Centric Maps
and
Realization of Diagrams in the Homotopy Category
W. G. Dwyer and D. M. Kan
University of Notre Dame
Massachusetts Institute of Technology
x1. Introduction
Let D be a small category. Suppose that X is a D-diagram in the
homotopy category (in other words, a functor from D to the homotopy
category of simplicial sets). The question of whether or not X can be
realized by a D-diagram of simplicial sets has been treated by [5]. The
purpose of this note is to study a special situation in which the treat-
ment can be simplified quite a bit. We look at two examples to which
this simplified treatment is applicable; both examples involve homotopy
decomposition diagrams for compact Lie groups. Our results show that
in at least one of these examples ([13]) the decomposition diagram is
completely determined by its underlying diagram in the homotopy cat-
egory. It is possible that this "rigidity" result will eventually contribute
to a general homotopy theoretic characterization theorem for classifying
spaces of compact Lie groups (cf. [8]).
Before going any further we have to introduce some notation. The
symbol S will denote the category of simplicial sets and ho S the as-
sociated homotopy category obtained by localizing with respect to (i.e.
formally inverting) weak equivalences. If C is a category and D is a
small category, then CD will stand for the category of D-diagrams in C;
the objects of CD are functors D ! C and the maps of CD are natural
transformations. There is a projection functor ss : S ! ho S ; we will use
the same symbol for induced functors SD ! (ho S )D . Both S and SD
admit closed simplicial model category structures [19, II] [20, p. 233]
[4, x2] , and we will sometimes require without loss of generality that
chosen objects in these categories be fibrant.
Given a small category D and a diagram X in (ho S )D , a realization
of X [5, 3.1] is a pair (Y; f ) such that Y is an object of SD and f is an
isomorphism f : ssY ~=X in (ho S )D . A weak equivalence between two
______________________________________
This work was supported in part by the National Science Foundation.
2 Dwyer and Kan
such realizations (Y; fY ) and (Z; fZ ) is a weak equivalence h : Y ! Z 2
SD such that fZ O (ssh) = fY .
If f : A ! B is a map of simplicial sets, let Map (A; B)f denote
the component of the simplicial mapping space Map (A; B) containing
f [17, p. 17]. Call a map f : A ! B between fibrant simplicial
sets centric if precomposition with f induces a homotopy equivalence
Map (A; A)1 ! Map (A; B)f . Here "1" stands for the identity map of A
and Map (A; A)1, also denoted haut (A)1, is the identity component of
the simplicial monoid of homotopy self-equivalences of A. It is clear that
whether or not a map is centric is a homotopy property. We will say
that a homotopy class of maps OE : A ! B is centric if any representative
is, and that a diagram X in SD (resp. X in (ho S )D ) is centric if X(g)
(resp. X (g)) is centric for each morphism g of D.
Remark: A map i : G ! H of discrete groups induces a centric map
BG ! BH of classifying spaces iff i induces an isomorphism between
the center of G and the centralizer in H of i(G). This explains the choice
of the term "centric".
Given a centric diagram X in (ho S )D construct functors ffiX : Dop !
Ab (where Ab is the category of abelian groups) by setting ffiX (D) =
ssi haut (X (D))1 for D an object of D; there is no basepoint problem with
these homotopy groups because haut (X (D))1 is a simple space. For each
map g : D0 ! D1 in D the induced map ffiX (g) : ffiX (D1) ! ffiX (D0)
is the composite
# )-1
ssi haut (X (D1))1 -fl#!ssi Map (X (D0); X (D1))fl(fl-! ssi haut (X (D0)*
*)1
where fl is a representative of X (g), fl# is induced by postcomposition
with fl and fl# by precomposition.
Our main technical result is the following one.
Theorem 1.1. Suppose that D is a small category and that X 2
(ho S )D is a centric diagram.
(1) If the groups lim i+2 ffiX vanish for i 1 then at least one real-
ization of X exists.
(2) If in addition the groups lim i+1 ffiX vanish for i 1 then any
two realizations of X are in the same weak equivalence class.
Remark: See [3, XI, x6] for a discussion of the right derived functors
lim i of the inverse limit functor lim : Ab D ! Ab .
Theorem 1.1 is actually a consequence of a sharper and more geometric
calculation which is described in x2.
Centric Maps 3
We study two examples, both involving diagrams which are used to
express the classifying space of a compact Lie group as a homotopy di-
rect limit (or at least as a homotopy direct limit up to Z=p homology).
We first show that a slight modification of the centralizer decomposition
diagram of [12] is centric. We then show that the p-toral decomposi-
tion diagram of [13] is centric, and point out that the results of [13]
imply that this diagram satisfies the hypotheses of parts (1) and (2)
of Theorem 1.1. It follows from 1.1(2), then, that the diagram of [13]
is determined up to weak equivalence by its underlying diagram in the
homotopy category.
Organization of the Paper: Section 2 contains a discussion of re-
alization complexes and some calculations in the centric case. Theorem
1.1 is proved in x3. Finally, x 4 deals with the centrializer diagram of
[12] and x5 with the p-toral diagram of [13].
Notation and Terminology: Some of the simplicial sets used in this
note as in [5] are nerves of categories which are not necessarily small.
Nevertheless these nerves are homotopically small and it is possible to
make standard homotopy-theoretic constructions with them.
Since [3, XI] homotopy inverse limits only have homotopy meaning
when applied to fibrant diagrams, we sometimes have to replace a given
diagram Y 2 SD by a weakly equivalent fibrant one, such as, for in-
stance, Ex 1 Y , where Ex 1 is the functor of [14]. To simplify notation,
we will write Y f instead of Ex 1 Y .
x2. Realization complexes
Let D be a small category, and suppose that X is a diagram in (ho S )D .
Recall from [5, x3] that there is a natural realization complex rX asso-
ciated to X such that the components of rX are in 1-1 correspondence
with weak equivalence classes {(Y; f )} of realizations of X and such
that the component corresponding to a particular (Y; f ) is equivalent
to the classifying space of an appropriate complex [5, 3.5] of homotopy
automorphisms of Y . The complex rX is defined as the nerve of the
category whose objects are the realizations of X and whose morphisms
are weak equivalences between these realizations. Remark 2.4 below in-
dicates that there is a simple characterization of centric maps in terms
of realization complexes.
For any object D of D let D # D denote the over category [3, XI,
x2] [16, p. 46] of the identity functor. The objects of this category
are pairs (D0; f ) where D0 is an object of D and f : D0 ! D is a
map; a morphism (D0; f ) ! (D00; g) is a map h : D0 ! D00 in D such
that gh = f . There is a forgetful functor OD : D # D ! D which
4 Dwyer and Kan
maps a pair (D0; f ) to D0. Given a diagram X 2 (ho S )D , there is an
induced diagram O*DX 2 (ho S )D#D for each object D of D; there is
also a functor rDop X : Dop ! S which maps an object D 2 Dop to the
realization complex r(O*DX ).
The following theorem is proven below.
Theorem 2.1. Suppose that D is a small category andopthat X is
a diagramoinpho S D . Then the natural map rX = lim D rDop X !
ho lim D (rDop X )f is a weak equivalence.
The main result of this section is a description of the diagram rDop X
in the special case in which X is centric.
Theorem 2.2. Suppose that D is a small category and that X 2
(ho S )D is a centric diagram. Then for each object D 2 D the com-
plex rDop X (D) has the weak homotopy type of the classifying complex
W haut (X (D))1, and for each i 2 the functor ssi(rDop X ) : Dop ! Ab
is naturally equivalent to ffi-1 X .
The rest of this section is taken up with proofs.
Proof of 2.1: For every integer n 0 let n denote the category with
the integers 0; : : :; n as objects and with exactly one map i ! j whenever
i j. The division dD of D is the category which has as objects the
functors n ! D (n 0) and as maps (J1 : n1 ! D) ! (J2 : n2 ! D)
the functors K : n2 ! n1 such that J1 O K = J2. There is a functor q :
dD ! Dop given by the formula (J : n ! D) 7! J(n). According to [5,
3.7] the natural map rX = lim dD rdD X ! ho lim dD (rdD X )f is a weak
equivalence, where rdD X is the functor which assigns to J : n ! D the
classificationocomplexpr(J* X ). Define two objects q# rdD X and q*rdD X
of SD by the formulas
q# rdD X (D) = lim D#q OE*rdD X
q*rdD X (D) = ho lim D#q OE*(rdD X )f
where D # q is an under category associated to the functor q [3, XI, x2]
and OE : D # q ! dD is the forgetful functor. There are two things to
notice here: first, that the diagram q# rdD X is naturally isomorphic to
rDop X and second, that the natural map rDop X ~= q# rdD X ! q*rdD X
is a weak equivalence. The first statement follows from inspection, and
the second from [5, 3.7] and the fact that for any object D 2 Dop the
category D # q is naturally isomorphic to the division d(D # D) in
such a way that the the functor OE*rdD X agrees with the classification
functor rd(D#D) O*DX . By [5, 3.7] and the pushdown theorem for homo-
topy inverse limits [6, 9.8] the composite map rX ! ho lim dD (rdD )f !
Centric Maps 5
op
ho lim D q*rdD X is a weak equivalence. The desired result follows im-
mediately.
Lemma 2.3. Suppose that D is a small category with a terminal object
E, and that X is a centric diagram in (ho S )D . Let E be the trivial
category with one object and one identity morphism, and ffl : E ! D
the functor which takes the single object of E to E. Then the map
rX ! r(ffl*X ) induced by ffl is a weak equivalence.
Proof: We will use some of the notation from the proof of 2.1. First
consider the case in which D is n for some integer n. Let Xi be the
space X (i), i = 1; : : :; n, and assume without loss of generality that
each Xi is fibrant. Let fli be a map Xi-1 ! Xi which represents the
homotopy class X ((i - 1) ! (i)). The argument of [7, 9.3] then shows
that the classification complex rX is naturally weakly equivalent to the
bar construction [7, 9.1]
B(haut (X0)1; Map (X0; X1)fl1; haut (X1)1; : : :; haut (Xn )1) :
A straightforward induction using [7, 9.2(vii)], the definition of the bar
construction, and the fact that X is centric gives the desired result.
Now assume that D is an arbitrary small category with a terminal
object. The result just proven for the categories n, n 0 shows that the
classification diagram rdD X has the property that rdD X (f ) is a weak
equivalence whenever f is a morphism of dD such that q(f ) is an identity
map of Dop . Let sdD be the subdivision of D [4, x5]; this is the category
obtained from dD by turning all of the "degeneracy maps" (ie. maps
(J1 : n1 ! D) ! (J2 : n2 ! D) which correspond to an epimorphism
K : n2 ! n1) into identity maps. The functor q can be factored as a
composite q1 O q0, where q0 : dD ! sdD is the natural projection and
q1 : sdD ! Dop is defined like q. According to [6, 6.11] the functor
q0 is L-cofinal [6, 6.2], and according to [4, 5.5 and 5.6] or [6, 6.10(ii)]
the functor q1 is R-cofinal [6, 6.15].opBy [6, 6.5(iv)] and [6, 6.15], then,
there is some fibrant diagram Y in SD such that q* (Y ) is in the same
weak equivalence class as the classification diagram rdD X . Moreover,
since a functor which is either L-cofinal or R-cofinal is left cofinal ([6,
6.6] and opposite of [6, 6.7]) it follows fromothepcofinality theorem for
homotopy inverse limits [6, 9.3] that ho lim D Y is weakly equivalent to
ho lim dD rdD X . The lemma now follows in a straightforward way from
the factothatpE is an initial object of Dop , so that the natural map
ho lim D Y ! Y (E) is a weak equivalence [3, XI, 4.1(iii)].
Proof of 2.2: The over category D # D has a terminal object, so the
first assertion is a consequence of Lemma 2.3 and [5, 3.5]. The second
6 Dwyer and Kan
assertion is a routine calculation which uses the bar constructions which
appear in the proof of Lemma 2.3; to make this calculation it is enough
to restrict to the case in which D is the category n for n = 1.
Remark 2.4: Let f : A ! B be a map between fibrant simplicial sets
and let 0 and 1 denote the categories n above for n = 0, 1. Consider the
functor f : 1 ! ho S which sends the map 0 ! 1 in 1 to the homotopy
class represented by f and the functor B : 0 ! ho S which sends the
object 0 to B. It is an immediate consequence of the proof of Lemma
2.3 that f is centric iff the evident realization complex map rf ! rB is
an equivalence.
x3. Proof of 1.1
Given the definition of the realization complex rX in x 2, part (1)
of Theorem 1.1 comes down to the statement that rX is non-empty
and part (2) to the statement that rX is connected. According to
2.1 the complexorXp is weakly equivalent to the homotopy inverse lim-
it ho lim D (rDop X )f . Hence rX is also weakly equivalent to Tot Y o,
where Y o is the cosimplicial replacement of (rDop X )f [3, XI, x5] and
Tot is the "total space" or codiagonal functor [3, X, x 3]. There is a
spectral sequence
{Es;t2= ssssstY o} =) sst-s Tot Y o
which has been studied by Bousfield [1] in great detail, especially in
connection with its low-dimensional behavior. It is not necessary to
specify basepoints in the above formula for the E2-term because, in
view of 2.2 and the way in which Y o is constructed [3, XI, 5.1], each
space Y s is 1-connected. To work with the spectral sequence beyond
E2 it is necessary to make successive choices which amount in the limit
to choice of a basepoint for Tot Y o, but this consideration has been
suppressed from the above notation for the abutment of the spectral
sequence. By 2.2, [3, XI, 6.2] and [1, 2.4] there are isomorphisms
8
>< * (s; t) = (0; 0) or (1; 1)
Es:t2= > {1} (s; t) = (0; 1)
: lim s
fft-1 X s 0 and t 2
and in fact these are the only cases in which Es;t2is in general defined.
Statement (1) of Theorem 1.1 in now a consequence of the next-to-last
sentence of [1, 6.1] in the special case r = 2, while (2) follows from [1,
6.5] in the special case q = 1.
Centric Maps 7
x 4. The centralizer diagram
Fix a prime number p. Let G be a compact Lie group, and let AG
be the category whose objects are the non-trivial elementary abelian
p-subgroups of G; a morphism A ! A0 in AG is a homomorphism
f : A ! A0 of abelian groups with the property that there exists an
element g 2 G such that f (x) = gxg-1 for all x 2 A. There is a functor
f^fGfrom AopGto the category of topological spaces which sends A to the
Borel construction EG xG (G=i(A)), where i(A) denotes the centralizer
of A in G. (Notice that this Borel construction has the homotopy type
of the classifying space Bi(A).) The main result of [12] states that the
natural map ho lim!^ffG ! EG xG * = BG is an isomorphism on Z=p
homology. In this section we will prove that a slight modification of this
"centralizer diagram" ^ffGis centric. We do not know whether ^ffGor its
modification satisfy the hypotheses of Theorem 1.1.
We will tacitly assume in this section as well as in x5 that the spaces
which appear have been replaced by their singular complexes.
For any connected space X let CX denote the partial Z=p -completion
of X in the sense of [3, 6.8]; there is a map X ! CX which in-
duces an isomorphism on Z=p homology, an isomorphism on fundamen-
tal groups, and the ordinary Z=p -completion map on universal covers.
If G is a compact Lie group, then, there are maps BG ! C(BG) and
ho lim!C(f^fG) ! C(BG) which induce isomorphisms on Z=p homology.
Theorem 4.1. For any compact Lie group G the diagram C(f^fG) is
centric.
In view of the definition of ^ffG, Theorem 4.1 is an easy consequence
of the following more general assertion.
Theorem 4.2. Let A be a finite abelian p-group which is a subgroup of
the compact Lie group G, H the centralizer of A in G, and f : C(BH) !
C(BG) the map derived from the inclusion H ! G. Then f is centric.
The proof of 4.2 depends on a number of lemmas.
Lemma 4.3. If X is a 1-connected Z=p -complete space [3, I, x5] then
the homotopy groups of X are uniquely q-divisible for every prime q 6= p.
If Y is a nilpotent space which has the Z=p homology of a point, then
the homotopy groups of Y are uniquely p-divisible.
Proof: The first statment follows from [3, VI, 5.4(ii)]. The second can
be proved by a standard induction on the refined Postnikov tower [3,
III, 5.3] of Y , or by combining the algebraic calculation of the homotopy
groups of the Z=p -completion of Y [3, VI, 5.1] with the fact that this
Z=p -completion is contractible if Y has the Z=p homology of a point.
8 Dwyer and Kan
Lemma 4.4. Let X and Y be connected spaces, f : X ! C(Y ) a map,
and W the universal cover of Map (X; C(Y ))f . Then W is Z=p -complete.
Proof: Let g : X ! K(ss1Y; 1) be the composite of f with the map
Y ! K(ss1Y; 1) which classifies the universal covering fibration "Y ! Y .
For elementary reasons there is up to homotopy a fibration sequence
! Map (X; C(Y ))f ! Map (X; K(ss1Y; 1))g
in which is a union of components of the space of sections of a cer-
tain fibration E ! X with homotopy fibre C(Y"). Since the space
Map (X; K(ss1Y; 1)) has no homotopy above dimension 1, the space W
is homotopy equivalent to the universal cover of one component of .
The space C(Y ) is Z=p -complete and therefore H*(-; Z=p )-local in the
sense of Bousfield [2, x4]. A slight variation on the construction of [3,
X, 2.2(ii) and 3.3(i)] shows that the space of sections of E ! X can be
expressed as Tot of a fibrant cosimplicial space in which each constituent
space has the homotopy type of a product of copies of C(Y"). Therefore
[3, XI, 4.4] the space of sections of E ! X is equivalent to the homo-
topy inverse limit of a diagram of H*(-; Z=p )-local spaces and is itself
H*(-; Z=p )-local [2, 12.9]. It follows from [2, 5.5] that the universal
cover of any component of is H*(-; Z=p )-local and therefore [2, x4]
Z=p -complete.
Lemma 4.5. Let A be a finite abelian p-group which is a subgroup of
the compact Lie group G, H the centralizer of A in G, and v : BA ~=
C(BA) ! C(BG) the map derived from the inclusion A ! G. Then
the natural map (see below)
C(BH) ! Map (BA; C(BG))v
is an equivalence
Remark: The multiplication map A x H ! G induces a map BA x
BH ! BG and hence a map BA x C(BH) ~= C(BA x BH) ! C(BG).
It is the adjoint of this last map which appears in 4.5.
Proof of 4.5: Let G0 be the identity component of G and F the
homotopy fibre of the completion map BG0 ! C(BG0). The space F is
simple, because the action of ss1F on the homotopy groups of F extends
to an action of the trivial group ss1BG0. The Serre spectral sequence of
the fibration F ! BG0 ! C(BG0) immediately shows that F has the
Z=p -homology of a point. It follows from Lemma 4.3 that the homotopy
groups of F are uniquely p-divisible and that ss1F is q-divisible for any
Centric Maps 9
prime q 6= p. It is clear that F is also the homotopy fibre of the partial
completion map BG ! C(BG).
Let u : BA ! BG be derived from the inclusion A ! G and let
F 0 be the homotopy fibre of the natural map U ! V , where U is
Map (BA; BG)u and V is Map (BA; C(BG))v. It follows from obstruc-
tion theory (cf. [10, proof of 2.3]) that F 0is connected and that ssiF 0
is a direct summand of ssiF for i 1; in particular, ssiF 0is a unique-
ly p-divisible abelian group for i 1 and a divisible abelian group for
i = 1. Consider the exact sequence ss1F 0! ss1U ! ss1V ! {1}: By
[10, 1.1] the group ss1U is finite and so cannot accept a non-trivial map
from the divisible group ss1F 0. It follows that ss1U ~= ss1V , that F 0is
the homotopy fibre of the map U" ! V" of universal covers, and hence
that U" ! V" induces an isomorphism on Z=p homology. By [10, 1.1],
then, the natural map C(BH) ! V induces an isomorphism on fun-
damental groups and a Z=p homology equivalence on universal covers.
The lemma follows [3, I, 5.5] from the fact that the universal cover of
C(BH) is Z=p -complete by construction, while the universal cover of
V = Map (BA; C(BG))v is Z=p -complete by 4.4.
Proof of 4.2: Let K be the quotient group H=A. By the fibre lemma
[3, II] and a short low-dimensional calculation the principal fibration
sequence BA ! BH ! BK gives rise to an induced principal fibration
sequence BA ! C(BH) ! C(BK). Since the fibre BA in this principal
fibration is connected, the monodromy action of ss1C(BK) on BA is
trivial up to homotopy. It follows by elementary homotopy theory that
there is an associated fibration e1 : E1 ! C(BK) with homotopy fibre
Map (BA; C(BG))v (v is as in Lemma 4.5) such that the space of sections
(e1) is naturally homotopy equivalent to the space of maps C(BH) !
C(BG) which restrict to v on BA.
Let w : BA ! C(BH) be induced by the inclusion A ! H. In a
similar way there is a fibration e2 : E2 ! C(BK) with homotopy fibre
Map (BA; C(BH))w such that the space of sections (e2) is naturally
homotopy equivalent to the space of maps C(BH) ! C(BH) which
restrict to w on BA. Composition with f gives a map E2 ! E1 over
C(BK) which, after passage to spaces of sections and appropriate com-
ponent selection, induces the precomposition map haut (C(BH))1 !
Map (C(BH); C(BG))f . Now the natural map from the centralizer of A
in H to the centralizer of A in G is an isomorphism (both groups are
H); by Lemma 4.5, this implies that the map Map (BA; C(BH))w !
Map (BA; C(BG))v is a homotopy equivalence and hence that the map-
s E2 ! E1 and (e2) ! (e1) are also homotopy equivalences. This
proves the theorem.
10 Dwyer and Kan
x5. The p-toral diagram
Let p be a fixed prime as in x4. A compact Lie group H is said to
be toral if its connected component H0 is a torus; H is p-toral if H
is toral and H=H0 is a p-group. If G is a compact Lie group, P a p-
toral subgroup of G, and N (P ) the normalizer of P in G, then P is
said to be p-stubborn if N (P )=P is finite and has no nontrivial normal
p-subgroups. Let Rp(G) be the full subcategory of the category of G-
spaces whose objects are the orbits G=P for p-stubborn P G. There
is a functor f^iGfrom Rp(G) to spaces which assigns to G=P the Borel
construction EG xG G=P ~ BP . One of the results in [13] states that if
G is a compact Lie group the natural map ho lim!^fiG! EG xG * = BG
induces an isomorphism on Z=p homology.
Theorem 5.1. For any connected compact Lie group G the "p-toral"
diagram ^fiGis centric.
By [13, 2.7, 7.1], if G is connected the diagram ss(f^iG) in (ho S )Rp(G)
satisfies the hypotheses of parts (1) and (2) of Theorem 1.1. The conclu-
sion of 1.1(1) is not interesting in this case, since ss(f^iG) is already known
to have a realization, namely f^iG. Statement 1.1(2), though, gives the
following corollary.
Corollary 5.2. If G is a connected compact Lie group then any re-
alization of the diagram ss(f^iG) is weakly equivalent, as an object of
SRp(G) , to ^fiGitself.
This corollary implies that if G is a connected compact Lie group
then BG is determined up to Z=p homology type by a certain relatively
small and well-controlled diagram in the homotopy category. There is
some hope that in the long run this observation will contribute to a
generalization of the uniqueness result in [8].
The proof of Theorem 5.1 depends on the following proposition.
Proposition 5.3. Let j : K ! L be a map of p-toral compact Lie
groups and H the centralizer of j(K) in L. Then the natural map (cf.
4.5, [10, x1])
BH ! Map (BK; BL)Bj
is an equivalence.
Remark: It seems likely that the conclusion of this proposition remains
valid under the weaker hypotheses that K is compact Lie and L is toral
(see [15]). A slightly modified form of the proposition holds if K is
assumed p-toral and L is assumed compact Lie [21] [18].
Centric Maps 11
Proof of 5.3: The group L acts on itself by conjugation, and com-
position of this action with the homomorphism j gives a conjugation
action of K on L. The fixed point set LK of this action is the centralizer
H of j(K) in L. As in [10, proof of 4.1], the proposition can be proved
by showing that the natural map from LK to the homotopy fixed point
set LhK of this conjugation action is an equivalence. (Here LhK is the
space of sections of the fibration over BK with fibre L associated to the
conjugation action.)
Assume then that L and K are p-toral, and that K acts on L via
group homomorphisms. We will show that the map LK ! LhK is an
equivalence. We can assume [10, proof of 2.2] that the action of K on
the component group ss0L is trivial, since any component of L which
is moved by the action of K contributes neither to LK nor to LhK .
It is enough, then to show that for each component Li of L the map
LKi ! LhKi is an equivalence. There is no difficulty if LhKi is empty,
since then LKi is empty too. Assume then that LhKi is non-empty. It
follows that for any finite p-subgroup o of K the homotopy fixed point
set Lhoi is non-empty and hence by [11] that the fixed point set Loi is
non-empty. Since K is p-toral it is possibleSto find a dense subgroup
o1 of K which is an increasing union n on of finite p-subgroups; by a
compactness argument, then, Lo1i is non-empty and hence LKi is non-
empty. Pick x 2 LKi. Multiplication by x then gives a K-equivariant
homeomorphism between Li and the identity component L0 of L, so we
will be done if we can prove that LK0 ! LhK0 is an equivalence. Assume
for this purpose that L is connected. The group L is then a torus (S1 )n
and the automorphism group of L is the discrete group GL(n; Z). The
identity component of K thus acts trivially on L, so that the action of
K on L factors through the quotient map K ! ss0K = oe. It is obvious
that the natural map Loe! LK is an isomorphism, and an elementary
homotopy theory argument shows that the natural map Lhoe ! LhK is
an equivalence (this comes down to the fact that the space of sections
of the trivial fibration over B(K0) with fibre L is homotopy equivalent
to L itself). Let L" = Rn be the universal cover of L. The group oe
acts on the short exact sequence {1} ! ss1L ! "L ! L ! {1} and the
low dimensional end of the induced twisted cohomology exact sequence
gives isomorphisms ss0(Loe) ~= H1 (oe; ss1L), ss1(Loe) ~= H0 (oe; ss1L), and
ssk (Loe) ~= 0 for k 2. A straightforward calculation now shows that the
homotopy groups of Lhoe are given by the same formulas and that the
map Loe! Lhoe in fact induces an isomorphism on homotopy.
Proof of 5.1: By [13, 5.1(i)] the centralizer in G of a p-stubborn
P G is the center of P . The desired result follows from 5.3.
12 Dwyer and Kan
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University of Notre Dame, Notre Dame, Indiana 46556
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139