-1)+1^ Dn(X) ~ (1 P }n^ X^n )"h :
p n
Comparing this with 1.13 gives the following result, which was first
proved by N. Kuhn with another technique.
1.17. Theorem. Suppose that S0 is the stable zero sphere and that
n = pk. Then after p-completion there is an equivalence
S2(k-1)+1^ Dn(S1) ~ SP n(S0)= SPn-1 (S0) :
p
We also obtain a retraction (9.6) involving the p-completion of DpkS2`+1.
The existence of a retraction like this was conjectured by Mahowald.
Organization of the paper. In x2 we set up a general scheme for
approximating G-spaces (G a finite group) and then in x3 study its
homological properties. Section 4 describes a particular approximation
to the partition complex, and x5 identifies it more or less as a Tits
building. (To get the Tits building, it is actually necessary to work in
a relative sense with respect to the approximation of a one-point space.)
Section 6 has proofs of the main homological results and x7 deals with
symmetric powers. The next two sections establish the stable self-
duality of the Tits building and then combine this with earlier results
to study layers in the Goodwillie tower of the identity. Finally, x10
has some algebraic material which is needed to deal with the Spanier-
Whitehead duality which appears in 1.8.
1.18. Notation and terminology. As a technical convenience, we
take the term space on its own to mean simplicial set [15 ] [6]. A map
of spaces is an equivalence or weak equivalence if it becomes an ordinary
weak equivalence of topological spaces after geometric realization. The
union of a space X with a disjoint basepoint is X+ ; the suspension
spectrum associated to a pointed space X is 1 X.
If G is a finite group, a map f : X ! Y of G-spaces is said to
be a G-equivalence or weak G-equivalence if f induces a weak equiv-
alence XK ! Y K for each subgroup K of G. If M is a G-module,
f is said to be an H G*(- ; M)-equivalence if it induces an isomorphism
H *(XhG ; M) ! H *(YhG ; M). Note that the coefficients here are lo-
cal coefficients. We will sometimes encounter G-spectra, by which we
mean ordinary spectra with an action of G (these are called naive G-
spectra in [14 ]). For instance, if X is a pointed space then 1 X^n is
a n-spectrum. If X is a G-spectrum, the spectrum X"hGis defined by
stabilizing the usual Borel construction [14 , I, 3.7]. A map of G-spectra
PARTITION COMPLEXES AND TITS BUILDINGS 7
is said to be an H G*(- ; M)-equivalence if the induced map of Borel con-
structions gives an isomorphism on homology with local coefficients
in M. (The local coefficient homology of a stable Borel construction is
defined by stabilizing the reduced local coefficient homology of unstable
Borel constructions.)
The equivariant cohomology H *G(X; M) of a G-space is the cohomol-
ogy of XhG with local coefficients in M. Equivariant cohomology of
a G spectrum, reduced equivariant cohomology of a pointed G-space,
etc., are defined in obvious ways. Note that we use the canonical an-
tiautomorphism g 7! g-1 of G to switch back and forth if necessary
between right G-modules and left G-modules, so that the same module
might be used as coefficients both for equivariant homology and for
equivariant cohomology.
If X is a spectrum or a pointed space, then X# = Map *(X; S0) de-
notes the Spanier-Whitehead dual of X; here S0 is the sphere spectrum.
If X is a G-spectrum, so is X# .
If P is a poset, then |P| denotes the space associated to P; this is the
simplicial set whose nondegenerate m-simplices correspond to (m + 1)-
tuples of elements of P which are totally ordered by the partial order
relation. More precisely, this is the nerve [3, XI x2] of the category
whose objects are the elements of P, and in which there is exactly
one morphism x ! y if y x. (Note the reversal, which is just
for convenience [4, 2.10]). If P has a maximal element or a minimal
element then |P| is contractible [4, 2.6].
We are grateful to N. Kuhn for reading through an initial version of
this manuscript and making some very useful suggestions. In partic-
ular, he helped us to formulate some of our results in a much better
way.
2. Approximations
Suppose that G is a finite group and that X is a G-space (in other
words, a simplicial set with an action of G). Recall from [5, 1.3] that a
collection C of subgroups of G is a set of subgroups closed under conju-
gation. In this section we construct for each such C an approximation
XC of X and a G-map aC : XC ! X. The space XC is in a sense the
best approximation to X which can be built up from the orbits G=K,
K 2 C. We give examples of the approximations, and describe some
properties of the construction.
2.1. Remark. A family F of subgroups of G is a collection which is
closed under inheritance, in the sense that if K 2 F and H K then
8 G. Z. ARONE AND W. G. DWYER
H 2 F. Some of the collections we look at are families, and some of
them aren't.
For the rest of this section, C denotes a fixed collection of subgroups
of G.
Constructing the approximation. For a G-space X, let Iso(X) de-
note the collection of all subgroups of G which appear as isotropy sub-
groups of simplices of X. A G-space X is said to have C-isotropy if
Iso(X) C. A G-map X ! Y is said to be a C-equivalence if the
induced map XK ! Y K is an equivalence of spaces for each K 2 C.
2.2. Definition. Suppose that X is a G-space. A G-map f : Y ! X
is said to be a C-approximation to X if Y has C-isotropy and f is a
C-equivalence.
2.3. Proposition. Let X be a G-space. Then there exists a func-
torial C-approximation to X, denoted aC : XC ! X. Any two C-
approximations to X are canonically G-equivalent.
The uniqueness part of the proposition depends on the following
well-known result (see for instance [4, 4.1]).
2.4. Lemma. Suppose that the G-map f : X ! Y is a C-equivalence
for C = Iso(X) [ Iso(Y ). Then f is a G-equivalence.
Proof of 2.3.We sketch one construction, which is related to ideas from
[7] in the same way as in [5] the subgroup decomposition is related
to the centralizer decomposition. Let O *(C) be the category whose
objects consists of pairs (G=K; c), where K 2 C and c 2 G=K; a
map (G=K; c) ! (G=K0; c0) is a G-map f : G=K ! G=K0 such that
f(c) = c0. For any G-space X there is a functor from O *(C)op to the
category of spaces given by
CX(G=K; c) = Map G (G=K; X) ~=XK :
We define XC to be the homotopy colimit [3] of CX. Evaluating ele-
ments of CX(G=K; c) at c leads to a map aC : XC ! X. This map is
G-equivariant, where the action of G on XC is induced by the action of
G on O *(C) defined by letting g 2 G send (G=K; c) to (G=K; gc). It is
clear by inspection that Iso(XC) C. It is also easy to see that for any
subgroup K G, (XC)K is the homotopy colimit of CX over the full
subcategory of O *(C)op containing the objects (G=H; c) with the prop-
erty that K fixes c. If K 2 C, then this subcategory has (G=K; eK) as
a terminal object, and so the homotopy colimit of CXover this subcat-
egory is equivalent to CX(G=K; eK) = XK . The construction of XC
PARTITION COMPLEXES AND TITS BUILDINGS 9
is clearly functorial. If Y ! X is another C-approximation of X then
there is a commutative diagram
YC -- - ! XC
? ?
? ?
y y
Y -- - ! X
in which by 2.4 the upper arrow and the left vertical arrow are G- __
equivalences. |__|
2.5. Remark. In later sections we will mostly be interested in the col-
lection E all nontrivial elementary abelian p-subgroups of G. The E-
approximation to a G-space X is closely related to the Henn's work
in [7]. Henn builds an approximation to the Borel construction on
X; we essentially work with the fibres over B G and build XE as the
corresponding approximation to X itself.
We will now give some examples and properties of the C-approximation
construction.
2.6. Elementary examples. If C is the collection of all subgroups
of G, then XC is G-equivalent to X. If C = {G}, then XC is XG .
If C = {{e}}, then XC is E G x X. If C is the collection of all non-
trivial subgroups of G, then XC is G-equivalent to the G-singular set
SingG (X), i.e., to the subspace of X consisting of all simplices which
have a nontrivial isotropy subgroup (to see this note that SingG (X) has
C-isotropy and that the inclusion SingG (X) ! X is a C-equivalence).
2.7. The universal space E C. Let * be the trivial one-point G-space.
2.8. Definition. The universal space for C, denoted E C, is the C-
approximation (*)C to *. The classifying space for C, denoted B C, is
the quotient (E C)=G.
2.9. Remark. If C = {{e}}, then E C is E G and B C = B G. More
generally, if C is a family of subgroups of G, then E C is the ordinary
universal space [14 ] of C and B C is the associated classifying space.
2.10. Homotopy type of E C. The collection C can be treated as a poset
under subgroup inclusion, and the associated space (1.18) is denoted
|C|. (This is not the same as BC !) The action of G on C by conjugation
induces an action of G on |C|. The space E C is isomorphic to the nerve
of the category O *(C)op, and so there is a map E C ! |C| induced by
the functor E C ! C which sends (G=K; c) to the isotropy subgroup
Gc. This map is G-equivariant and is a weak equivalence of spaces [5,
3.8] [4, 4.12]. In particular E C is weakly equivalent to a finite complex.
10 G. Z. ARONE AND W. G. DWYER
2.11. Fixed point sets of E C.If K is a subgroup of G we let C # K
denote the set {H 2 C | H K} and K # C the set {H 2 C | K H}.
Both of these sets are sub-posets of C; the first one is also a collection
of subgroups of K. For any subgroup K of G the fixed point set (E C)K
is equivalent to |K # C| [4, 2.14]. Note that if K 2 C then K # C has
K as a minimal element and so (E C)K is contractible, as it must be.
By construction Iso(E C) = C. If C0 is a collection of subgroups of G
with C0 C, then there is a natural G-inclusion E C0! E C.
2.12. Building XC for general X. Suppose that K is a subgroup
of G, and that C0 = C # K. If X is a K-space with C0-isotropy, then
G xK X is a G-space with C-isotropy. If the K-map Y ! X is a C0-
equivalence, then G xK Y ! G xK X is a C-equivalence. Both of these
statements are easy to prove by inspection. If follows that if Y ! X
is a C0-approximation of X as K-space, then G xK Y ! G xK X is a
C-approximation of G xK X as a G-space.
In particular (G=K)C can be identified up to G-equivalence with
G xK (*)C0 = G xK EC0, and so (G=K)C is weakly equivalent as a
space to the finite complex G xK |C0|. It is clear that (G=K x [n])C
is isomorphic to (G=K)C x [n]. Since the functor X 7! XC commutes
with homotopy pushouts and directed colimits, this calculation for the
G-cells G=K x [n] gives an inductive approach to understanding the
G-homotopy type of XC for any G-space X.
2.13. Preservation of equivalences. Say that a G-map f : X ! Y
is a (G; n)-equivalence if it induces an n-equivalence XK ! Y K for
each subgroup K of G; more generally, f is a (C; n)-equivalence if it
induces an n-equivalence XK ! Y K for each K 2 C. The following
statement is easy to prove by the inductive method used in the proof
of [4, 4.1].
2.14. Proposition. If X ! Y is a (C; n)-equivalence, then the in-
duced map XC ! YC is a (G; n)-equivalence.
2.15. Preservation of dimension. It follows from 2.7 and 2.12 that
if X is a finite G-space, that is, one with a finite number of nondegen-
erate simplices, then XC is equivalent as a space to a finite complex.
Moreover, there is a integer dC such that the homotopy dimension of
XC is bounded above by dim (X) + dC. The integer dC can be taken to
be the dimension of |C|.
Note however that XC is not necessarily G-equivalent to a finite
complex, even if X = * (see 2.6).
PARTITION COMPLEXES AND TITS BUILDINGS 11
3. Homological properties of approximations
Let G be a finite group, C a collection of subgroups of G, and M a
fixed Fp[G]-module. In this section we study the faithfulness of the C-
approximation of a G-space (x2) from the point of view of equivariant
M-homology. In particular, we look for ways to identify G-spaces X
such that the map aC : XC ! X or its Spanier-Whitehead dual a#C :
X# ! XC # are H G*(- ; M)-equivalences (1.18).
Testing with a forward arrow. Before looking at whether the map
XC ! X is an H G*(- ; M)-equivalence, we will establish some terminol-
ogy.
3.1. Definition. If X is a G-space, the collection C is M-ample for
X if the map XC ! X is an H G*(- ; M)-equivalence.
If there is more than one group around, we will sometime emphasize
the role of G by saying that C is (G; M)-ample for X.
3.2. Proposition. Suppose that X is a G-space and that, for each
K 2 Iso(X), C is (G; M)-ample for G=K. Then C is (G; M)-ample for
X.
This is proved by a Mayer-Vietoris argument (2.12).
3.3. Proposition. Let K be a subgroup of G. If C # K is (K; M)-
ample for *, then C is (G; M)-ample for G=K.
Proof. See [7, 2.8]. Let C0 = C # K. Since (G=K)C ~= G xK (*)C0 (cf.
2.12) the result follows from the form of Shapiro's lemma which says if__
Y is a K-space, then (G xK Y )hG is equivalent to YhK . |__|
The following proposition is the starting point we will use in applying
the previous two results.
3.4. Proposition. Let E be the collection of nontrivial elementary
abelian p-subgroups of G. If p divides the order of the kernel of the
action map G ! Aut (M), then E is M-ample for *.
Proof. By the remarks in 2.10, E is M-ample for * if and only if the
map |E|! * is an H G*(- ; M)-equivalence. The proposition then follows__
from [5, x8]. |__|
3.5. Remark. See [5, 1.4] for examples involving other collections.
Testing with a reversed arrow. Our second measure of the effec-
tiveness of C-approximation uses Spanier-Whitehead duality.
12 G. Z. ARONE AND W. G. DWYER
3.6. Definition. Let X be a G-space and M a G-module. The collec-
tion C is reverse M-ample for X (or reverse (G; M)-ample for X) if the
map (XC) #+ X#+ induced by XC ! X is an H G*(- ; M)-equivalence.
3.7. Proposition. Suppose that X is a G-space with finite skeleta
such that, for each K 2 Iso(X), C is reverse (G; M)-ample for G=K.
Assume in addition that the spaces X and XC have the homotopy type
of finite complexes. Then C is reverse (G; M)-ample for X.
Proof. Let Z be the cofibre of the map XC ! X and Z(n) the cofibre
of the map (skn X)C ! sknX. (Here sknX is the n-skeleton of X.) Let
dZ be the homotopy dimension of Z (as a space, not as a G-space); by
assumption, dZ is finite. Let D = Z# , D(n) = Z(n)# . By an inductive
Mayer-Vietoris argument (cf. 2.12) it is easy to see that C is reverse
(G; M)-ample for skn X, or equivalently that H G*(D(n); M) vanishes;
we want to prove that H G*(D; M) also vanishes. The homology groups
of the spectra D and D(n) are concentrated in dimensions 0. The
group H iD vanishes unless 0 i -dZ , and H iD(n) vanishes unless
0 i -(n + dC) (see 2.15). In fact, if n >> dZ , then by 2.14 there
are isomorphisms
8
> i > -n
>:
0 i < -(n + dC + 1)
Now there is a spectral sequence
E2i;j(D) = H i(G; Hj(D; M)) ) H Gi+j(D; M)
which maps to parallel spectral sequences Eri;j(D(n)). The E2-page
E2i;j(D) is concentrated in the horizontal band 0 j -dZ , while
E2i;j(D(n)) is concentrated in an upper band 0 j -dZ and a lower
band (-n) j -(n + dC + 1); the map D ! D(n) induces an
isomorphism on E2-pages in the upper band. Since Er(D(n) converges
to zero, we can conclude that Er(D) converges to zero if we can pick
n in such a way that no nontrivial differential in Er(D(n)) jumps from
the lower band to the upper band (remember that these are fourth
quadrant homology spectral sequences, so differentials move up and to
the left). Choose n1 and n2 so that n1 > n2 + dC + 1 >> dZ . Then
the natural map E2(D(n1)) ! E2(D(n2)) is an isomorphism on upper
bands but zero on lower bands, since the lower bands in these two E2-
pages lie in different strips in the plane. By naturality all differentials
in Er(D(n1)) from the lower band to the upper band are zero, which __
as above implies that Er(D) converges to zero. |__|
PARTITION COMPLEXES AND TITS BUILDINGS 13
3.8. Proposition. Let K be a subgroup of G. If C # K is reverse
(K; M)-ample for *, then C is reverse (G; M)-ample for G=K.
__
Proof. Given 10.2, this follows from the proof of 3.3 |__|.
3.9. Proposition. Let E be the collection of nontrivial elementary
abelian p-subgroups of G. Suppose that p divides the order of G, that
M is a finite Fp[G]-module, and that every element of order p in G acts
trivially on M. Then E is reverse M-ample for *.
This will be proved below in x10.
4. An approximation to the partition complex
In this section we apply the approximation machinery of x2 in a
special case. The finite group in question is the symmetric group n,
with its collection E of nontrivial elementary abelian subgroups. The
n-space X to be approximated is a model for the partition complex
P n, where we use the term model to mean that there is a n-map
X ! P n which is a weak equivalence of spaces. Let M denote the
sign representation Fp of n on Fp. We find a model X such that E is
both M-ample and reverse M-ample for X. It will later turn out that
if n = pk, the space XE can be described in terms of the Tits building
T k.
For the rest of this section we let denote n and M the -module
Fp .
4.1. A model for P n. For each equivalence relation on n, let K
denote the subgroup of consisting of all oe 2 which preserve in the
strong sense that for each j 2 n, oe(j) ~ j. Let P denote the collection
consisting of all the subgroups K for nontrivial equivalence relations .
Up to conjugacy, the elements of PPare exactly the subgroups of of
the form n1 x . .x.nj with j > 1, ni = n, and ni > 1 for at least
one i. The association 7! K is bijective and order preserving, and
so |P| is isomorphic as a -space to the partition complex P n.
By 2.10, there is a -map E P ! |P| = P n which is an equivalence
of spaces. The space E P is the model for P n that we will work with.
4.2. Another description. There is another way to construct E P
which gives the same result up to -equivalence. Let F denote the fam-
ily of all non-transitive subgroups of , i.e. the family of all subgroups
of which do not act transitively on the set n, and let FO F be
the subcollection obtained by deleting the trivial subgroup {e}. Since
P FO, there is a -inclusion E P ! E FO.
4.3. Lemma. The inclusion E P ! E FO is a -equivalence.
14 G. Z. ARONE AND W. G. DWYER
Proof. By 2.4, it is enough to show that for each H 2 FO the map
(E P)H ! (E FO)H is an equivalence. By 2.11 (see [4, 2.12]), this
amounts to showing that the inclusion |H # P| ! |H # FO| is an equiv-
alence, or, what is the same thing (2.11), that |H # P| is contractible.
Let (H) be the equivalence relation on n in which the equivalence
classes are the H-orbits, and let K = K(H) . Then K 2 P and K is a __
minimal element of H # P, so the result follows. |__|
4.4. Remark. By the remarks in 2.7 there is a commutative diagram
of -spaces
E P --- ! E FO
? ?
? ?
y y
|P| --- ! |FO|
in which the vertical arrows are weak equivalences. By 4.3 the upper
arrow is a -equivalence, and so the lower arrow is a weak equivalence.
The construction in the proof of 4.3 gives for each H 2 FO an element
K(H) 2 P. The assignment H 7! K(H) is order-preserving and -
equivariant; it induces a -map |FO| ! |P| which is a left inverse to
the lower arrow in the above square, and so is a weak equivalence.
The approximation. Let E0 denote the collection of subgroups of
given by E \ FO. This is the collection obtained from E by deleting
all elementary abelian p-subgroups of which act transitively on n.
In terms of E0, we can now give a preliminary calculation of the E-
approximation (E P)E ' (E FO)E.
4.5. Lemma. The natural map E (E0) ! E FO is an E-approximation.
Proof. By construction the -space E (E0) has E0-isotropy, and so a
fortiori it has E-isotropy. It remains to show that the indicated map is
an E-equivalence. However, the V -fixed points of both spaces involved__
are contractible if V 2 E0 and empty if V 2 E \ E0. |__|
Ampleness and reverse ampleness. Recall that M is the sign rep-
resentation of on Fp.
4.6. Lemma. If K is an element of P, then the collection E # K of
subgroups of K is (K; M)-ample for the trivial K-space *.
Proof. The collection E # K is the collection of nontrivial elementary
abelian p-subgroups of K. If p divides the order of K it is clear that
p also divides the order of the kernel of the action map K ! Aut (M),
and so by 3.4 the collection E # K is (K; M)-ample for *. If p does
not divide the order of K then p must be odd and by calculation the
PARTITION COMPLEXES AND TITS BUILDINGS 15
group H i(K; M) vanishes for i 0. By inspection the poset E # K is
empty. For trivial reasons, then, the map |E # K| ! * is an H K*(- ; M)-
equivalence, and so (by the proof of 3.4) E # K is (K; M)-ample for __
the space *. |__|
4.7. Proposition. The collection E is M-ample for E P.
Proof. This follows from 4.6, 3.3, 3.2 and the fact that Iso(E P) = __
P. |__|
4.8. Lemma. If K is an element of P, then E # K is reverse (K; M)-
ample for the trivial K-space *.
__
Proof. This is very similar to the proof of 4.6; it depends on 3.9. |__|
4.9. Proposition. The collection E is reverse M-ample for E P.
Proof. This follows from 3:7, 3.8, and 4.8. It is necessary to check
that the -spaces E (E) and * have finite skeleta, but this is clear by
inspection. It is also necessary to check that E P, (E P)E and (*)E all
have the homotopy types of finite complexes. By 2.10 and 4.5, these
spaces are equivalent, respectively, to the finite simplicial complexes__
|P|, |E0|, and |E|. |__|
4.10. Remark. Similar arguments show that the collection E is both
M-ample and reverse M-ample for the trivial -space *.
5. Computing with the approximation
In this section we draw some specific conclusions from the approx-
imation construction in x4. We continue to use the notation of that
section.
We are going to concentrate on the commutative square
|E0| --- ! |E|
? ?
? ?
(5.1) y y
|P| --- ! *
of -spaces. The left vertical arrow here requires a little comment:
it is constructed by composing the inclusion E0 ! FO with the poset
retraction FO ! P from 4.4. Recall that M denotes the -module Fp .
We are interested in proving the following three statements.
5.2. Proposition. Both of the vertical arrows in 5.1 are H *(- ; M)-
equivalences.
5.3. Proposition. If n is not a power of p, then the homotopy cofibre
of the upper horizontal arrow in 5.1 is contractible.
16 G. Z. ARONE AND W. G. DWYER
Note that the homotopy cofibre of the lower horizontal arrow in 5.1
is P }n(see 4.1).
5.4. Proposition. Suppose that n = pk. Let C be the homotopy cofi-
bre of the upper horizontal arrow in 5.1. Then there is a -map
C ! + ^Affk;pT}k
which is a weak equivalence of spaces. Under this equivalence the map
C ! P }ninduced by the square 5.1 corresponds to the map
+ ^Affk;pT}k! P }n
induced by the Affk;p-equivariant inclusion T k ! P n.
Proof of 5.2.It is enough show that in the square
|E0| --- ! |E|
? ?
? ?
y y
|FO| --- ! *
both of the vertical arrows are H *(- ; M)-equivalences. Consider the
square
E(E0) --- ! E(E)
? ?
? ?
y y
E FO --- ! *
Both vertical arrow are E-approximations, the right by construction
and the left by 4.5. By 4.7 and 4.10, both vertical arrows are H *(- ; M)-
equivalences. The proposition follows from the construction of 2.10,
which produces a -map from the second square to the first giving a __
weak equivalence at each of the four corners. |__|
Proof of 5.3.This is obvious; E0 = E because for cardinality reasons__
no elementary abelian subgroup of can act transitively on n. |__|
Suppose that n = pk. It is not hard to see that up to conjugacy there
is a unique elementary abelian subgroup of which acts transitively
on n. Such a subgroup is obtained using an identification of (Fp)k with
n and then letting this additive group act on itself by translation. Let
denote a chosen one of these subgroups. The normalizer N() of
in is isomorphic to the semidirect product Aut () n ~=Aff k;p, and
the quotient N()= is isomorphic to Aut () ~=GL k;p.
PARTITION COMPLEXES AND TITS BUILDINGS 17
5.5. Lemma. Suppose that n = pk. Let and N be as above. Then
there is a homotopy pushout square of -spaces
xN |E0 # | --- ! xN |E # |
? ?
? ?
y y :
|E0| --- ! |E|
Proof. By inspection this is a pushout square, and the upper horizontal_
arrow is an inclusion. |__|
Proof of 5.4.Consider the square of 5.5. The poset E0 # is the poset
of all subgroups of other than the trivial subgroup and itself, and
so |E0# | is the Tits building T k. The poset E # is the poset of
all subgroups of other than the trivial subgroup. This poset has
as a maximal element, and so |E # | is contractible; in fact, |E # |
is combinatorially the cone on |E0# | . Both horizontal maps in this
square thus have homotopy cofibre + ^N T }k. It is easy to check that
__
the map from this space to P }ngiven by the 5.1 is the obvious one. |__|
6. The homological results
In this section we give proofs of the main homological results from
the introduction, namely, 1.1, 1.2, 1.5, 1.7, and 1.8. All of these results
follow directly from the calculations in x5. We continue to denote the
symmetric group n by .
Proof of 1.1.Let M be the -module Fp . By 5.2, both of the vertical
arrows in the square 5.1 are H *(- ; M)-equivalences. By a long exact
sequence argument, the induced map of horizontal homotopy cofibres
is also an H *(- ; M)-equivalence. The proof is finished by identifying
the map between these cofibres (5.3, 5.4), and, if n = pk, by applying_
Shapiro's lemma. |__|
Proof of 1.2.If G is a finite group, X is a G-space, and M is a G-
module, then there is a Serre sequence
E2i;j= H i(G; "Hj(X} ; M)) ) "HGi+j(X} ; M) :
If X is spherical and has homotopy dimension d, the spectral sequence
collapses at E2 into isomorphisms
"HGi(X} ; M) ~=H i-d-1(G; "Hd(X; M)) :
The theorem is proved by starting with 1.1 and applying this collapse
observation to the spherical space P n and, if n = pk, to the spherical_
space T k. |__|
18 G. Z. ARONE AND W. G. DWYER
Proof of 1.5.If M is a module over Affk;p, then associated to the group
extension ! Affk;p! GL k;pis a Serre spectral sequence
E2i;j= H i(GL k;p; Hj(; M)) ) H i+j(Affk;p; M) :
In the special case M = Fp Stk, the action of on M is trivial, and
so the spectral sequence takes the form
E2i;j= H i(GL k;p; Hj(; Fp) Stk) ) H i+j(Affk;p; M) ;
where the action of GL k;pon H j(; Fp) St kis a diagonal one. This ac-
tion can be identified with the diagonal action of GL k;pon H j(; Fp )
Stk. Since Fp Stk is projective as a module over GL k;p, the group
H i(GL k;p; N Stk) vanishes for any i > 0 and any Fp[GL k;p]-module
N. By formula 1.4, then, the above spectral sequence collapses into
isomorphisms
H i(Affk;p; Fp Stk) ~=fflStk. Hi(; Fp )
__
and the proof is finished by combining these isomorphisms with 1.2. |__|
Proof of 1.7.Since ` is odd, the action of on H `n(S`n; Fp) gives the
-module Fp . Essentially by the Thom isomorphism theorem (or by a
Serre spectral sequence argument) there are natural isomorphisms
"Hi((S`n ^ P }n)"h; Fp) ~=H"i-`n(P }n; Fp )
as well as similar natural isomorphisms for "H*((S`n ^ T }k)"hAffk;p; Fp).
__
The theorem is proved by combining these isomorphisms with 1.1. |__|
Proof of 1.8.Let M denote the -module Fp . Consider the square
|E0|#+-- - |E|#+
x x
? ?
(6.1) ? ?
|P|#+- - - (*)#+
obtained by applying Spanier-Whitehead duality to 5.1. The line of
argument in the proof of 5.2 shows that the vertical arrows in this
square are H *(- ; M)-equivalences; it is only necessary to replace the
appeal to 4.7 by an appeal to 4.9. By a long exact sequence argument,
it follows that the induced map of horizontal homotopy fibres is an
H *(- ; M)-equivalence. The homotopy fibre of the lower horizontal map
is (P }n)# . Let C be the homotopy fibre of the upper map. If n is not
a power of p then C is contractible (5.3). If n = pk then by 5.4 C
is equivalent to (+ ^Affk;pT}k)# , which by 10.2 is in turn equivalent
PARTITION COMPLEXES AND TITS BUILDINGS 19
to + ^Affk;p(T }k)# . In any case, the kind of Thom isomorphism that
figures in the proof of 1.7 shows that the induced map
(S`n ^ (P }n)# )"h ! (S`n ^ C)"h
gives an isomorphism on mod p homology. If n is not a power of p
the target of this map is contractible, while if n = pk it follows from
the stable form of Shapiro's lemma that the target is equivalent to __
(S`n ^ (T }k)# )"hAffk;p. |__|
6.2. Remark. Theorem 1.8 leads to a cohomological version of 1.2.
Suppose that n = pk. The spectrum (P }n)# is spherical; its only non-
trivial mod p homology group is in dimension (2-n) and is isomorphic
to Fp Lien as a module over n. Similarly, (T }k)# is spherical; its
only nontrivial mod p homology group is in dimension 1 - k and is
isomorphic to Hom (Stk; Fp) as a module over Affk;p. By the collaps-
ing spectral sequence argument in the above proof of 1.2, the mod p
homology equivalence of 1.8 gives isomorphisms
(6.3) H i(Affk;p; Hom (Stk ; Fp)) ~=H i+n-k-1(n; Fp Lien) :
If G is a finite group and M is a module over Fp[G], there is a natural
isomorphism between H *(G; Hom (M; Fp)) and the Fp-dual of H *(G; M)
(see the proof of 10.1). By this duality, 6.3 gives isomorphisms
H i(Affk;p; Fp Stk) ~=H i+n-k-1(n; Fp Lie*n) :
There is another consequence of 6.3. By the self-duality of Fp Stk
described in x8, Hom (Stk ; Fp)is isomorphic as an Aff k;p-module to
Fp Stk. In combination with 1.2, then, 6.3 gives isomorphisms
Hi-n+k+1(n; Fp Lie*n) ~=H i+n-k-1(n; Fp Lien) :
It is not hard to see that the groups involved are not all zero (1.13),
and it follows that if n is a power of p then Fp Lie*nand Fp Lienare
not isomorphic as n-modules.
The above arguments also show that if n is not a power of p then
the groups H *(n; Fp Lie*n) and H *(n; Fp Lien) vanish.
7. Symmetric powers
The goal of this section is to prove Theorems 1.11 and 1.13. There
are three auxiliary propositions. We continue to use the notation of
section 4, so that in particular = n, FO is the collection of all
nontrivial subgroups of which do not act transitively on n, and P
is the collection of subgroups of described in 4.1. In the first two
statements below, acts on the sphere S`n by permuting the factors
of S`n = (S`)^n.
20 G. Z. ARONE AND W. G. DWYER
7.1. Proposition. The space SP n(S`)= SPn-1 (S`) is homeomorphic to
the quotient space S`n=
Let P" be the collection of subgroups of given by adding to P the
trivial subgroup {e}. Recall from 2.7 that B "P is defined to be the
quotient space (E P")=.
7.2. Proposition. In the stable range with respect to ` there is an
equivalence S`n= ' S` ^ BP"}
The following proposition is interesting because it relates an orbit
space (B "P) on the left to a homotopy orbit space on the right. In the
statement, acts on Sn = (S1)^n by permuting smash factors.
7.3. Proposition. There is an equivalence of spaces
S1 ^ BP"} ' S1 ^ (Sn ^ E P} )"h
Theorem 1.11 is now proved by combining 7.1, 7.2, and 7.3. Theorem
1.13 is proved by first checking that it is possible to pass to the limit
in ` with 1.11 and then copying the spectral sequence calculation from
the proof of 1.2.
7.4. Remark. Recall from 4.1 that F is the family of nontransitive
subgroups of . It is clear that "P F, and the argument of 4.3 shows
that the induced inclusion E P" ! E F is a -equivalence. It follows
that the quotient map B "P! B F is an equivalence. Combining this
with 7.1 and 7.2, and passing to the limit in ` gives the following
calculation of Lesh.
7.5. Corollary. [13 ] Let S0 be the stable zero sphere, and let F be the
family of nontransitive subgroups of . Then there is an equivalence
SP n(S0)= SPn-1 (S0) ' 1 B F} :
Proposition 7.1 is clear by inspection. We now go on to the proofs
of 7.2 and 7.3. The first proof depends on a few lemmas.
7.6. Lemma. Suppose that G is a finite group, that X is a G-space,
and that C0 is a collection of subgroups of G. Let C = C0[ {G}. Then
the natural G-map XG ! X gives rise to a homotopy pushout diagram
of G-spaces
(XG )C0 --- ! (XG )C
? ?
? ?
y y
XC0 --- ! XC
PARTITION COMPLEXES AND TITS BUILDINGS 21
Proof. Assume G 2= C0, since otherwise the statement is trivial. By
2.4, it is only necessary to check that for each K 2 C the square be-
comes a homotopy pushout square when the fixed point functor (-)K
is applied. This is obvious if K 2 C0, since the horizontal arrows are
C0-equivalences. The remaining case is K = G. The G-fixed point sets
of the spaces on the left are empty, since these spaces have C0-isotropy.
Since G 2 C, the G-fixed points of the spaces on the right are both __
equivalent to XG . |__|
7.7. Remark. In the situation of 7.6, it is easy to see that (XG )C0 is
isomorphic to XG x (*)C0, and similarly that (XG )C is isomorphic to
XG x (*)C. This last product is G-equivalent to XG , because * has
C-isotropy.
The following lemma can be proved by the inductive technique used
to prove [4, 4.1]. The notion of (G; n)-equivalence is from 2.13.
7.8. Lemma. Suppose that G is a finite group. If X ! Y is a (G; n)-
equivalence of G-spaces, then the quotient map X=G ! Y =G is an
n-equivalence of spaces.
Proof of 7.2.Let X = S`n and let C = "P[ {}. Consider the map of
squares
(X )P" --- ! (X )C (X )P" --- ! (X )C
? ? ? ?
? ? ? ?
y y -! y y :
XP" --- ! XC (*)P" --- ! Y
Here the map XP"! (*)P"is induced by X ! * and Y is defined so that
the left-hand square is a homotopy pushout square. The map XC !
Y is well-defined because (7.6) the right hand square is a homotopy
pushout square. If K is a subgroup of then XK is a sphere So(K)`,
where o(K) is the number of orbits of the action of K on n. If K 2 "P,
then o(K) 2, so that XK ! (*)K = * is a (2` - 1)-equivalence. By
2.14, the map XP"! (*)P"is a (; 2` - 1)-equivalence. It follows that
XC ! Y is also a (; 2` - 1)-equivalence. Since X has C-isotropy, the
map XC ! X is a -equivalence. The fixed point set X is S`, so by
7.7 the right-hand square can be identified up to -equivalence as the
square
S` x E "P -- - ! S`
? ?
? ?
y y :
E "P -- - ! Y
22 G. Z. ARONE AND W. G. DWYER
It is easy to argue by naturality that the upper arrow and the left-hand
vertical arrow are the obvious projections. This implies that Y is -
equivalent to the join S`# E "P, which is itself -equivalent to S`^E P"}.
Since X is -equivalent to Y in the stable range with respect to `, it
follows from 7.8 that X= is equivalent to Y = ' S` ^ B "P} in the __
same stable range. |__|
The proof of 7.3 also depends on a few lemmas. Recall (2.6) that if G
is a finite group and X is a G-space, SingG (X) is the G-subspace of X
consisting of all simplicies which have a nontrivial isotropy subgroup.
7.9. Lemma. Suppose that G is a finite group, X is a pointed G-
space, and Y X is any G-subspace of X containing SingG (X). Then
the diagram
Y"hG -- - ! X"hG
? ?
? ?
y y
Y =G -- - ! X=G
is a homotopy pushout diagram.
Proof. The statement is clear if X = Y . The fact that SingG (X) Y
implies that X is obtained from Y by adding free G-cells of the form
Ck = (G x [k]; G x @[k]). For any one of these cells Ck the map
(Ck)hG ! Ck=G is clearly an equivalence of pairs. The lemma is proved
by induction on the number of added cells, if this number is finite, and_
then in general by passage to a sequential colimit. |__|
7.10. Lemma. If k 2 the quotient Sk=k is contractible.
Proof. We will give a topological argument. Take S1 to be the unit
circle in the complex plane, and use the basepoint 1 2 S1 to obtain
inclusions SP i(S1) ! SP j(S1), i < j. Then Sk=k is homeomorphic to
SP k(S1)= SPk-1 (S1) (see 7.1), so to obtain the desired contractibility
it is enough to show that the map S1 = SP 1(S1) ! SP k(S1) is an
equivalence. Now SP k(S1) is equivalent to SP k(C \ {0}), which, by
the fundamental theorem of algebra, is homeomorphic to the space of
monic polynomials of degree k with complex coefficients and nonzero
constant term. The result follows easily from the fact that using the
coefficients of the polynomials as coordinates gives a homeomorphism __
SP k(C \ {0}) ~=Cn-1 x (C \ {0}). |__|
7.11. Lemma. If X is a pointed -space with Iso(X) P [{}, then
the quotient (X ^ Sn)= is contractible.
PARTITION COMPLEXES AND TITS BUILDINGS 23
Proof. The statement is clear if X = *. By induction on the number
of -cells of X (and eventual passage to a sequential colimit) it is
enough to show that if (Y ^ Sn)= is contractible and X is obtained
from Y by adding a single -cell of the form =K x ([m]; @[m]),
K 2 P [ {}, then (X ^ Sn)= is contractible. There is a (homotopy)
pushout diagram
@[m]+ ^ (Sn=K) -- - ! Y =
? ?
? ?
y y
[m]+ ^ (Sn=K) -- - ! X=
P
where K is of the form n1x . .x.nj, with ni = n and at least one
integer ni greater than one. By 7.10 the quotient Sn=K ~= Sn1=n1 ^ __
. .^.Snj=nj is contractible, and the lemma follows. |__|
7.12. Lemma. The fixed point inclusion S1 = (Sn) ! Sn induces a
-equivalence
E "P} ^ S1 ! EP"} ^ Sn :
Proof. As in 7.4, it is enough it is enough to prove the result with
E "P replaced by E F. We check that for each subgroup K of the
fixed point map (E F} ^ S1)K ! (E F} ^ Sn)K is an equivalence. If
K acts transitively on n, this is the map (;} ) ^ S1 ! (;} ) ^ (Sn)K =
(;} ) ^ S1 and so it is a homeomorphism. If K does not act transitively
on n, the space (E F)K is contractible, hence (E F} )K is contractible,
and hence both domain and range of the above fixed point map are __
contractible. |__|
Proof of 7.3.Let X denote the pointed -space E "P}. Then Sing (X)
is E P} . It is clear that Sing (X ^ Sn) is contained in Sing (X) ^ Sn,
and so by 7.9 there is a homotopy pushout diagram
(E P} ^ Sn)"h -- - ! (X ^ Sn)"h
? ?
? ?
y y :
(E P} ^ Sn)= -- - ! (X ^ Sn)=
Since Iso(E P} ) = P [ {}, the lower left-hand space is contractible by
7.11. The space X is contractible because E P" is contractible, where
this last follows (cf. 2.10) from the fact that the poset P" contains
the minimal element {e}; it follows that the upper right-hand space
is also contractible. The homotopy pushout diagram then shows that
(X ^ Sn)= is equivalent to S1 ^ (E P} ^ Sn)"h. By 7.12, (X ^ Sn)=
__
is equivalent to B "P^ S1. |__|
24 G. Z. ARONE AND W. G. DWYER
8. duality
It is known that mod p reduction of the Steinberg module is self-dual.
In this section we describe an explicit duality isomorphism and use it
to construct a geometric duality map for the suspension spectrum of
the Tits building.
Fix n = pk, and assume k > 2; the cases with k 2 are simpler but
require some adjustments in the notation. Let be the elementary
abelian p-group (Fp)k, T kthe associated Tits building, and G the group
GL k;p= Aut (). Let C* denote the reduced normalized simplicial
chain complex of T }kwith coefficients in Fp. We will use the simplicial
model for T }k from [15 , 27.6] (adjusted in an evident way to omit
baspoint identifications), so that the nondegenerate m-simplices of T }k,
m > 0, correspond bijectively to the nondegenerate (m - 1)-simplices
of T k; there are two zero-simplices. Thus
( L
Fp[G=PI] 0 < m k - 1
Cm ~= I
0 m > k - 1
where I ranges through the set of ordered partitions of k with (m + 1)
constituents and PI is the parabolic subgroupPof G associated with I.
Recall that if I = with ij = k and ij 1, then PI
is the subgroup of G which preserves the flag
P
(Fp)i1 (Fp)i1+i2 . . .(Fp) ij= :
In particular, Ck-1 ~= Fp[G=B], where BL is the Borel subgroup of
upper triangular matrices, and Ck-2 ~= k-1i=1Fp[G=Pi], where Pi is
the parabolic subgroup associated with the partition <1; : :;:2; : :;:1>
with (k - 1) constituents and 2 in the i-th place. The boundary map
@k-1 :Ck-1 ! Ck-2 is the sum of components (-1)idi, where the map
di: Fp[G=B] ! Fp[G=Pi] is induced by the inclusion B ! Pi. Note
that there are only (k - 1) terms in the sum for @k-1 because one
of the simplicial face operators induces the zero map on normalized
chains. TheThomology of C* is concentrated in degree k - 1 and
H k-1(C*) ~= k-1i=1ker(di) ~=H"k-2(T k; Fp) is the mod p Steinberg repre-
sentation of G.
Let C* be the cochain complex dual to C*. Each group Cj is a direct
sum of permutation modules and so has a basis preserved by G. It
follows that Cj = hom (Cj; Fp) is isomorphic to Cj as a module over G.
The cohomology of C* is concentrated in degree k - 1 and H k-1(C*) is
abstractly isomorphic to H k-1(C*), at least as a vector space over Fp.
The group H k-1(C*) is the dual of the mod p Steinberg module. The
coboundary map @k-1 :Ck-2 ! Ck-1 is given by the alternating sum
PARTITION COMPLEXES AND TITS BUILDINGS 25
of maps di, where di is dual to di. The map di can be interpreted as a
transfer associated with the inclusion B ! Pi.
We use permutation bases to identify Cj with Cj. To emphasize this,
we will use the notation C(j) for both of these groups. Thus we view
the boundary and cobundary maps @j and @j as maps C(j) ! C(j - 1)
and C(j - 1) ! C(j) respectively.
Consider the composite homomorphism
S :H k-1(C*) ! C(k - 1) ! H k-1(C*)
where the first map is the inclusion of the kernel of @k-1 and the second
map is projection to the cokernel of @k-1. Clearly, S is a G-equivariant
map.
8.1. Lemma. The map S is an isomorphism between the mod p Stein-
berg module and its dual.
Proof. Since the source and the target of S are finite dimensional vector
spaces over Fp of the same dimension, it is enough to show that S is
surjective. To do this, we consider the maps ei: C(k - 1) ! C(k - 1)
defined by ei = didi for i = 1; : :;:k - 1. Let wi be the transposition
(i; i + 1) in the symmetric group k, which is the Weyl group of the
standard split (B; N)-pair structure on G. It is not hard to see that
ei can be identified with 1 + fwi, where fwi is the G-endomorphism of
Fp[G=B] referred to in [10 , 2.7]. Since (fwi)2 = -fwi, it follows that
(ei)2 = ei; this also follows directly from the fact that the index of B
in Pi is congruent to 1 mod p . The endomorphisms fwi satisfy braid
relations due to Iwahori [10 , 2.4], and (as a consequence of the fact
that (fwi)2 = -fwi) the maps ei satisfy the same relations:
eiej= ejei if|i - j| 2
eiei+1ei= ei+1eiei+1 :
To prove that S is surjective it is enough to prove that for each
u 2 C(k - 1) there exists an element v in the image of @k-1 such
that u + v 2 ker(@k-1). Let u 2 C(k - 1). Let ei = 1 - ei and let
Ei = e1e2. .e.i. Define
w = Ek-1 Ek-2 . .E.1u
and let v = w - u.
Clearly, v is in the subspace of C(k - 1) generated by the images of
the maps ei, and thus v is in the image of @k-1. (To derive this last
conclusion, note that djdi = 0 if i 6= j, so that ei = @k-1diei.) The
idempotents eisatisfy the same braid relations that the idempotents ei
do, and from this it is easy to see that eiw = w and thus that eiw = 0
26 G. Z. ARONE AND W. G. DWYER
for i = 1; : :;:k - 1. By inspection, eix = 0 only if dix = 0, so it follows_
that w 2 ker(@k-1). |__|
Our next step will be to construct a map of spectra that realizes the
map S on homology. Consider the map
ff :1 T }k-! 1 (Sk-1 ^ (G=B)+ )
given by collapsing the (k - 2)-skeleton of T }kto a point. The mod p
homology of both spectra is concentrated in dimension (k - 1) and
it is clear that the induced homology map in this dimension is the
inclusion of the Steinberg module in C(k - 2). In the case of the
Spanier-Whitehead dual map
k-1 # } #
ff# : S ^ (G=B)+ -! T k
the mod p homology of both spectra is concentrated in degree (-k + 1)
and the induced homology map in this dimension is the projection of
C(k - 1) on the dual of the Steinberg module. By 10.2 there is a
G-equivariant map
k-1 #
fi :1 (Sk-1 ^ (G=B)+ ) -! S2(k-1)^ S ^ (G=B)+
inducing on mod p homology the isomorphism Ck-1 ~=Ck-1 .
Now let Stopbe the composed map
} #
(S2(k-1)^ ff# ). fi . ff : 1 T }k-! S2(k-1)^ T k :
8.2. Theorem. The map Stopis a GL k;p-equivariant map that induces
an isomorphism in mod p homology.
Proof. It is clear that Stop is equivariant and that it induces the al-
gebraic map S on the only non-trivial homology group. The theorem __
follows from lemma 8.1. |__|
9. Layers in the Goodwillie tower of the identity
In this section we prove Theorems 1.16 and 1.17. We begin with an
immediate consequence of Theorem 8.2. Let = (Fp)k, and observe
that the map Stop from 8.2 is equivariant with respect to Aff k;p=
GL k;pn, where as usual acts trivially on the Tits building and its
dual.
9.1. Proposition. For any Affk;p-spectrum W, the map
1 } 2(k-1) } #
T k ^ W "hAffk;p! S ^ (T k) ^ W "hAffk;p
induced by Stop is a mod p homology isomorphism.
PARTITION COMPLEXES AND TITS BUILDINGS 27
Now suppose that n = pk and embed Affk;pin n in the usual way
(x1, x5). Let X be a based space, so that X^n has a natural action of
n, and apply 9.1 to W = 1 X^n with the induced action of Affk;p.
Since (T }k)# ^ W ~ Map *(T }k; W), the statement becomes the follow-
ing.
9.2. Corollary. For any based space X, the map
1 } ^n 2(k-1) } 1 ^n
T k ^ X "hAffk;p! S ^ Map * Tk ; X h"Affk;p
induced by Stop is a mod p homology isomorphism.
9.3. Remark. If W is a spectrum with an action of Affk;p, there is an
equivalence W"hAffk;p~ (W"h )"hGLk;p(cf. [4, 8.5]). Let X be as in 9.2,
and let U denote 1 (X^n )"h . It follows that the map of 9.2 can be
interpreted as the mod p homology isomorphism
(9.4) (1 T }k^ U)"hGLk;p! S2(k-1)^ Map *(T }k; U)"hGLk;p
induced by Stop. Let B GL k;pbe the group of upper-triangular
matrices. By way in which Stop is constructed in x8, the map in 9.4
factors through the spectrum
(9.5) (Sk-1 ^ (GL k;p=B)+ ^ U) "hGLk;p~ Sk-1 ^ U"hB:
This implies that after p-completion the spectra in 9.2 are retracts of
the p-completion of the spectrum in 9.5.
Proof of 1.16. Let X be an odd-dimensional sphere. In this case, by
theorems 1.7 and 1.8 there are mod p equivalences
1 } ^n 1 } ^n
T k ^ X "hAffk;p! P n ^ X "hn
and
} 1 ^n
Map *(P }n; 1 X^n )"hn ! Map * T k; X "hAffk;p:
__
Combining these equivalences with 9.2 we obtain 1.16. |__|
The Steinberg idempotent fflStk2 Fp[GL k;p] (1.4) lifts to an element
"fflStk2 Z[GL k;p] = ss01 (GL k;p)+ and in this way can be made to act
up to homotopy on any G-spectrum. A straightforward telescope con-
struction shows that after p-completion a G-spectrum W can be split
as a wedge W1 _ W2, where H *(W1; Fp) ~= fflStk. H *(W; Fp). The spec-
trum W1 is well-defined up to homotopy and we will denote it fflStk. W.
A homology calculation using the sphericity of the Tits building shows
that up to p-completion there is an equivalence
fflStk. W ~ S1-k ^ (T }k^ W)"hGL :
p k;p
28 G. Z. ARONE AND W. G. DWYER
In view of the remarks above in 9.3, Theorem 1.16 gives the following
statement.
9.6. Corollary. Let X be an odd-dimensional sphere and let n = pk.
Then up to p-completion there is an equivalence
Sk ^ Dn(X) ~ fflStk. 1 (X^n )"h :
p
k
Note that since X is a sphere, (X^p )"h is the Thom space of a vector
bundle over B .
Proof of 1.17. By Theorem 1.13 there is an equivalence of spectra
1 } n n 0 n-1 0
P n ^ S "h ' SP (S )= SP (S )
On the other hand, by theorem 1.16 the left-hand side is equivalent __
after p-completion to S2(k-1)+1^ Dpk(S1). This proves 1.17. |__|
9.7. Remark. That corollary 9.6 and theorem 1.17 are true had been
suggested by M. Mahowald to the first-named author a few years prior
to the writing of this paper. Theorem 1.17 and Corollary 9.6 (in the
case X = S1) have been demonstrated by N. Kuhn, using different
methods. He shows that the mod p cohomologies of the spectra in-
volved are isomorphic as modules over the Steenrod algebra. In this
particular case it turns out that the homotopy types of the spectra are
determined up to p-completion by their mod p cohomology modules.
10. Reverse ampleness
The purpose of this section is to prove 3.9. We will follow a line of
argument similar to the one used in [5] to prove the results in section 8
of that paper. We first describe a kind of acyclicity which implies
reverse ampleness, prove a general acyclicity theorem, and then apply
it to the collection of nontrivial elementary abelian subgroups. For the
rest of the section G denotes a particular finite group.
We first need some constructions from [5]. Recall that a coefficient
system H for G [5, 4.1] is a functor from the category of Fp[G]-modules
to the category of vector spaces over Fp which preserves arbitrary di-
rect sums. If X is a G-space, the Bredon homology of X with coeffi-
cients in H, denoted HBr G*(X; H), is the homology of the chain complex
CBr G*(X; H) obtained by the following three-step process:
1. Apply the free Fp-module functor dimensionwise to X, to obtain
a simplicial Fp[G]-module Fp[X].
2. Apply H dimensionwise to Fp[X] to obtain a simplicial vector
space H(Fp[X]).
3. Normalize H(Fp[X]) to obtain CBr G*(X; H).
PARTITION COMPLEXES AND TITS BUILDINGS 29
A G-space X is said to be acyclic for H if the map X ! * induces an
isomorphism HBr G*(X; H) ! HBr G*(*; H).
If M is a module over Fp[G], let HiM denote the coefficient system
for G which assigns to an Fp[G]-module A the Fp-module H i(G; A
Hom (M; Fp)). The following proposition is an analog of [5, 6.2].
10.1. Proposition. Suppose that M is a finite Fp[G]-module. If the
G-space |C| is acyclic for the coefficient systems HiM, i 0, then C is
reverse M-ample for *.
10.2. Lemma. Suppose that K is a subgroup of G and that X is a
K-space. Then there is a G-map
G+ ^K (X#+) ! (G xK X)#+
which is an equivalence of spectra.
Proof. There is a G-module isomorphism Z[G] ! Hom (Z[G]; Z) which
takes thePbasis element e of Z[G] to the map Z[G] ! Z obtained by
sending x2G cxx to ce. This extends easily to a G-map 1 G+ ! G#+
which is a weak equivalence of spectra; note that both of these spectra
are wedges of S0; their respective zero-dimensional homology groups
are isomorphic respectively to Z[G] and Hom (Z[G]; Z). The proof is
finished by observing that because 1 G+ is a finite K-spectrum there
is for any K-spectrum Y an equivalence
__
1 G+ ^K Y # ~=G#+^K Y # ! (1 G+ ^K Y )# : |__|
10.3. Remark. An algebraic reflection of the above lemma is the fact
that if K is a subgroup of G, then H j(G; M Z[G=K]) is naturally
isomorphic to H j(K; M). We note for future reference that if H K,
then under this isomorphism the projection G=H ! G=K induces the
cohomological transfer map H j(H; M) ! H j(K; M). For example, the
map H 0(G; M Z[G=H]) ! H 0(G; M Z[G=K]) induced by G=H !
G=K is the map MH ! MK given by averaging an element of MH
over coset representatives of H in K.
Proof of 10.1. By 2.10, it is enough to show that under the given as-
sumptions the map S0 = (*)#+ ! |C|#+induced by |C| ! * is an
H G*(- ; M)-equivalence. For any Fp[G]-module A there is a natural iso-
morphism Hom G(A; Hom (M; Fp)) ~=Hom (A Fp[G]M; Fp); this implies
that for any G-space or G-spectrum Y , there are isomorphisms
H *G(Y ; Hom (M; Fp)) ~=Hom (H G*(Y ; M); Fp) :
Note that left and right module structures are being implicitly switched
here (1.18). Given this duality formula, it is enough to show that the
map S0 ! |C|#+induces an isomorphism on H *G(- ; Hom (M; Fp)).
30 G. Z. ARONE AND W. G. DWYER
Let K be a subgroup of G. By inspection (see 10.3 and 10.2) there
are natural isomorphisms
(
H jG((G=K)#+; Hom (M; Fp)) i = 0
HBr Gi(G=K; HjM) ~= :
0 i > 0
Dualizing the skeletal filtration of |C|thus gives a spectral sequence
Ei;j2= HBr Gi(|C|; HjM) ) H j-iG(|C|#+; Hom (M; Fp)) :
This spectral sequence converges because the skeletal filtration of |C|is
finite. Under the assumed acyclicity condition, the spectral sequence__
collapses onto the j-axis and gives the desired isomorphism. |__|
The next proposition is like [5, 6.8], but a little more awkward to
formulate. In the statement, H is a coefficient system for G. If K is
a subgroup of G, H|K is the coefficient system for K which assigns to
the K-module N the abelian group H(Z[G] Z[K]N).
10.4. Proposition. Let X be a G-space, K a subgroup of G of index
prime to p, and Y a subspace of X which is closed under the action of
K. Assume that Y is acyclic for H|K , and that for any x 2 X \ Y the
following three conditions hold:
1. the map H(Fp[G=Kx]) ! H(Fp[G=Gx) is zero,
2. the map H(Fp[G=Kx]) ! H(Fp[G=G]) = H(Fp) is zero, and
3. for any y 2 Y with Kx Ky, the map H(Fp[G=Kx]) ! H(Fp[G=Ky])
is zero.
Then X is acyclic for H.
Proof. Compare this with [5, proof of 6.8], but note that we use slightly
different notation. For instance, what we denote CBr G*(X; H) is de-
noted CG*(X; H) in [5].
The transfers associated to the maps q : G xK Xn ! Xn provide a
map t : CBr G*(X; H) ! CBr K (X; H|K ) [5, 4.1]. By [5, 5.10] this map
commutes with differentials, and there is a commutative diagram
q G
CBr G*(X; H) -- t-! CBr K*(X; H|K ) --- ! CBr *(X; H)
? ? ?
u?y v?y w?y
CBr G*(*; H) -- - ! CBr K*(*; H|K )--- ! CBr G*(*; H)
in which the lower arrows are induced by similar transfers and projec-
tions. The index assumption assures that the horizontal composites are
isomorphisms. Let D be the graded submodule of CBr K*(X; H|K ) which
in dimension m is given by H|K (Fp[Xm \ Ym ]). By assumption (3), D
is actually a subcomplex; the quotient complex Q = CBr K*(X; H|K )=D
PARTITION COMPLEXES AND TITS BUILDINGS 31
is clearly isomorphic to CBr K*(Y ; H|K ). By assumption (1) the map q
in the above diagram factors through a map Q ! CBr G*(X; H), and by
assumption (2) the map v factors through a map Q ! CBr K*(*; H|K ).
This implies that the homology map HBr G*(X; H) ! HBr G*(*; H) is a
retract of the isomorphism HBr K*(Y ; H|K ) ! HBr K*(*; H|K ), and the
theorem follows from the fact that a retract of an isomorphism is an__
isomorphism. |__|
This is a variation on Webb's theorem (see [22 ] or [5, 6.0]), but in
this cohomological situation the hypotheses are stronger.
10.5. Proposition. Let X be a G-space, P a Sylow p-subgroup of G,
and M a module over Fp[G]. Suppose that for any nonidentity subgroup
Q of P the fixed point set XQ is contractible. Assume in addition that
for each simplex x 2 X the order of Gx is divisible by p, and that every
element of order p in G acts trivially on M. Then X is acyclic for the
functors H j(G; M -), j 0.
Proof. We will use the remarks in 10.3. Let Y be the P -subspace of X
consisting of simplices which are fixed by a nonidentity element of P .
By [5, 4.7] the map Y ! * is a P -equivalence, and so by [5, 4.8] the
space Y is acyclic for the functors H j(P ; M -). Let x be a simplex
of X which is not in Y , so that Px = {e}. We now check the three
conditions of 10.4. For (1), the map
H j({e}; M) = H j(Px; M) ! H j(Gx; M)
is trivial for j > 0 because the domain group vanishes, and for j = 0
because it can be identified with the norm or transfer map
X
g : M ! MGx :
g2Gx
This norm map vanishes because M is a vector space over Fp and there
is an element of order p in Gx which acts trivially on M. For similar
reasons the map H j({e}; M) ! H j(G; M) vanishes (condition (2)), as
well as the maps H j({e}; M) ! H j(Py; M) for each y 2 Y (condition __
(3)). |__|
Proof of 3.9.By 10.1, it is enough to prove that the G-space |E| is
acyclic for the coefficient systems HiM. This will be a consequence
of 10.5 if we can check that for every simplex x 2 |E|, Gx has order
divisible by p, and that for every nonidentity p-subgroup Q of G, |E|Q
is contractible. Both of these conditions are verified in [5, x8], where_
the space |E|is denoted XffiE. |__|
32 G. Z. ARONE AND W. G. DWYER
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Department of Mathematics, University of Chicago, Chicago, IL
60637 USA
Department of Mathematics, University of Notre Dame, Notre Dame,
IN 46556 USA
E-mail address: arone@math.uchicago.edu
E-mail address: dwyer.1@nd.edu