ON SPACES OF SELF HOMOTOPY EQUIVALENCES OF
pCOMPLETED CLASSIFYING SPACES OF FINITE GROUPS AND
HOMOTOPY GROUP EXTENSIONS
CARLOS BROTO AND RAN LEVI
1.introduction
Let G and ss be groups and let p be a prime number. A modp homotopy group
extension of ss by G is a fibration with base space Bss^pand fibre BG^p, where *
*()^p
denotes the BousfieldKan pcompletion functor.
In this paper we study homotopy extensions of finite groups. Thus if X is the*
* total
space in a modp homotopy group extension of ss by G, where both ss and G are
finite, we shall call X a finite homotopy group extension. As one would expect *
*from
a classification problem of a class of fibrations, the project involves studyin*
*g spaces
of homotopy equivalences of the fibres under consideration. This problem forms *
*one
of the main cores of the paper and is of independent interest. Indeed, spaces o*
*f self
homotopy equivalences for compact connected simple Lie groups are well understo*
*od
as a byproduct of [8]. Our results are partially the finite group analogue.
Let p be a prime and let G be a finite group. Then Op0G is defined to be the
maximal normal subgroup of G of order prime to p. Define the modp0 reduction
of G to be the quotient G=Op0G. If Op0G is trivial then G is said to be p0redu*
*ced.
Notice that the natural projection from G to G=Op0G induces a modp homology
isomorphism and hence a plocal homotopy equivalence on classifying spaces.
For an arbitrary space X, let Aut (X) denote topological monoid of all self h*
*o
motopy equivalences of X. Let SAut (X) denote the identity component of Aut (X)
and let Out (X) denote the group of components ss0(Aut (X)). By general theory,
fibre homotopy equivalence classes of Modp homotopy group extensions of ss by G
are in 11 correspondence with the set of homotopy classes of maps from Bss^pto
B Aut(BG^p). The next theorem describes the homotopy type of the identity compo
nent SAut (BG^p).
Theorem 1.1. Let G be a finite group. Then SAut (BG^p) ' BZ(G=Op0G).
Recall that a group G is said to be pperfect if its first modp homology gro*
*up
is trivial and psuperperfect if in addition its second modp homology vanishes*
*. For
every discrete group G there exists a maximal normal pperfect subgroup OpG < G
and BOpG^pis the 1connected cover of BG^p. Furthermore, if G is itself pperfe*
*ct then
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q5*
*2.
Key words and phrases. Classifying spaces, self equivalences, finite groups.
C. Broto is partially supported by DGICYT grant PB970203.
1
2 CARLOS BROTO AND RAN LEVI
there is a central extension UpG of G by H2(BG; Z(p)) and BUpG^pis the 2connec*
*ted
cover of BG^p. For simplicity of notation we shall denote Up(OpG) by UpG, even *
*if G
itself is not pprefect. As corollaries of Theorem 1.1 we obtain
Corollary 1.2. Let X be a modp homotopy extension of ss by G. Then there is a
homotopy equivalence
X<2> ' BUpss^px BUpG^p;
where X<2> is the 2connected cover of X.
Corollary 1.3. Assume ss is pperfect and let X be a modp homotopy extension of
ss by G. Then, there is a homotopy equivalence
X ' BH^p
where H is an extension of ss by G with trivial action ss ____ Out(G).
Corollary 1.4. Assume ss is psuperperfect and let G be any finite group. Then *
*every
modp homotopy extension of ss by G is trivial.
The general case might be easily reduced to the case in which the base is the
classifying space of a finite pgroup ss. In fact, if ss is not a p group, the*
*n there is
a fibration BOp(ss)^p_____Bss^p_____Bss=Op(ss), where ss=Op(ss) = ss1(Bss^p) *
*is a p
group. Then, if X is a modp homotopy extension of ss by G, one has a diagram of
fibrations
BG^p==== BG^p
 
 
? ?
Y _______X ____Bss=Op(ss)
  ww
  ww
  w
? ?
BOp(ss)^p___Bss^p__Bss=Op(ss)
Now, Corollary 1.3 applies to the fibration in the left column and Y ' BH^p, he*
*nce
X is expressed in the middle row fibration as a modp homotopy groups extension*
* of
a pgroup, ss=Op(ss), by a group H.
We now turn to the group of components Out (BG^p). There is an obvious group
homomorphism
flG : Out(G) ____ Out(BG^p);
which, as we shall see below, is not generally an isomorphism. A different appr*
*oxi
mation of the group Out(BG^p) is given as follows. Let Cp denote the category w*
*hose
objects are finite pgroups and whose morphisms are conjugacy classes of monomo*
*r
phisms ss _____ss0. For a finite group G, let IG : Copp_____Sets denote the *
*functor
which takes a finite pgroup ss to the set Inj(ss; G) of conjugacy classes of m*
*onomor
ae
phism ss _____G and which takes a morphism ss0 _____ss in Cp to the induced m*
*ap
SELF HOMOTOPY EQUIVALENCES 3
Inj(ss; G) ____ Inj(ss0; G). The set of all natural equivalences of IG forms a*
* group un
der composition, which we denote by Aut(IG ). Let Ocp(G) denote the orbit categ*
*ory
of pcentric subgroups of G. Let Z : Ocp(G) ____Ab denote the contravariant fu*
*nctor
which associate with an orbit G=P the center Z(P ).
Theorem 1.5. Let G be a finite group. Then there is an exact sequence
G
0 ____ lim1 Z ____ Out(BG^p) ____ Aut(IG ):
Ocp(G)op
Furthermore, the obstruction for G being onto is an element of lim2 Z. In par*
*tic
Ocp(G)op
ular if both higher limits vanish then G is an isomorphism.
The general behavior of the maps flG and G is not well understood. One class
of examples where flG is always an isomorphism is given by groups G, whose Sylow
psubgroup is normal. However, different groups with the same modp homotopy
type may well have different outer automorphism groups (e.g. (BM11)^2' BSL3(3)^2
but Z=2 ~= Out(SL2(3)) 6= Out (M11) = {1}) and so flG cannot be an isomorphism
in general. The map G is known not to be a monomorphism in general as the
next theorem demonstrates, but there is no known example where G fails to be an
epimorphism.
s
Theorem 1.6. Let q = 32 , s 1. Let G = SL2(q) and let H = P SL2(q). Then
1. There is a commutative diagram of isomorphisms
~=
Out (G) _____Out (H)
~= ~=
? ~= ?
Out (BG^2)___Out(BH^2)
and Out (G) ~=Z=2Z x Z=2sZ.
2. limiZ = 0 for all i > 0. Thus Out (BH^2) ~=Aut (IH ).
Ocp(G)
3. limiZ ~=Z=2Z if i = 1, but vanishes otherwise. Thus there is a nonsplit s*
*hort
Ocp(H)
exact sequence
0 ____Z=2Z ____ Out(BH^2) ____Aut(IH ) ____0:
A complete identification of Out(BG^p) for an arbitrary finite group G will a*
*ppear
in a future paper joint with R. Oliver.
The authors would like to express their warm gratitude to Bill Dwyer for shar*
*ing
with them his insightful observations. We also thank Bob Oliver for many useful
discussions and for his continuing interest in the project. Finally both autho*
*rs are
grateful to the Centre de Recerca Matemtica in Barcelona for allowing them to m*
*eet
frequently enough to complete this work.
4 CARLOS BROTO AND RAN LEVI
2. The Space Map (Bss; BG^p)
In this section we record the following
Proposition 2.1. Let ss be a finite pgroup and let G be a finite group. Then
Map (Bss; BG^p) is pcomplete and the natural map
(1) Map (Bss; BG) ____ Map (Bss; BG^p) ;
induces a modp homology isomorphism.
The proposition is known to the experts, but as we are not aware of a suitable
reference, a proof is included here.
Proof.The case where ss is an elementary abelian pgroup is due to Lannes [9]. *
*By
induction we may thus assume the statement of the proposition for every pgroup
of order less than pn.
Let ss a group of order pn. There is an extension
____ss ____V
where V is a nontrivial elementary abelian pgroup. The group V acts on fB = E*
*ss=
and one has a homotopy equivalence
Map (Bss; BG^p) ' Map (Bf ; BG^p)hV :
Recall that for a general V space X, the homotopy fixed points space is defi*
*ned as
the space of V equivariant maps Map V(EV; X) and this can be reinterpreted as *
*the
space of sections of the projection XhV ____BV . In fact, using the action of *
*BV on
Map (BV; XhV) one obtains in general a homotopy equivalence over BV
'
BV x XhV ____ Map (BV; XhV)1
where Map (BV; XhV)1 denotes the set of components of Map (BV; XhV) over the co*
*m
ponent of the identity Map (BV; BV )1 ' BV .
In our case we take X to be either Map (Bf ; BG^p) or Map (Bf ; BG). Both spa*
*ces
are related by the induction hypothesis, and after taking homotopy quotients, we
have
^ ^
(2) Map (Bf ; BG^p)hV ' Map (Bf ; BG)p hV ' Map (Bf ; BG)hV p :
Now, notice that Map (Bf ; BG)hV has the homotopy type of a union of a finite n*
*umber
of classifying spaces of finite groups. Hence we have homotopy equivalences ove*
*r BV
^
(3) BV x Map (Bf ; BG^p)hV ' Map BV; Map (Bf ; BGp)hV 1 '
^ hV ^
Map BV; Map (Bf ; BG)hV 1 p ' BV x Map (Bf ; BG) p ;
from which we obtain Map (Bss; BG^p) ' Map (Bss; BG)^p.
Notice that Equation (2) implies that Map (Bf ; BG^p)hV ' (Map (Bf ; BG^p)hV)*
*^p
and the same arguments as above yield
Map (Bss; BG^p) ' Map (Bss; BG^p)^p:
SELF HOMOTOPY EQUIVALENCES 5
*
* __
Hence Map (Bss; BG^p) is pcomplete. This finishes the proof of the proposition*
*. __
3. The Topological Monoid SAut (BG^p)
In this section we prove the following
Theorem 3.1. Let G be a finite group. Then
SAut (BG^p) ' BZ(G=Op0G);
where Op0G is the maximal normal subgroup of G of order prime to p.
Proof.Notice that since G=Op0G is p0reduced, its center is automatically an ab*
*elian
pgroup. Also BG^p' B(G=Op0G)^pand so SAut (BG^p) ' SAut(B(G=Op0G)^p). Thus
to prove the theorem it suffices to show that if G is p0reduced then SAut (BG^*
*p) '
BZ(G).
This is done in three steps. In Proposition 3.6 below it is shown that SAut (*
*BG^p)
has at most one nontrivial homotopy group
ss1(SAut (BG^p); id) = L ;
where L is a finite pgroup and furthermore, that there is a monomorphism i :
L ____G such that Z(G) i(L) Z(ss), where ss is any Sylow psubgroup of G con
taining L. Next, identify L with its image i(L) < G. In Proposition 3.7 it is p*
*roved
that the inclusion of the centralizer of L in G, CG (L) G, induces an isomorph*
*ism
in modp cohomology; that is to say L is pcohomologically central in G. Finall*
*y one
uses a theorem of Mislin, stated below, which shows that L is central in G. Si*
*nce
L Z(G), the two subgroups have to coincide, and therefore SAut (BG^p) ' BZ(G).
Theorem 3.2 (Mislin [12]). Let G be a finite p0reduced group. Then, every pco*
*ho
mologically central subgroup of G is central.
Remark 3.3. This statement is not explicit in [12] but the necessary arguments
are contained in there. We sketch here the proof for convenience of the reader.*
* By
definition, if P is pcohomologically central, then the inclusion of its centra*
*lizer CG (P )
in G induces a modp cohomology isomorphism and therefore an equivalence of the
corresponding Frobenius categories of psubgroups. In particular CG (P ) contro*
*ls p
fusion in G and according to the Z*theorem this implies CG (P ) = G, because G
is p0reduced. It might be worth pointing out that the Z*theorem for finite gr*
*oups __
depends on the classification of finite simple groups [3] *
* __
For a finite group G and a collection C of subgroups of G, which is closed un*
*der
conjugacy, the Corbit category is the category whose objects are the Gsets gi*
*ven by
left cosets G=H with H 2 C and whose morphisms are given by all Gmaps. Notice
that the action of G on a coset is given by g0(gH) := g0gH. Thus any morphism
' from G=H to G=K in the orbit category is determined by '(H) = gK for some
g 2 G, which conjugates H into K. Moreover this element g is unique up to a rig*
*ht
shift by an element of K. Thus each map ' : G=H ____G=K gives rise to a unique
faithful representation of H in K (a conjugacy class of monomorphisms). If g 2 G
conjugates H to K then g1 conjugates CG (K) to CG (H). Furthermore, if g1 and
6 CARLOS BROTO AND RAN LEVI
g2 represent the same morphism G=H ____G=K then conjugation by g11and g12
coincides on CG (K). Recall the following
Definition 3.4. A psubgroup ss of a finite group G is said to be pcentric if
CG (ss) ~=Z(ss) x W;
where W is a group of order prime to p. Let Ocp(G) denote the full subcategory *
*of
the orbit category whose objects are orbits of pcentric subgroups of G.
Our main tool is a homology decomposition theorem which expresses the modp
homotopy type of BG by means of classifying spaces of pcentric subgroups. The
technique is originally due to Jackowski, McClure and Oliver [7, 8]. A very coh*
*erent
account of these methods by Dwyer, with some significant improvements and uni
fication of methods appears in [4, 5]. The following is a particular case, whi*
*ch is
all we need here. Let OEG : Ocp(G) _____G  Sets be the functor associating w*
*ith
an orbit G=H the Gset given by G=H itself. For a Gspace X let XhG denote the
homotopy orbit space X xG EG, where EG is a free contractible G space, thus,
OEG (G=H)hG = G=H xG EG ' BH
Theorem 3.5. Let G be a finite group. Then the natural map
G :hocolim(OEG )hG ____BG
Ocp(G)
induces a modp homology isomorphism.
Thus one has for any finite group G a homotopy equivalence
^
(4) Map (BG^p; BG^p) ' holimMap (OEG )hG; BGp :
Ocp(G)
Proposition 3.6. Let G be a finite p0reduced group and ss a pSylow subgroup o*
*f G.
Then
(
L i=1
ssi(SAut (BG^p)) =
0 otherwise
where L is an abelian pgroup which can be identified with a subgroup L0 < G su*
*ch
that Z(G) L0 Z(ss).
Proof.We write SAut (BG^p) = Map (BG^p; BG^p)idand then use the BousfieldKan
spectral sequence for computing the homotopy groups of a homotopy limit associa*
*ted
to equation (4)
Es;t2= limssst(Map ((G=H)hG; BG^p)) ) ssts(Map (BG^p; BG^p)):
Ocp(G)
The differentials in this spectral sequence have the form dr : Es;tr___Es+r;t+*
*r1r.
Take the identity map of BG^pas a base point. It restricts to the standard in*
*clusion
inc: (G=H)hG = BH ____BG^p
SELF HOMOTOPY EQUIVALENCES 7
for any centric psubgroup H < G. One has
Map ((G=H)hG; BG^p)inc' Map (BH; BG^p)inc'
(Map (BH; BG)inc)^p' BCG (H)^p' BZ(H):
Let Z : Ocp(G) _____Ab denote the contravariant functor which associates to ea*
*ch
orbit G=H the center Z(H). Then, for our choice of base point, the E2 term of t*
*he
spectral sequence has the form
8
< limsZ ; t = 1
Es;t2~=limssst(Map ((G=H)hG; BG^p)inc) = Ocp(G)
Ocp(G) : 0 ; otherwise.
Hence the only relevant value is obtained in the spectral sequence for s = 0. C*
*onse
quently ssi(SAut (BG^p)) = 0 if i > 1 and
ss1(SAut (BG^p)) = L := lim0Z:
Ocp(G)
We proceed by proving the properties of L claimed in the proposition. By defi*
*nition
the inverse limit is the kernel of the map
Y Y
OE : Z(H) ____ Z(range('))
G=H2Obj(Ocp(G)) '2Mor(Ocp(G))
defined by its projection to the ' component as xP  BCG (')(xQ ), where ' :
P _____Q, xP 2 Z(P ) and xQ 2 Z(Q). Notice that for every G=P 2 Ocp(G),
Z(P ) > Z(G), since Z(G) is a pgroup. Since all maps in Ocp(G) are induced by
translation in G, the induced maps on centralizers are given by sub conjugation*
* and
it follows at once that Z(G) is a subgroup of L. An obvious embeding is given *
*by
including Z(G) diagonally in the domain of OE.
Finally, notice that if x 2 L is represented by a sequence of elements of G i*
*ndexed
by objects of Ocp(G) and xP is the projection of x to the component of P , then*
* xP is
invariant under the action of the normalizer NG (P ) and hence is central in P *
*. Let ss
be a Sylow psubgroup of G containing P . Then xP = xssand so xssis central in *
*ss.
Notice that projection from L to the component of each one of the groups involv*
*ed is
a monomorphism. Thus L can be embedded in the center of each Sylow psubgroup. *
* __
In particular L is an abelian pgroup and the proof is complete. *
* __
The inclusion of L in G obtained in the above Proposition factors through a p*
*Sylow
subgroup ss of G. This is represented up to conjugation by the composition
ev ^
BL ____BZ(ss) ' Map ((G=ss)hG; BG^p) ____BGp
and therefore by
ev ^
BL ' Map (BG^p; BG^p)id____BGp
This last map is canonically defined. However a representing homomorphism is on*
*ly
well defined up to conjugation. The next Proposition however shows that L is p
cohomologically central in G and hence central in G, by Mislin theorem 3.2. Th*
*us
8 CARLOS BROTO AND RAN LEVI
L coincides with all its conjugates. Hence we conclude that the identification*
* of
L = ss1Map (BG^p; BG^p)idwith a subgroup of G is canonically induced by evalua*
*tion.
In the next Proposition identify the group L with its image L0under some appr*
*o
priate embedding.
Proposition 3.7. Fix an embeding of L in G and identify L with its image. Then
the modp cohomology restriction
Res : H*(BG) ____H*(BCG (L))
is an isomorphism.
Proof.Let ss be a Sylow psubgroup of G containing L. Then L < Z(ss) and so
ss < CG (Z(ss)) < CG (L). Thus the restriction map is a monomorphism.
Next notice that there is a homotopy equivalence
OE : BL ____ SAut(BG^p);
and a homotopy commutative diagram
OE ^ ^
BL ___SAut (BGp) = Map (BG; BGp)id
@@ Bi*
@R ?
BZ(ss) ' Map (Bss; BG)Bi
where i : ss ____G is the inclusion. Taking adjoints we get a commutative diag*
*ram
BL x BG ___BG^p
1xBi6

BL x Bss
Taking adjoints the other way and writing BCG (L)^pfor Map (BL; BG^p) gives a
commutative diagram
BG ___BCG (L)^p
Bi6

Bss
Let : BG _____BCG (L)^pdenote the resulting map. By commutativity this
map induces a monomorphism on modp cohomology. Composed with the restric
tion BCG (L) _____BG we get a self map of BCG (L) which induces a cohomology
monomorphism. Since H*(BCG (L)) is a graded group of finite type any monomor
phism is an isomorphism. It follows then that the restriction map is an epimorp*
*hism_
and thus an isomorphism and the proof is complete. *
*__
Corollary 3.8. For every finite group G the space B Aut(BG^p) is a 2stage Post
nikov tower with homotopy groups in dimensions 1 and 2. In particular
ss2(B Aut(BG^p)) = Z(G=Op0G):
SELF HOMOTOPY EQUIVALENCES 9
Let a modp homotopy group extension of ss by G be given. Then one has the
following commutative diagram of fibrations, where the bottom square is a homot*
*opy
pullback.
BG^p ____BG^p
 
 
? ?
Y ______X
 
 
 
? ?
BUpss^p___Bss^p
But the left column is also a modp homotopy group extension, where the base
space is 2connected. Since B Aut(BG^p) is a 2stage Postnikov tower with homot*
*opy
in dimension 1 and 2, the classifying map for this extension is trivial. Hence*
* the
extension is split, namely
Y ' BUpss^px BG^p:
But Y is the fibre of the composition
X ____Bss^p____Bss^p[2];
where Bss^p[2] denotes the second Postnikov section of Bss^p. Hence Y and X hav*
*e the
same two connected covers and Corollary 1.2 follows.
Let a finite modp homotopy extension of ss by G be given, with ss a pperfect
group and assume without loss of generality that G is p0reduced. Then Bss^pis
simplyconnected and hence the classifying map Bss^p____B Aut(BG^p) is trivial*
* on
fundamental groups. It follows that the action of ss1(Bss^p) on BG^pis trivial *
*and so
there exists at least one modp homotopy extension of ss by G with this action,*
* namely
the trivial extension. Hence fibre homotopy equivalence classes of modp homoto*
*py
extensions of ss by G with a trivial action of ss1(Bss^p) are in 11 correspond*
*ence with
H2(Bss^p; Z(G)). Now, ordinary group extensions of ss by G with trivial outer a*
*ction of
ss on G are classified by H2(ss; Z(G)) ~=H2(Bss^p; Z(G)), since ss is finite an*
*d Z(G) is
a pgroup. But given any such ordinary group extension, pcompletion of the fib*
*ration
resulting from applying the classifying space construction to the extension rem*
*ains a
fibration. Corollary 1.3 follows.
Finally assume that ss is psuperperfect. Then the classifying map
Bss^p____B Aut(BG^p)
is trivial. Corollary 1.4 follows.
4. The Group of Components Out (BG^p)
Let G be a finite group. As before we shall assume that G is p0reduced. In
this section we give an approximation for Out (BG^p). We shall essentially red*
*uce
the calculation to the computation of two higher limits. In the case where the*
*se
10 CARLOS BROTO AND RAN LEVI
limits vanish we shall obtain an identification of Out(BG^p) with the group of *
*natural
equivalences of the functor IG defined in the introduction.
Recall that the category Cp has all finite pgroups as objects and conjugacy *
*classes
of monomorphisms ss ____ss0 as morphisms. The functor IG : Copp____Sets takes*
* a
finite pgroup ss to the set Inj(ss; G) of conjugacy classes of monomorphism ss*
* ____G
ae 0 op 0
and a morphism ss ____ss in Cp to the induced map Inj(ss; G) ____ Inj(ss ; *
*G).
Lemma 4.1. For any finite group G, there are homomorphisms of groups
ffiG : Out(G) ____ Aut(IG )
and
flG : Out(G) ____ Out(BG^p):
Proof.For [ff] 2 Out(G) and a finite pgroup ss define
ffiG ([ff])ss: IG (ss) ____IG (ss)
to be the map taking [f] 2 IG (ss) to [ff O f], where ff is any automorphism of*
* G
representing its class. That ffiG is a homomorphism is clear from the definiti*
*on and
naturality follows from functoriality of Inj(ss; ).
The definition of flG is as the map induced by a representative for an outer *
*auto_
morphism by applying it to the second variable in the respective mapping space.*
* __
Let G be a finite group and let ss be a finite pgroup. By Proposition 2.1 th*
*ere are
isomorphisms of sets
ss0(Map (Bss; BG^p)) ~=ss0(Map (Bss; BG)) ~=Rep (ss; G) :
The following lemma is well known.
Lemma 4.2. Let f : ss ____G be a homomorphism. Then the following are equiva
lent:
1. f is a monomorphism
2. H*Bss is a finitely generated module over Bf*(H*(BG)).
Lemma 4.3. There is a group homomorphism
G : Out(BG^p) ____ Aut(IG )
Proof.Let [f] 2 Out(BG^p) be the homotopy class of some self equivalence
f : BG^p____BG^p:
We must present a natural equivalence G (f) of IG . Thus for each ss 2 Cp let
G (f)ss: IG (ss) ____IG (ss)
be the function which takes the conjugacy class of a monomorphism ' : ss ____G*
* to
the conjugacy class of a homomorphism inducing the composition
B' ^ f ^
Bss ____BGp ____BGp
SELF HOMOTOPY EQUIVALENCES 11
up to homotopy. By the Proposition 2.1 such a class exists. Furthermore, by t*
*he
Lemma 4.2 it follows that G (f)ss['] is the conjugacy class of a monomorphism.*
* The
definition does not depend on the choice of f or ', but rather only on their ho*
*motopy
classes and so G (f)ssis well defined.
Naturality of G (f) follows at once by similar considerations. Also, to see *
*that G
is a group homomorphism observe that multiplication in both domain and range is_
given by composition. __
Definition 4.4. Let G be a finite group and ss a finite pgroup. Let Bss denote*
* any
model for the classifying space for ss. Define Inj(Bss; BG^p) to be the subset*
* of the
mapping space Map (Bss; BG^p) consisting of all maps which are homotopic to Bf *
*for
some monomorphism f : ss ____G.
Let CGpdenote the full subcategory of Cp whose objects are all psubgroups of*
* G.
Consider the functor
KG : CGp____Sets
taking a psubgroup ss of G to the set ss0(Inj((G=ss)hG; BG^p)) and a morphism *
*in CGp
to the induced map. Then there is a natural transformation of functors
fi : IG ____KG ;
which by Proposition 2.1 is a natural equivalence. Here IG is considered as a f*
*unctor
defined on CGp.
Consider the homology decomposition of Theorem 3.5
G : hocolim(OEG )hG ____BG:
Ocp(G)
Thus G induces a modp homology equivalence and so its pcompletion is a homoto*
*py
equivalence. Thus one has an isomorphism of monoids
2 3
4(hocolim(OE ) )^; (hocolim(OE ) ~)^5=BG^; BG^ :
! G hG p ! G hG p p p
Ocp(G) Ocp(G)
The proof of Theorem 1.5 thus amounts to showing that given any natural equiv*
*a
lence ff 2 Aut(IG ), the obstruction to existence of a lift of ff to a self map*
* "ffof BG^p
such that G ("ff) = ff is an element of lim2 Z and if a lift exists then any t*
*wo differ
Ocp(G)
by an element of lim1 Z.
Ocp(G)
Lemma 4.5. Let G be a finite p0reduced group. For any psubgroup ss of G, t*
*he
following are equivalent
1. ss is a pcentric subgroup of G.
2. ss is a pcentric subgroup of every pSylow subgroup of G that contains ss.
Proof.Assume first that ss is pcentric in G. That is, CG (ss) ~=Z(ss) x W , wh*
*ere W
is a p0group. For any subgroup H of G containing ss, Z(ss) < CH (ss) < CG (ss)*
*, hence
12 CARLOS BROTO AND RAN LEVI
CH (ss) ~=Z(ss) x W 0, where W 0= CH (ss) \ W is a p0groups and so ss is pcen*
*tric in
H.
Conversely, assume that ss is pcentric in every Sylow p subgroup of G contai*
*ning it.
Let R be a Sylow p subgroup of CG (ss). We must show that R = Z(ss). Notice that
ss . R < G is a psubgroup. Let S be a Sylow psubgroup of G containing ss . R.*
* Since
ss is pcentric in S, one has CS(ss) = Z(ss). Then R < CS(ss), since it is cont*
*ained in
S and centralizes ss. On the other hand CS(ss) = Z(ss) is contained in every p*
*Sylow __
of CG (ss), in particular, it is contained in R and so R = Z(ss) as claimed. *
* __
Lemma 4.6. Let ss be a finite p group and assume that there exists a monomorp*
*hism
f : ss ____G, including ss in G as a pcentric subgroup. Let [f] denote the co*
*njugacy
class of f. Then for any natural equivalence ' 2 Aut(IG ) any representative of*
* the
class 'ss[f] is a monomorphism t : ss ____G such that t(ss) is pcentric in G.
Proof.First note that if t : ss _____G is a monomorphism such that t(ss) < G is
pcentric, then any conjugate of t also includes ss in G as a pcentric subgrou*
*p. Let
t : ss _____G be any representative for 'ss[f]. It suffices to show that t(ss*
*) < G is
pcentric.
Let S < G be a Sylow psubgroup containing t(ss). Let S : S _____G and t :
ss ____S denote the respective inclusions, such that t = S O t.
Since ' is a natural equivalence, there is a commutative diagram
'S
Inj(S; G) ___Inj(S; G)
t* t*
? 'ss ?
Inj(ss; G)___Inj(ss; G)
Thus t*[S] = [t]. By commutativity and the fact that ' is an equivalence one h*
*as
t*'1S[S] = [f]. Since S is a Sylow psubgroup in G, there exists an automorphi*
*sm ff
of S, such that '1S[S] = [S O ff] and it follows that S O ff O tis conjugate t*
*o f in G.
Since f includes ss in G as a pcentric subgroup, it follows by Lemma 4.5 tha*
*t ff O t
includes ss as a pcentric subgroup of S. Since ff is an automorphism, t inclu*
*des ss
in S as a pcentric subgroup. Since this applies to any Sylow subgroup S contai*
*ning
t(ss), it follows, again by Lemma 4.5, that t(ss) is pcentric in G. __
__
Lemma 4.7. For each ' 2 Aut (IG ) and each pcentric subgroup ss < G there is*
* a
map
O'ss: (G=ss)hG ____BG^p
SELF HOMOTOPY EQUIVALENCES 13
such that if f : G=ss ____G=ss0 is a morphism in Ocp(G) then the diagram
(G=ss)hG
@@O'ss
 @R
(5) fhG BG^p


? O'ss0
(G=ss0)hG
commutes up to homotopy.
Proof.By the remarks above, we have a natural equivalence of functors
fi : IG ____KG ;
where IG is restricted to CGp. Thus each natural equivalence ' of IG induces a *
*natural
equivalence of KG , which by abuse of notation, we also denote by '.
For two psubgroups ss; ss0< G, the set of morphisms Mor Ocp(G)(G=ss; G=ss0) *
*consists
of all elements of G conjugating ss into ss0 modulo the right action of ss0. O*
*n the
other hand the group of inner automorphisms of ss0 operates from the right on t*
*he
set of all injective homomorphisms from from ss to ss0 and the orbits of this a*
*ction
form the morphism set Mor CGp(ss; ss0). In other words this is the set of those*
* faithful
representations of ss in ss0 induced by conjugation in G. Hence there is a fun*
*ctor
ff : Ocp(G) _____CGp, which sends an orbit of a pcentric G=ss to the group ss*
* and a
morphism in Ocp(G) to the induced representation. Let K"G denote the composition
KG O ff. Then every natural self equivalence ' of KG induces a self equivalenc*
*e of
"KG, which again we denote by '.
For each pcentric ss < G let [ss] 2 "KG(G=ss) denote the class of the map in*
*duced
by inclusion. Let
O'ss: (G=ss)hG ____BG^p
be any representative for the homotopy class of 'ss[ss].
Let f : G=ss _____G=ss0 be a morphism in Ocp(G). Then, since ' is a natural
equivalence we have "KG(f)O'ss0= 'ssOK"G(f). Homotopy commutativity of Diagram
(5) is equivalent to the following equation holding:
"KG(f)[O'ss0] = [O'ss]:
But since Diagram (5) commutes up to homotopy if O'ssand O'ss0are replaced by ss
and ss0respectively, we have "KG(f)[ss0] = [ss]. Hence
"KG(f)[O'ss0] = "KG(f)'ss0[ss0] = 'ss"KG(f)[ss0] = '[ss] = [O'ss];
*
* __
as required. This completes the proof. *
*__
Lemma 4.7 implies that each natural equivalence ' 2 Aut(IG ) gives rise to a *
*map
E1': hocolim (1)(OEG )hG ____BG^p;
Ocp(G)
14 CARLOS BROTO AND RAN LEVI
where the superscript on the left hand side means the 1skeleton of the homotopy
colimit.
Let G be a finite group and let ss be a pcentric subgroup. For a natural equ*
*ivalence
' 2 Aut(IG ) let t'ss: ss ____G denote any choice of a monomorphism such that
B(t'ss)^p: Bss ____BG^p
is homotopic to O'ss, under the usual identification of (G=ss)hG with Bss. By L*
*emma
4.5, t'ssincludes ss in G as a pcentric subgroup.
For every j 1, define a functor
'j: Ocp(G) ____Ab
by
'j(G=ss) = ssi(Map((G=ss)hG; BG^p)O'ss):
Notice that
Map((G=ss)hG; BG^p)O'ss' Map((G=ss)hG; BG^p)B(t'ss)^p' BCG (t'ss(ss))^p' BZ(s*
*s):
Thus 'jis the zero functor for all j except possibly for j = 1 and '1(G=ss) ~=Z*
*(ss).
General obstruction theory for maps out of a homotopy colimit [13] gives that*
* if
En': hocolim!(n)(OEG )hG ____BG^p
Ocp(G)
has been constructed, the obstructions to extending it to a map En+1'out of the
(n + 1)st skeleton of the homotopy colimit are in the group limn+1'nand if any
Ocp(G)
extension exists then any two differ by an element in limn 'n. For a self equiv*
*alence
Ocp(G)
' of the functor IG , let
ij(') = limi'j:
Ocp(G)
We proceed by analyzing the obstruction groups. For a finite (preduced) grou*
*p G,
let
Z :Ocp(G)op ____Ab
denote the functor, which associates to a pcentric subgroup ss of G its center*
* Z(ss).
Definition 4.8. For any finite group G and i 0, define
Ji(G) = limi Z
Ocp(G)op
Notice in particular that the discussion in section 3 implies that J0(G) = Z(*
*G) if
G is p0reduced.
Proposition 4.9. For every natural equivalence ' 2 Aut (IG ), we have ij(') = 0
for j 2 and
Ji(G) ~=i1(')
for all i 0.
SELF HOMOTOPY EQUIVALENCES 15
Proof.Let t'ss: ss ____G be the homomorphisms chosen above. Let 'ssdenote the
composition
'ss
Z(ss) x ss_________G
@@
@R t'ss
ss
where denotes the multiplication map. Then the maps 'ssinduce homotopy equiv
alences
gB'ss:BZ(ss) ____'Map (Bss; BG^ '
p)(Btss)^p:
Furthermore, by Lemma 4.7, if f :G=ss ____G=ss0 is a morphism in Ocp(G), whi*
*ch
is determined by an inner automorphism c of G sub conjugating ss to ss0, then t*
*here
is a homotopy commutative diagram
B
BZ(ss) x Bss____Bss
Bc1x1 @ B(t'ss)^p
 @@R
BZ(ss0) x Bss Bc BG^p

@ @ 
1xBc @R ? B(t'ss0)^p
BZ(ss0) x Bss0___Bss0
B
Taking respective adjoints one gets a homotopy commutative diagram
' ^
BZ(ss) ___Map (Bss; BGp)(Bt'ss)^p
6 6
Bc1 Bc*
 
BZ(ss0)___Map(Bss0; BG^p)(Bt' )^
' ss0p
which in turn induces a commutative diagram of homotopy groups for every j 1
~= '
ssj(BZ(ss))___ssj(Map (Bss; BG^p)(Bt'ss)^p) = j(G=ss)
6 6
(Bc1)# (Bc*)#
 
ssj(BZ(ss0))__~ssj(Map (Bss0; BG^p)(Bt' 0)^p) = 'j(G=ss0)
= ss
But 'j(G=ss) = 0 for all ss in Ocp(G) and every j 2. Thus for j 2 one gets
ij(') = 0. On the other hand, for j = 1, one obtains a natural equivalence of
functors on Ocp(G)
~= '
Z ____1
*
* __
which induces the required isomorphism upon taking higher limits. *
*__
16 CARLOS BROTO AND RAN LEVI
Let G : Out (BG^p) _____ Aut(IG ) denote the map constructed in Lemma 4.3.
The discussion above now implies the following theorem, which in particular giv*
*es
Theorem 1.5.
Theorem 4.10. Let G be a finite group and let ' 2 Aut(IG ) be any natural equ*
*iva
lence. Let
E1': hocolim!(1)()hG ____BG^p
Ocp(G)
be a map constructed by the procedure described above. Then
1. The obstruction to lifting E1'to a map
E' : hocolim(OEG )hG ____BG^p
Ocp(G)
such that G ([(E')^p]) = ' is an element in J2(G).
2. If some lifting E' of ' exists then all homotopy classes of lifts are in 1*
*1 corre
spondence with J1(G).
In other words the sequence
G
0 ____J1(G) ____ Out(BG^p) ____ Aut(IG )
is exact and if J2(G) = 0 then G is an epimorphism.
Remark 4.11. The group of automorphisms of the functor IG is strongly related*
* to
self equivalences of the Frobenius category of G, namely, the category whose ob*
*jects
are all psubgroups of G and whose morphisms are all homomorphisms induced by
inclusions and conjugations. These equivalences are frequently quite easy to de*
*scribe.
Thus one may wonder about the case when the map G is in fact an isomorphism.
By the theorem this is the case when the groups Ji(G) vanish for i = 1; 2. Belo*
*w we
compute an example where J1(G) 6= 0. This may suggest that Ji(G) is nonvanishi*
*ng
in general. However, it would be interesting to examine conditions on the group*
* G
which insure vanishing of the obstruction groups.
5. Sample Calculations
5.1. Normal Sylow psubgroup. Let G be a finite p0reduced group with a Sylow
psubgroup ss and assume that ss is normal in G.
Lemma 5.1. Let G be a p0reduced finite group with a Sylow psubgroup ss C G *
*then
Out(BG^p) ~=Aut (IG ).
Proof.Since ss C G, it is the only psubgroup which is both pcentric and pstu*
*bborn.
Thus we may compute higher limits over the category containing ss alone as an o*
*bject
and W = G=ss as morphisms. But in this case all higher limits vanish and so The*
*orem_
1.5 applies to give the result. *
* __
Proposition 5.2. Let G be a finite p0reduced group with a normal Sylow psubgr*
*oup
ss. Then
Out (BG^p) ' Out(G):
SELF HOMOTOPY EQUIVALENCES 17
Proof.Let [BG; BG^p]* denote the classes of maps which correspond to equivalenc*
*es
of BG^p. Let W = G=ss. Then W operates on EG=ss and one has
(6) [BG; BG^p] = ss0(Map (BG; BG^p)) ~=
ss0(Map ((EG=ss)hW ; BG^p)) ~=ss0((Map (EGss; BG^p))hW ):
To compute [BG; BG^p]* we only need to consider components of inclusions of ss
in G. Let i denote the collection of all faithful representations ss ____G. Th*
*en for
every i 2 iMap (EG=ss; BG)Bi ' BCG (i(ss)) ' BZ(ss) is pcomplete. Thus
Map (EG=ss; BG^p)i' (Map (EG=ss; BG)i)^p' Map (EG=ss; BG)i
so
(7) ss0((Map (EG=ss; BG^p)i)hW ) ' ss0((Map (EG=ss; BG)i)hW ) '
ss0(Map (BG; BG)equiv) = Out(BG) = Out(G):
__
This completes the proof. _*
*_
5.2. The Groups SL2(q) and P SL2(q) at 2. Let SL2(q) denote the special linear
group over Fq, where q = pn is an odd prime power. Then SL2(q) < GL2(q), the
general linear group over Fq and the quotients of both by their respective cent*
*ers
give the projective groups P SL2(q) and P GL2(q) respectively. Diagrammatically*
* one
has the following diagram, where rows are group extensions and columns are cent*
*ral
extensions.
Z=2 ________F*q_____Z=q1_2Z
  
  
? ? det ?
(8) SL2(q) ____GL2(q) ______F*q
  
  
? ? ?
P SL2(q)___P GL2(q)____Z=2
det *
Observe that the composition F*q____GL2(q) ____Fq is the squaring map.
The outer automorphism group of P SL2(q) is given by
Out (P SL2(q)) ~=Z=2 x Z=n
where Z=2 is generated by ff given as the outer action defined by the extension*
* in the
bottom row of the above diagram, and Z=n ~=Gal(FqFp) if q = pn, generated by t*
*he
Frobenius automorphism OE, acts in the obvious way.
The Sylow 2subgroup of P SL2(q) is a dihedral 2group
s1 2 2
D2s=
of order 2s depending on q. For s = 2, D4 is elementary abelian of rank two, he*
*nce
Aut(D4) = Out(D4) ~=3. The automorphism group of D2s is also easily described
18 CARLOS BROTO AND RAN LEVI
as the semidirect product
Aut(D2s) = Z=2s1o (Z=2s1)*;
where an element l 2 Z=2s1 corresponds to the automorphism glwith gl(x) = x and
gl(y) = xly, and an element a 2 (Z=2s1)* corresponds to the automorphism fa su*
*ch
that fa(x) = xa and fa(y) = y. One verifies that glwith l even and f1 are the *
*inner
automorphisms and since (Z=4)* ~= (Z=2) and (Z=2s1)* ~= Z=2 x Z=2s3, s 4,
generated by 1 and 3 modulo 2s1 one obtains
Out(D4) = 3
Out(D8) ~=Z=2
Out(D2s) ~=Z=2 x Z=2s3; s 4
generated by the classes g1(order 2) and f3( order 2s3) of g1 and f3 respectiv*
*ely.
Proposition 5.3. Assume that q 1 (mod 8), q = pn, p an odd prime number, and
let s be the largest integer such that 2s  q  1. There is a homomorphism defi*
*ned by
restriction
Out (P SL2(q)) ____ Out(D2s)
that sends ff to g1 and OE to fp.
s2
In particular, if q = 32 , s 3, then there is an extension
1 ____Z=2 ____ Out(P SL2(q)) ____ Out(D2s) ____1
s3
where Z=2 in the kernel is represented by OE2 .
Proof.Let ffl be a 2s root of unity in Fq. The 2Sylow subgroup of P SL2(q) is *
*D2s,
generated by the classes of the matrices X = ffl00ffl1and Y = 0110:
s1 2 2
D2s~= P SL2(Fq) :
Since ffl is not a square in Fq, the class matrix A = ffl001provides a set t*
*heoretic
section of the bottom extension in diagram (8), and therefore the outer automor*
*phism
ff of P SL2(Fq) is described as conjugation by A in P GL2(Fq). One can now check
that this conjugation leaves the given 2Sylow subgroup stable and induces the *
*outer
automorphism g1of D2s. On the other hand the Frobenius OE is defined as pth po*
*wer
on Fq and then it induces fpon D2s.
s2 s3
In case q = 32 , s 3, f3is a generator of Z=2 Out(D2s), so the restrict*
*ion
s3 *
* __
is an epimorphism with kernel clearly generated by OE2 that has order 2. *
* __
Lemma 5.4. There are isomorphism
~=
Aut(SL2(q)) ____ Aut(P SL2(q))
and
~=
Out (SL2(q)) ____ Out(P SL2(q)) :
SELF HOMOTOPY EQUIVALENCES 19
Proof.Since P SL2(q) is the quotient of SL2(q) by its center, any automorphism
of SL2(q) induces an automorphism of P SL2(q). Conversely, any automorphism of
P SL2(q) preserves the extension class for SL2(q) and thus induces an automorph*
*ism
of SL2(q). This proves the first statement. For the second statement, notice th*
*at
__
Inn(SL2(q)) ~=SL2(q)=Z(SL2(q)) ~=P SL2(q) ~=Inn(P SL2(q)): __
Next, compute the automorphisms of the generalized quaternion 2groups,
s1 2 2s 1 1
Q2s+1= ;
which appear as the Sylow 2subgroups of SL2(q), q odd. Thus for Q2s+1one has t*
*he
automorphisms gldefined by gl(x) = x and gl(y) = xly for l 2 Z=2s and fa such t*
*hat
fa(x) = xa and fa(y) = y for a 2 (Z=2s)*. This gives Aut(Q2s+1) ~=Z=2so (Z=2s)**
* if
s 3. s1
The center of Q2s+1is cyclic of order 2 generated by x2 = y2 and the quotie*
*nt
of Q2s+1by its center is isomorphic to D2s. There is an induced homomorphism
Aut(Q2s+1) ____ Aut(D2s), which is an epimorphism with kernel Z=2xZ=2 generated
by f2s1+1:
s1+1
f2s1+1(x) = x2 ; f2s1+1(y) = y
and g2s1:
s1
g2s1(x) = x; g2s1(y) = x2 y :
Notice that g2s1is inner while f2s1+1it not inner unless s = 2, and therefore
Out (Q8) ~=Out (D4) ~=3
and for s 3, we obtain an extension
1 ____Z=2 ____ Out(Q2s+1) ____ Out(D2s) ____1:
The following Proposition describes thesrelationships between the outer automor
phism groups computed above for q = 32 , s 1.
s2
Proposition 5.5. Assume that q = 32 , s 3. There is an isomorphism
ss : Out(SL2(Fq)) ____ Out(Q2s+1)
and a commutative diagram
Z=2
?


~= ?
Out(SL2(q)) ____Out(Q2s+1)

~= 
? ??
Z=2___Out (P SL2(q))___Out(D2s)
*
*__
where the bottom row and right column are exact. *
*__
20 CARLOS BROTO AND RAN LEVI
Homology decomposition of SL2(q) and P SL2(q). A homology decomposition
of P SL2(q) is described in detail in [1]. We refer to [4] for the general theo*
*ry. Restrict
s2
attention to the case where q = 32 , s 3. Choose a Sylow subgroup S ~=D2s. Th*
*en
there are subgroups Z < S of order 2 given by the center of S and two nonconju*
*gate
elementary abelian 2subgroups V; W < S of rank 2, which give an ample collecti*
*on
E2 = {Z; V; W } of elementary abelian 2subgroups of P SL2(Fq). The associated
conjugacy category AE2 can be described by means of the following diagram
oe____ _____
3 W oe____oe_Z ______V 3 :
3=2 3=2
The centralizer diagram ffE2: AopE2___Spaces is up to homotopy and 2completi*
*on
described by ____ oe___
3 BW _____BD2s oe___oe_BV 3:
and the natural map
(9) aE2: hocolimffE2 ____BP SL2(Fq)
induces a mod2 homology isomorphism.
For SL2(q) one obtains a homology decomposition by pulling back ffE2 along the
s2
projection SL2(q) _____P SL2(q) (q = 32 , s 3 as above). We obtain a new
strictly commutative diagram fiE2 of the form
____ oe__
3 BQ8 _____BQ2s+1 oe__oe_BQ8 3:
One can view fiE2 as a functor from AE2 to Spaces and there is a map bE2 out of*
* the
diagram to BSL2(q) given by the pullback process described above.
Lemma 5.6. The map
bE2: hocolimfiE2 ____BSL2(q)
AE2
induces a mod2 equivalence.
*
* __
Proof.This follows from the homology decomposition in equation (9). *
*__
The group Out (BG^2) for G = SL2(q) and P SL2(q). The left column of diagram
(8) induces a principal fibration
Bi ^ Bp ^
(10) BZ=2 ____BSL2(q)2 ____BP SL2(q)2:
Lemma 5.7. There is an isomorphism
:Out (BSL2(q)^2) ____ Out(BP SL2(q)^2) :
Proof.There is a homotopy equivalence Map (BZ=2; BP SL2(q)^2)c ' BP SL2(q)^2,
where c denotes the constant map (cf. [1]). Then the Zabrodsky lemma [14, 10]
applies to the principal fibration (10) to give a homotopy equivalence
Map (BSL2(q)^2; BP SL2(q)^2){ffOBi'*}' Map (BP SL2(q)^2; BP SL2(q)^2) :
Now, any homotopy equivalence g 2 Out (BSL2(q)^2) induces the identity in mod2
cohomology. Hence g O Bi ' Bi and so Bp O g O Bi ' *. It follows that there exi*
*sts
g 2 Out(BP SL2(q)^2) satisfying Bp O g ' gO Bp. Define (g) = g.
SELF HOMOTOPY EQUIVALENCES 21
Finally, we observe that [BSL2(q)^2); BZ=2] ~= [BSL2(q)^2); B2Z=2] ~= 0, henc*
*e_
turns out to be an isomorphism. _*
*_
Lemma 5.8. Let G be a finite group and let S be its Sylow psubgroup. Assume *
*that S
res
is self normalizing in G. Then there is a homomorphism res : Out(G) ____ Out(S*
*),
which factors through Out (BG^p).
Proof.Let OEbe an outer automorphism of G represented by some automorphism OE.
Then OE carries S into another Sylow psubgroup S0. But there is an inner auto
morphism cg of G which carries S0 back to S. Define res(OE) to be the class of*
* the
composition cg O OE in Out (S). If g02 G is another element conjugating S0 to S*
* then
g0g1 2 NG (S) = S: Hence cg O OE and cg0O OE differ only by an inner automorph*
*ism of
S. Also if OE0is another representative for OEthen OE and OE0differ by an inner*
* automor
phism of G and the procedure carries the difference again into an inner automor*
*phism
of S. Thus the restriction map is well defined and obviously a group homomorphi*
*sm.
Similarly, if 2 Out(BG^p) then the composition
B ^ OE ^
BS ____BGp ____BGp;
where denotes the inclusion, is homotopic to a map induced by a homomorphism
(see Proposition 2.1). Thus the same argument implies that there is an automorp*
*hism
ff of S such that Bff ' B.
Finally if 2 Out(BG^p) is induced by an automorphism OE of G, then the proc*
*edure
*
* __
described above gives that ff is conjugate to res(OE) and the lemma follows. *
* __
Remark 5.9. Lemma 5.8 applies to G = SL2(q) and G = P SL2(q), q 1
(mod 8). In fact, the centralizer in P SL2(q) of the center Z ~= Z=2 of its S*
*ylow
2subgroup is isomorphic to Dq1, the dihedral group of order 2q  2, generated*
* by
the classes of the matrices i0i01and 0110, where i is a generator of F*q(Pr*
*opo
sition 4.2 of [1]). Then NPSL2(q)(D2s) Dq1 and a quick calculation shows that
NPSL2(q)(D2s) = D2s. From this equality it follows that NSL2(q)(Q2s+1) = Q2s+1*
*as
well. s2
In particular, for q = 32 , s 3, we have a commutative diagram
~=
Out (SL2(q))____Out (Q2s+1)
(11)  res
?
Out (BSL2(q)^2)
s2
Proposition 5.10. For q = 32 , s 3, the natural map
B :Out (SL2(q)) ____ Out(BSL2(q)^2)
is an isomorphism.
Proof.It suffices to show that
res: Out (BSL2(q)^2) ____ Out(BQ2s+1)
22 CARLOS BROTO AND RAN LEVI
is an isomorphism. It is clear from Diagram (11) that res is an epimorphism. We*
* use
the homology decomposition for BSL2(q)^2of Lemma 5.6 in order to prove injectiv*
*ity.
The class of a self equivalence f of BSL2(q)^2is in the kernel the restrictio*
*n map if
and only if the diagram
BQ2s+1===== BQ2s+1
B  B
? f ?
BSL2(q)^2___BSL2(q)^2
is homotopy commutative, where denotes the inclusion map. Then, for every obje*
*ct
in the diagram fiE2 we have
fBQ2s+1' IdBQ2s+1
fBQ8 ' IdBQ8:
The obstructions for f to be homotopic to the identity lie in the groups
^
limissj Map (fiE2; BSL2(q)2)B. ; i 1
AE2
and according to Proposition 5.12 these groups are trivial. Hence f ' Id, that *
*is_res
is injective. *
*__
s2
Proposition 5.11. For q = 32 , s 3,
1. Map (BQ8; BSL2(q)^2)Bffl' BZ=2, where ffl = 1; 2 denotes the two different*
* in
clusions.
2. Map (BQ2s+1; BSL2(q)^2)B ' BZ=2.
Proof.By Proposition 2.1 these spaces are equivalent to the 2completion of the
respective centralizer. The claim now follows by direct calculation using Rema*
*rk_
5.9 __
s2
Proposition 5.12. For q = 32 , s 3,
(
Z=2 i = 0; j = 1
limissjMap (fiE2; BSL2(q)^2)B. ~=
AE2 0 otherwise.
*
* __
Proof.This follows at once from Proposition 5.11 and [1, x10]. *
* __
The discussion above implies Theorem 1.6. Specifically, part 1 follows by com*
*bining
Proposition 5.5, Lemma 5.7 and Proposition 5.10. Part 2 follows from the calcul*
*ation
of Proposition 5.12 and part 3 follows from [1, Lemma 6.4].
6. Modp Homotopy Group Extensions
In this final section we return to the motivation for our study and discuss m*
*odp
homotopy group extensions. The following table compares the classification of g*
*roup
extensions to the classification of homotopy group extensions. The last is of c*
*ourse
just a special case of the general classification problem for fibrations.
SELF HOMOTOPY EQUIVALENCES 23
____________________________________________________________________________*
*
  Group Extensions  Modp Homotopy Group Extensions  
_____________ss_by_G____________________________ss_by_G_________________
  1Ahomomorphism ss ____ffOut(G)  A map Bss^ ____ffB Out(BG^)  
______________________________________________p________________p________*
*
 Obstructionsto the existence Obstructionsto the existence of any *
*
2 of any3extension of ss by G modphomotopy3extension^of ss by G in*
* 
 inH (ss; Z(G)), where Z(G) H(Bssp;Z(G=Op0G)),^where Z(G=Op0G)   *
* 
_____becomes_a_ssmodule_via_ff_____becomes_a_ss1(Bssp)module_via_ff___*
*
 Ifan extension exists then all  If an extension exists *
*   
3 extensions2are classified by  then all2the^others are classifiedby*
*  
___________H_(ss;_Z(G))___________________H_(Bssp;_Z(G=Op0G))___________
Corollary 6.1. Let G be a finite group such that the natural map
flG : Out(G=Op0G) ____ Out(BG^p)
is an isomorphism. Let ss be a finite pgroup. Then there is a 11 corresponden*
*ce be
tween fibre homotopy classes of modp homotopy extensions of ss by G and equiva*
*lence
classes of ordinary group extensions of ss by G=Op0G.
Proof.Under our hypotheses, there is a homotopy equivalence
' ^
B Aut(B(G=Op0G)) ____B Aut(BGp):
__
The result follows. _*
*_
Recall that a pgroup P is called a Swan group if for any finite group G cont*
*aining
P as a Sylow psubgroup, the inclusion of the normalizer NG (P ) ____G induces*
* a
modp homology isomorphism.
Corollary 6.2. Let G be a finite group with a Sylow psubgroup P and assume eit*
*her
1. P is normal in G or
2. the inclusion NG (P ) ____G induces a modp homology equivalence or
3. P is a Swan group.
In either case let H denote the modp0 reduction of NG (P ) and let ss be any f*
*inite
pgroup. Then fibre homotopy classes of modp homotopy extensions of ss by G ar*
*e in
11 correspondence with ordinary group extensions of ss by H.
Proof.Under either one of 1, 2 or 3, we may replace BG^pby BH^pup to homotopy. *
* __
The result follows at once from Corollary 6.1. *
* __
Our calculation for the special linear groups implies a similar result. Speci*
*fically
we have
s2
Proposition 6.3. Assume q = 32 , s 3, then there is a homotopy equivalence
' ^
B Aut(BSL2(q)) ____B Aut(BSL2(q)2) :
24 CARLOS BROTO AND RAN LEVI
Proof.Since pcompletion is a continuous functor, there is a diagram of fibrati*
*ons
B SAut(BSL2(q)) ____B Aut(BSL2(q)) _____B Out(SL2(q))
  
  
? ? ?
B SAut(BSL2(q)^2) ___B Aut(BSL2(q)^2)___B Out(BSL2(Fq)^2)
where the left vertical arrow is a homotopy equivalence by Theorem 1.1 and the
right vertical arrow is a homotopy equivalence by Proposition 5.10, hence the_r*
*esult_
follows. __
Remark 6.4. Notice that we have actually computed the spaces B Aut(BSL2(q)^2)
and B Aut(BP SL2(q)^2) for every odd prime power q = pk. In fact, according to *
*[1]
the homotopy types of BSL2(q)^2and BP SL2(q)^2depends only on the order of the
Sylow 2subgroup rather than on the concrete odd prime power q. Observesthat any
order of a Sylow 2subgroup in SL2(q) can be obtained by letting q = 32 for so*
*me
s 1 if q 1 mod 8 and that if q 3 mod 8 then BSL2(q)^2' BSL2(3)^2,
in which case the Sylow 2subgroup Q8, and respectively BP SL2(3)^2' (BA4)^2
with Sylow 2subgroup elementary abelian of rank 2. In these two cases the Syl*
*ow
2subgroups are Swan groups.
Corollary 6.5. Let ss be a finite 2group and let q be any odd prime power. Th*
*en
there is a 11 correspondence between fibre homotopy classes mod2 homotopy ext*
*en
sions of ss by SL2(q)sand equivalence classes of ordinary group extensions of s*
*s by
SL2(t) where t = 32 for some s such that SL2(q) and SL2(t) have Sylow 2subgrou*
*ps
of the same order if q 1 mod 8 and t = 3 otherwise. The corresponding result
applies to homotopy extensions of a finite 2group by P SL2(q).
Our interest in this project was motivated by a rather simple minded question,
namely homotopy uniqueness of the space BQ2n [2]. Our results here enable us to
give an easy solution of this problem.
Corollary 6.6. Let Q2rbe a generalized quaternion group of order 2r. Let X be a*
* 2
complete space with H*(X) ~=H*(BQ2r) as an algebra over the Steenrod algebra and
assume further that there is an isomorphism between the Bockstein spectral sequ*
*ences
of H*(X) and H*(BQ2r) in the sense of [2].Then there is a homotopy equivalence
X ' BQ2r
Proof.It is shown in [2] that under our hypotheses if X is not equivalent to BQ*
*2r
then ss1(X) ~=Q2sfor some s < r  2 and its universal cover "Xhas the cohomolog*
*y of
BSL2(q) for some appropriate q, again as an algebra over the Steenrod algebra a*
*nd
with the same Bockstein spectral sequence. By [1] it follows that X" ' BSL2(q)*
*^2.
The previous corollary thus implies that X is the 2completion of the classifyi*
*ng space
of an extension of Q2sby SL2(t) for an appropriate t. But one now easily checks*
* that_
no such extension has the cohomology assumed for X. The result follows. *
*__
To conclude this paper we comment that homotopy group extensions were defined
to be fibrations where both base and fibre are pcompleted classifying spaces. *
* In
SELF HOMOTOPY EQUIVALENCES 25
some contexts, in particular if whether or not the total space is pcomplete ha*
*s no
significance, it makes sense to consider fibrations with base Bss (rather than *
*Bss^p),
and fibre BG^p.
Corollary 6.7. Let G be a finite group and assume that the natural map
flG : Out(G=Op0G) ____ Out(BG^p)
is an isomorphism. Then for any discrete group ss there is a 11 correspondence
between fibre homotopy equivalence classes of fibrations with base Bss and fibr*
*e BG^p
and equivalence classes of ordinary group extensions of ss by G=Op0G.
Finally, notice that in the case of the foregoing corollary, the corresponden*
*ce is
given via the BousfieldKan fibrewise pcompletion functor.
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26 CARLOS BROTO AND RAN LEVI
Departament de Matematiques, Universitat Autonoma de Barcelona, E08193 Bel
laterra, Spain
Email address: broto@mat.uab.es
Department of Mathematical Sciences, University of Aberdeen, Meston Building
339, Aberdeen AB24 3UE, U.K.
Email address: ran@maths.aberdeen.ac.uk