DIAGRAMS UP TO COHOMOLOGY
W.G. Dwyer and C.W. Wilkerson
University of Notre Dame
Purdue University
1. Introduction
Let Sp denote the category of spaces, Ho the associated homotopy category, *
*and D a
small (index) category. A diagram in Ho with the shape of D is by definition a*
* functor
F : D!- Ho . Given such a diagram one can ask whether or not it has a realizat*
*ion, i.e., a
lift to a functor D!- Sp, and if so, how many realizations there are up to an *
*appropriate
kind of equivalence. This question is studied in [3] and [5] for general diagra*
*ms and in [6] for
the special case of "centric" diagrams (see x6). In this paper we look at a rel*
*ated question.
Let p be a fixed prime number and let H *denote the cohomology functor H *(-; F*
*p). If
F : D!- Ho is a functor, an H*-clone of F is by definition a collection (G; {s*
*d}) where
(1) G : D!- Ho is a functor,
(2) for each object d of D, sd : F (d)!- G(d) is an isomorphism, and
(3) for each morphism g : d!- e of D, the diagram
F(g)* *
H *F (d)---- H F (e)
x x
s*d?? s*e??
G(g)* *
H *G(d) ---- H G(e)
commutes.
In other words, an H *-clone of F is another diagram in the homotopy category w*
*hich is
built from essentially the same spaces as F and looks exactly the same as F fro*
*m the
point of view of mod p cohomology. An equivalence or isomorphism between two H**
*-clones
(G; {sd}) and (G0; {s0d}) is a natural equivalence t : G!- G0 with the propert*
*y that for
each object d of D, the composite
sd td 0 (s0d)-1
F (d) -! G(d) -! G (d) ----! F (d)
_____________
The authors were supported in part by the National Science Foundation.
Typeset by AM S-*
*TEX
1
2 W. DWYER AND C. WILKERSON
gives the identity map of H* F (d). Our main theorem (2.4) shows how to compute*
* the set
of isomorphism classes of H*-clones of F in the special case in which F satisfi*
*es a "centric"
condition (2.1) analogous to that of [6]. We also work out some examples (x3, *
*x5) and
explain (6.7) how our results relate to those of [6].
The motivation for this comes from the following example. Suppose that X is*
* a p-
compact group [10], for instance, X might be the p-completion G^pof a compact L*
*ie group
G such that ss0G is a finite p-group. Given X , the decomposition method of [11*
*] (see also
[8] and [12]) produces
(1) a category AX ,
(2) a functor ffX : AopX-!Sp , and
(3) a natural homotopy equivalence BX ~ (hocolim ffX )^p.
Let ffXdenote the diagram in the homotopy category which underlies ffX . We mak*
*e the
following conjecture, which is related to the question of whether or not the ho*
*motopy type
of BX is determined by the cohomology ring H* BX as an algebra over the mod p S*
*teenrod
algebra.
1.1 Conjecture. Let X be a p-compact group (perhaps connected). Then ffX has no
nontrivial H*-clones, i.e., any H*-clone of ffXis isomorphic to ffX.
We show below (x3) that the machinery of this paper applies to ffX; this giv*
*es a way to
check 1.1 in particular cases.
Organization of the paper. Section 2.4 enumerates the set of H *-clones of diag*
*rams of
a certain special type, and x3 shows that this enumeration applies to the case *
*of the
decomposition diagram of a p-compact group. In x4 there is a detailed computat*
*ional
analysis of the "lim1" which comes up in the examples from x3; we use this in x*
*5 to prove
Conjecture 1.1 for the F2-completion of the exceptional Lie group G2. Calculati*
*ons similar
to the ones in x5 also appear in the work of K. Premadasa [18]. Finally, x6 de*
*scribes a
homotopy limit which combines the ideas here with those of [6] to give, for a s*
*uitable
diagram in the "cohomology category" an efficient computation of the set of equ*
*ivalence
classes of its realizations. At the end of the paper (6.8) there is a conjectur*
*e which is in a
sense a generalization of 1.1.
Notation and terminology. The category Sp can be taken to be either the catego*
*ry of
topological spaces or of simplicial sets. When forming a function space Map (A*
*; B), we
assume in the first case that A has been replaced if necessary by a weakly equi*
*valent
cofibrant object (i.e. CW-complex ) and in the second that B has been replaced*
* if necessary
by a weakly equivalent fibrant object (i.e. Kan complex). The homotopy category*
* Ho is
the category obtained from Sp by formally inverting all weak equivalences.
If A is a space, A^pdenotes the Fp-completion of A in the sense of [1].
x2. The main theorem
If f : A!- B is a map in Ho , let [A; B]{f} denote the set of maps f0 : A!-*
* B in Ho
such that H*(f0) = H*(f).
2.1 Definition. A map f : A!- B in Ho is said to be H*-centric up to homotopy*
* if
(1) [A; A]{id}is a group under composition, and
DIAGRAMS 3
~=
(2) composition with f induces a bijection [A; A]{id}-! [A; B]{f}.
A functor F : D!- Ho is said to be H*-centric up to homotopy if F (g) satisfie*
*s the above
conditions for each morphism g of D.
2.2 Remarks. We will define the stronger notion of H*-centric later on (6.3). N*
*ote that if
A is a space with is Fp-complete [1] then A satisfies condition (1) above; this*
* follows from
the fact that a map between Fp-complete spaces is a weak equivalence if and onl*
*y if it
induces an isomorphism on mod p cohomology.
Example. If ff : G1!- G2 is a map of finite groups, denote by ff* the induced *
*cohomology
map H *(B ff). Let i : G!- H be an inclusion of finite p-groups, N(G) the nor*
*malizer
of G in H, and Nin(G) N(G) the subgroup of N(G) consisting of elements whose
conjugation action on G is via inner automorphisms. Since inner automorphisms o*
*f a group
act trivially on the cohomology of the classifying space, there is a natural ho*
*momorphism
N(G)=Nin(G)!- Aut (H *B G). It is not hard to see that the map B i : B G!- B*
* H is
H *-centric up to homotopy if and only if
(1) for any homomorphism j : G!- H with H *(B j) = H*(B i) the subgroup j(*
*G) of
H is conjugate to G, and
(2) the N(G)=Nin(G)!- Aut(H *B G) is injective, and contains ff* for every*
* automor-
phism ff of G such that ff*i* = i*.
It follows from 3.1 that these conditions are satisfied if G is the centralizer*
* in H of an
elementary abelian p-group.
Suppose that F : D!- Ho is a functor which is H *-centric up to homotopy. *
*Let Grp
denote the category of groups, and construct a functor OEF : Dop!- Grp by sett*
*ing
OEF (d) = [F (d); F (d)]{id}
for each object d of D. For a map g : d!- e of D the induced homomorphism OEF*
* (g) :
OEF (e)!- OEF (d) assigns to a map v 2 [F (e); F (e)]{id}the unique element u *
*of OEF (d) such
that the following diagram
u
F (d)----! F (d)
? ?
(2.3) F(g)?y F(g)?y
v
F (e)----! F (e)
commutes. The existence and uniqueness of u follow immediately from the fact th*
*at F (g)
is H*-centric up to homotopy. Given elements v and v0 in OEF (e), let u and u0d*
*enote their
respective images in OEF (d) and consider the following diagram
u u0
F (d) ----! F (d)----! F (d)
? ? ?
F(g)?y F(g)?y F(g)?y :
v v0
F (e) ----! F (e)----! F (e)
4 W. DWYER AND C. WILKERSON
By choice of u and u0 each small square commutes. It follows that the large sq*
*uare
commutes, which shows that the image under OEF (g) of v0v is u0u and thus that *
*OEF (g) is a
group homomorphism.
Given a functor F : D!- Ho , let C`(F ) denote the set of equivalence class*
*es (x1) of
H *-clones of F . This is a pointed set with the class of (F; {idd}) as basepoi*
*nt.
2.4 Theorem. If F : D!- Ho is a functor which is H *-centric up to homotopy, *
*then
there is a natural bijection of pointed sets
C`(F ) ~=lim1OEF
and a natural isomorphism of groups
Aut(F; {idd}) ~=limOEF :
2.5 Description of lim1. Before proving 2.4 we will give an explicit descriptio*
*n of the
functor "lim1" (also see [1, p. 307]). Suppose that D is a small category and *
*that OE :
Dop!- Grp is a functor. For convenience, if g : d0!- d1 is a morphism of D we*
* will write
g] : OE(d1)!- OE(d0) for the map OE(g). Let O denote the set of objects of D, *
*M the set of
morphisms of D, and S the disjoint union
[
S = OE(d) :
d2O
The object S is just a set, or, perhaps better, the collection of morphisms in *
*a groupoid
with object set O. Let C0(OE) denote the set of all functions c : O!- S such t*
*hat for each
d 2 O, c(d) 2 OE(d). It is clear that C0(OE) is a group under pointwise multipl*
*ication. Let
C1(OE) denote the set of all functions c : M!- S such that for each g : d0!- *
*d1 in M,
c(g) 2 OE(d0). An element c 2 C1(OE) is called a "normalized 1-cocycle" if
(1) for each identity map g : d0!- d1, c(g) is the identity element of OE(*
*d0), and
(2) for each pair g0 : d0!- d1 and g1 : d1!- d2 of composable elements of*
* M, the
product
g]0(c(g1))c(g1g0)-1c(g0)
is the identity element of OE(d0).
Let Z1(OE) C1(OE) denote the collection of normalized 1-cocycles; this is a po*
*inted set
with basepoint given by the element z 2 Z1(OE) which assigns to each g : d0!- *
*d1 in M
the identity element of OE(d0). There is an action of C0(OE) on Z1(OE) such tha*
*t if c 2 C0(OE)
and z 2 Z1(OE) then c . z = z0, where for each g : d0!- d1 in M,
z0(g) = c(d0)z(g)g](c(d1)-1) :
By definition, the orbit set of this action is lim1OE. The basepoint of Z1(OE) *
*projects to a
basepoint for lim1OE.
DIAGRAMS 5
Proof of 2.4. Say that an H*-clone (G; {sd}) of F is special if for each object*
* d of D, G(d) =
F (d) and sd is the identity map of F (d). It is immediate that any H *-clone (*
*G0; {s0d}) of
F is equivalent to a special H *-clone (G; {sd}); for instance, set G(g) = (s0d*
*1)-1G0(g)s0d0
for each morphism g : d0!- d1 of D. It follows that we can interpret C`(F ) *
*as the
set of equivalence classes of special H *-clones of F . If G is a special H *-*
*clone of F , let
zG 2 C1(OEF ) be the function which assigns to each morphism g : d0!- d1 of D *
*the unique
element h 2 [F (d0); F (d0)]{id}which makes the following diagram commute
h
F (d0)----! F (d0)
? ?
F(g)?y G(g)?y :
id
F (d1)----! F (d1)
The existence and uniqueness of h follows immediately from the fact that F (g) *
*is H*-centric
up to homotopy. Clearly zG (g) is the appropriate identity element if g is an i*
*dentity map.
The commutative diagram
g]0(zG(g1)) zG(g0)
F (d0)-------! F (d0)----! F (d0)
? ? ?
F(g0)?y F(g0)?y G(g0)?y
zG(g1) id
F (d1) ----! F (d1)----! F (d1)
? ? ?
F(g1)?y G(g1)?y G(g1)?y
id id
F (d2) ----! F (d2)----! F (d2)
shows that zG (g1g0) = zG (g0)g]0(zG (g1)) and thus that zG 2 Z1(OEF ). It is e*
*asy to check
directly that the image of zG in lim1OEF depends only on the equivalenc*
*e class of
the special H *-clone G, and that the assignment G 7! gives the desired b*
*ijection
C`(F )!- lim1OEF . The identification of limOEF is straightforward.
x3. The basic example
In this section we will give the motivating example of a diagram which is H**
*-centric up to
homotopy. Suppose that X is a p-compact group with classifying space BX . Let A*
*X denote
the category whose objects are the pairs (V; Bf), where V is a nontrivial eleme*
*ntary abelian
p-group and f : V!- X is a conjugacy class of monomorphisms [11, x8]. By def*
*inition,
giving f amounts to giving an ordinary homotopy class of maps Bf : BV!- B X su*
*ch that
H *B V is finitely generated as a module over (B f)*(H *B X ). A morphism (V; f*
*)!- (V 0; f0)
in AX is an injection i : V!- V 0such that f0. i is conjugate to f (equivalent*
*ly, (B f0) . (B i)
is homotopic to Bf). There is a functor ffX : AopX-!Sp given by
ffX (V; f) = Map (B V; BX )Bf
6 W. DWYER AND C. WILKERSON
where the subscript "B f" denotes the mapping space component corresponding to *
*the ho-
motopy class Bf. This diagram is interesting because there is a natural map hoc*
*olimffX!-
B X which induces an isomorphism on mod p homology as well as a weak equivalence
(hocolim ffX )^p~ BX [11, 8.1].
Let ffX: AopX-!Ho denote the diagram in the homotopy category which underli*
*es ffX .
3.1 Theorem. For any p-compact group X the diagram ffXis H*-centric up to homot*
*opy.
The proof depends on three lemmas. In these lemmas, Z denotes a p-compact gr*
*oup,
V an elementary abelian p-group, and i : V !- Z a monomorphism (see above). *
*Let
B Y denote the mapping space component Map (B V; BZ)Bi. (The space Y = BY *
*is a
p-compact group which is called the centralizer of V in Z.) Evaluation at the b*
*asepoint
of BV gives a map Bj : BY!- B Z. The map Bi lifts to a map Bi0: BV!- B Y such*
* that
(B j) . (B i0) = Bi. This lift has two key properties.
o The map Bi0 is central [11, 2.7] in the sense that evaluation at the ba*
*sepoint of
BV gives a weak equivalence
~
(3.2) Map (B V; BY )Bi0-! B Y :
o The map Bi0 extends to a principal fibration sequence [11, 2.8]
Bi0
(3.3) B V --! BY!- B(Y=V ) :
The lift Bi0is obtained from the usual abelian group structure on BV [10, 8.2],*
* and sends
y 2 BV to the map BV!- B X which takes x 2 BV to Bj(x + y).
3.4 Lemma. Consider the commutative diagram
v=Map(id;Bj)
Map (B V; BY )Bi0 ---------! B Y = Map (B V; BZ)Bi
? ?
(3.5) u?y ?yBj
Bj
B Y = Map (B V; BZ)Bi ----! BZ
in which the vertical maps are given by evaluation at the basepoint of BV . The*
*n the maps
u and v are homotopy equivalences and, if u-1 is the inverse of u in Ho , the c*
*omposite
vu-1 is equal in Ho to the identity map of Y .
Proof. Let k be the composite of Bi with the map : BV xB V = B(V xV )!- BV in*
*duced
by addition V x V!- V . The space in the upper left hand corner of 3.5 can be *
*identified
as Map (B V x BV; BZ)k. Under this identification the map u corresponds to rest*
*riction to
the factor * x BV and v corresponds to restriction to the factor BV x *. Compos*
*ition with
thus gives a map
w : Map (B V; BZ)Bi!- Map (B V x BV; BZ)k
DIAGRAMS 7
such that uw and vw are identity maps. The map u is an equivalence as above (3.*
*2); this
implies that w and hence v are also equivalences. The composite v . u-1 is then*
* the same
in Ho as the composite (vw)(uw)-1 of two identity maps. .
Let Map (B Y; BY )[Bi0]denote the space of all maps h : BY!- B Y such that *
*h . (B i0) is
homotopic to Bi0. Similarly, let Map (B Y; BZ)[Bi]denote the space of all maps *
*h : BY!-
B Z such that h . (B i) is homotopic to Bi.
3.6 Lemma. The map Bj : BY!- B Z induces an equivalence
~
Map (B Y; BY )[Bi0]-!Map (B Y; BZ)[Bi] :
Proof. Since (B j) . (B i0) = Bi, it is clear that composition with Bj gives a *
*map of the indi-
cated type. By elementary homotopy theory, the fibration sequence 3.3 gives ris*
*e to a fibra-
tion p1 : E1!- B(Y=V ) with fibre Map (B V; BY )Bi0and space of sections Map (*
*B Y; BY )[Bi0].
There is a similar fibration p2 : E2!- B(Y=V ) with fibre Map (B V; BZ)Bi and *
*space of sec-
tions Map (B Y; BZ)[Bi]. The map Bj induces a map between these two fibrations *
*which by
3.4 is a fibrewise equivalence; it follows that the corresponding map on spaces*
* of sections
is an equivalence.
3.7 Lemma. The map Bj : BY!- B Z is H*-centric up to homotopy.
Proof. Since homotopy classes of maps from BV to BZ or BY are detected by their*
* effect
on mod p cohomology [14, 3.1.4], it is clear that there is a commutative diagram
(Bj).(-)
[B Y; BY ]{id} -----! [B Y; BZ]{Bj}
?? ?
y ?y
~=
ss0 Map (B Y; BY )[Bi0]----!ss0 Map (B Y; BZ)[Bi]
in which the vertical arrows are monomorphisms. By 3.6, the lower horizontal ar*
*row is a
bijection. To show that B j is H *-centric up to homotopy it is enough to show*
* that the
upper horizontal arrow is a surjection.
Let h : BY!- B Z be a map which has the same effect on mod p cohomology as *
*Bj;
it is necessary to find a map w : B Y !- BY which induces the identity map on*
* mod p
cohomology and such that (B j) . w is homotopic to h. Consider the commutative *
*diagram
v0=Map(id;h)
Map (B V; BY )Bi0 --------! BY = Map (B V; BZ)Bi
? ?
(3.8) u?y ?yBj
h
BY = Map (B V; BZ)Bi ----! B Z
in which the vertical arrows are given by evaluation at the basepoint of BV . T*
*he map u
is an equivalence by 3.4. We will be done if we can show both that the map v0 g*
*ives an
8 W. DWYER AND C. WILKERSON
isomorphism on mod p cohomology and that the composite v0u-1 induces the identi*
*ty map
on H* BY ; the composite w = v0u-1 will then be the required map with (B j).w h*
*omotopic
to h. Let K denote the category of unstable algebras over the mod p Steenrod al*
*gebra and
T (V; -) : K!- K the functor which is left adjoint to tensor product with H *B*
*V . For a
map fl : H* BV!- R in K, let T (V; R)fldenote the summand or "component" of T *
*(V; R)
corresponding to fl [8, x3]. The inclusion 0!- V induces a natural map efl: *
*T (0; R) =
R!- T (V; R)fl. If A is a space, g : BV!- A is a map , and fl = g* : H* A!- *
*H*B V , there
is a natural commutative diagram
g
T (V; H*A)fl----! H* Map (B V; A)
x x
(3.9) efl?? ??
= *
H *A ----! H A
in which the right hand vertical arrow is induced by evaluation at the basepoin*
*t of BV .
The map g is an isomorphism if, for instance, A is the classifying space of a p*
*-compact
group [11, proof of 8.1]. It is a consequence of this fact and of the naturalit*
*y of 3.9 that
the diagram obtained by applying the functor H *(-) to 3.8 is identical to the *
*diagram
obtained by applying H *(-) to 3.5 (note that there are no choices of isomorphi*
*sms here;
the diagrams are the same). The desired properties of v0 follow from 3.4.
Proof of 3.1. The values of the functor ffX are Fp-complete spaces because they*
* are the
classifying spaces of p-compact groups [11, 2.5]. Let : (V; f)!- (V 0; f0) be*
* a morphism
in AX coming from an injection i : V!- V 0. Write V 0= V 00x i(V ), and let Bf*
*00: BV 00-!
Map (B V; BX )Bf be the map which is adjoint to Bf0 : BV 00x BV!- B X . The ma*
*p ffX ()
can be identified as the map
Map (B V 0; BX )Bf0 = Map (B V 00; Map (B V; BX )Bf)Bf00-! Map (B V; BX*
* )Bf
obtained by evaluating at the basepoint of B V 00. Since Map (B V; BX ) is the*
* classifying
space of a p-compact group [10, 5.1], it follows from 3.7 that ffX () is H *-ce*
*ntric up to
homotopy.
x4. Vanishing lim1
In this section we will give a simple way to check for vanishing of the lim1*
* sets that
arise in applications of 2.4 to the theory of p-compact groups. The formulas in*
* this section
can be interpreted as nonabelian generalizations of formulas of Oliver [17].
The first step, which is mostly for notational convenience, is to reduce to *
*certain very
explicit categories. For each i 1 let Ai denote the Fp vector space (Fp)i and *
*for n 1
let An denote the category whose objects are the vector spaces Ai, 1 i n, an*
*d whose
morphisms are Fp-module monomorphisms. Suppose that X is a p-compact group and *
*that
OE : AX!- Grp is a functor (for instance, OE might be the functor derived fro*
*m ffX (3.1)
by the procedure of x2). Let n be the largest integer such that there is a mono*
*morphism
DIAGRAMS 9
An!- X (see x3); for the existence of such an n see [11, 8.3]. Let : An!- Gr*
*p be the
functor given by the formula Y
(A) = OE(A; f) ;
f
in which the product is taken over the set Mono (A; X ) of all monomorphisms f *
*: A!-
X . The behavior of on a morphism i : A!- B of An is as follows: the (alge*
*braic)
monomorphism i induces a map
i* : Mono (B; X )!- Mono (A; X )
and, given x = {xf} 2 (A), the image of x under (i) is the element y = {yg} 2 (*
*B)
with yg = OE(i)(xi*(g)).
4.1 Lemma. In the above situation, there is a natural isomorphism of pointed se*
*ts be-
tween lim1 and lim1OE.
Proof. (cf. [8, 2.3]) Let OE0 be the restriction of OE to the full subcategory*
* A0X of AX
given by the objects (V; f) such that V is an object of An . It is clear that t*
*he inclusion
A0X-! AX is an equivalence of categories, and so lim1OE0is naturally isomorphic*
* to lim1OE.
By inspection, however, the set Z1(OE0) corresponds bijectively to Z1() in a wa*
*y which
respects the equivalence relations giving lim1.
From now on, then, we will deal only with functors : An !- Grp . Let 1 be*
* the
restriction of to the subcategory A1 of An . To simplify both the exposition*
* and the
formulas we will assume that n 3 and that lim11 = *. This last condition is a*
*lways
satisfied if p = 2, since in this case A1 is a trivial category; the condition *
*is also frequently
satisfied for p odd, since A1 is then the category of a group of order prime t*
*o p (i.e.
GL (1; Fp)) and in examples arising from 2.4 the group (A1) is usually p-comple*
*te in
some sense.
Let ei : Ai!- Ai+1 be the standard inclusion obtained by adding a zero as t*
*he last
coordinate and, for j > i, let ei;j: Ai-! Aj be the composite ej-1 . .e.i. For *
*j > i we will
identify Ai with its image in Aj under ei;j. Let P (i; j) GL (i + j; Fp) be th*
*e subgroup
of linear transformations which carry the subspace Ai to itself; these are bloc*
*k matrices of
the form
M1 M2
0 M3
in which M1 is square of size i x i. There is a surjection ri : P (i; j)!- GL *
*(i; Fp) which
maps the above block matrix to M1.
Recall that the Tits building T (A3) of A3 is the graph with a vertex for ea*
*ch 1-
dimensional subspace V or 2-dimensional subspace W of A3 and an edge for each *
*pair
(V; W ) with V W .
4.2 Definition. A 1-cycle in T (A3)is a circular sequence of vertices of T (A3)*
*such that
any two adjacent ones are connected by an edge; equivalently, C is a a sequence
(4.3) C = (V0; W1; V1; W2; V2; : :;:Wk; Vk)
10 W. DWYER AND C. WILKERSON
of subspaces of A3 such that each Vihas dimension 1, each Wihas dimension 2, Wi*
* Vi-1,
Wi Vi, and Vk = V0. A polarization of C is a collection {ffi; fii}ki=1of elem*
*ents of
GL (3; Fp) such that ffi carries the subspace pair (Wi; Vi-1) to the standard p*
*air (A2; A1),
and fii carries (Wi; Vi) to (A2; A1). The difference elements {ai; bi} (i = 1; *
*. .;.k) associ-
ated to the polarization are given by bi = fiiff-1i, ai = ffi+1fi-1i(i < k), an*
*d ak = ff1fi-1k.
Note that ai2 P (1; 2) and bi2 P (2; 1).
Suppose that n 3 and that : An!- Grp is a functor. If x 2 (Ai) and f : i*
*-! j
is a morphism of An , write f# (x) for the image of x under (f). Observe that G*
*L (i; Fp)
is the group of self-maps in An of Ai (i n) so that there is an action of GL *
*(i; Fp) on
(Ai) which sends (g; x) to g# (x).
4.4 Definition. The set Z1sp() of special 1-cocycles for is the collection of *
*set maps
i : GL (2; Fp)!- (A2) which have the following three properties.
(1) For g, h 2 GL (2; Fp), i(gh) = i(g)h# (i(h)).
(2) If g 2 P (1; 1), then i(g) = 1.
(3) Suppose that C is a 1-cycle of T (A3)(4.3) with polarization {ffi; fii}*
* and associated
difference elements {ai, bi}. Let "idenote the composite function
r2 i e2#
P (2; 1) -! GL (2; Fp)!- (A2) -! (A3)
and for each i let fli denote the map (ff1fi-1i)# . Then the following *
*equality holds
in (A3):
flk("i(bk))flk-1("i(bk-1)) . .f.l1("i(b1)) = 1 :
Remark. It can be checked that given conditions (1) and (2) of 4.4, condition (*
*3) holds for
all polarizations of a 1-cycle C if and only if it holds for any single polariz*
*ation.
Let x be an element of the fixed set F of the action of GL (1; Fp) on (A1), *
*and let
y 2 (A2) be the image of x under e1#. Given i 2 Z1sp(), there is another eleme*
*nt
i0 2 Z1sp() with i0(g) = yi(g)g# (y)-1 and the formula x . i = i0 gives an acti*
*on of F on
Z1sp(). Let Z1sp()=~ denote the orbit set of this action; this has a basepoin*
*t given by
the orbit of the element i 2 Z1sp() with i(g) = 1 for all g 2 GL (2; Fp).
4.5 Proposition. Suppose that : An !- Grp is a functor, n 2. Let 1 be the
restriction of to A1, and assume that lim11 = *. Then there is a natural monom*
*orphism
of pointed sets
lim1!- Z1sp()=~ :
Remark. In fact, the monomorphism in 4.5 is an isomorphism, although we will no*
*t write
down the proof of this. The argument consists in following what appears below *
*and
observing, with the help of the action of GL (3; Fp) on T (A3), that GL (3; Fp)*
* is the quotient
of an amalgamated sum P (1; 2) *P(1;1;1)P (2; 1) by a normal subgroup isomorphi*
*c to the
fundamental group of T (A3). For i > 3, GL (i; Fp) is isomorphic to the appropr*
*iate analog
of this amalgamated sum, essentially because T (Ai)is 1-connected.
The proof of 4.5 relies on the following more elementary fact from linear al*
*gebra.
DIAGRAMS 11
4.6 Lemma. If n 3 then the group GL (n; Fp) is generated by the subgroups P (i*
*; j),
i + j = n.
Proof of 4.5. Let z be an element of Z1(). According to 2.5 (adjusted for the f*
*act that
is a covariant functor) z is a function which assigns to each morphism g : Ai!*
*- Aj of
An an element z(g) 2 (Aj), such that z(g) = 1 if g is an identity morphism and
(4.7) z(gh) = z(g)g# z(h) :
If c 2 C0(), then z represents the same element of lim1 as the "cohomologous" c*
*ocycle
c . z = z0 with
(4.8) z0(g) = c(Aj)z(g)g# c(Ai)-1 :
Since lim11 = * the restriction of z to A1 is cohomologous to the trivial cocyc*
*le; in other
words, there is an element x 2 (A1) such that z(g) = xg# (x)-1 for each g 2 GL *
*(1; Fp).
Define an element c 2 C0() inductively by setting c(A1) = x-1 and c(Ai+1) equal*
* to the
inverse of z(ei)ei#(c(Ai)). A calculation with 4.8 shows that the cocycle c . z*
* = z0 satisfies
two conditions:
(1) z0(g) = 1 for each g 2 GL (1; Fp), and
(2) z0(ei) = 1 (i = 1; : :;:n - 1).
Let Z0 Z1() denote the collection of all cocycles which satisfy the above two *
*conditions.
Let F denote the fixed set of the action of GL (1; Fp) on (A1), and, given x 2 *
*F , let
cx 2 C0() be defined inductively by c(A1) = x, c(Ai+1) = ei#c(Ai). The constru*
*ction
(x; z) 7! cx . z gives an action of F on Z0, and by elementary calculation the *
*subgroup
{cx : x 2 F } of C0() is equal to the set of elements c 2 C0() such that Z0 and*
* c . Z0
intersect nontrivially. It follows immediately that lim1 is isomorphic to the o*
*rbit set of
the action of F on Z0. Given z 2 Z0, let i be the restriction of z to GL (2; Fp*
*). We will
show that i 2 Z1sp() and that the map z 7! i is injective; the proposition then*
* follows
from the obvious fact that this map respects the actions of F on the objects in*
*volved.
Condition 4.4(1) for i follows from the cocycle condition 4.7. Observe that*
* if g 2
P (i; j) GL (i + j; Fp) (i + j n), then there is an equality
(4.9) z(g) = ei;i+j#z(ri(g)) 2 (Ai+j)
which follows from the chain
z(g) = z(g)1= z(g)g# (z(ei;i+j)) = z(gei;i+j) = z(ei;i+jri(g))
:
= z(ei;i+j)ei;i+j#(ri(g))) = ei;i+j#(ri(g)))
Here we have used that z 2 Z0 and hence z(ei;i+j) = 1. Condition 4.4(2) is der*
*ived by
applying 4.9 to g 2 P (1; 1) and noting that z(g) = 1 for g 2 GL (1; Fp). Let C*
* be a 1-cycle
in T (A3)(4.3) with polarization {ffi; fii} and associated difference elements *
*{ai, bi}. It is
clear that there is an identity
(4.10) akbkak-1bk-1 . .b.1a1 = 1
12 W. DWYER AND C. WILKERSON
in GL (3; Fp). Since ai 2 P (1; 2) and bi 2 P (2; 1), it follows from 4.9 that *
*z(ai) = 1 and
z(bi) = "i(bi) (where the notation "iis from 4.4). Applying z to equation 4.10 *
*and using the
cocycle property of z (4.7) to expand the resulting expression now gives condit*
*ion 4.4(3);
part of the "expansion" is actually a contraction that uses the identity
akbk . .a.i+1bi+1ai= ff1fi-1i:
It remains to prove that the assignment z 7! i is injective. Suppose that z *
*and z0 are
two elements of Z0 which agree on GL (2; Fp). We will first prove by induction *
*on m that
for any 2 m n the cocycles z and z0 agree on GL (i; Fp) for all i m. In fac*
*t, if
g 2 P (i; j), i + j = m, then z(g) = ei;m#z(ri(g)) = ei;m#z0(ri(g)) = z0(g) by *
*4.9; thus z
and z0 agree on a set of elements which generate GL (m; Fp) (4.6), and hence ag*
*ree on the
whole group by the cocycle property 4.7. Suppose now that f : Ai!- Aj is an ar*
*bitrary
morphism in An with i < j. It is clear that there is an element g 2 GL (j; Fp*
*) with
gei;j= f, and hence
z(f) = z(gei;j) = z(g)g# z(ei;j) = z0(g)g# z0(ei;j) = z0(f) :
This shows that z and z0 are identical.
x5. The exceptional group G2
In this chapter we use 4.5 to sketch a proof of Conjecture 1.1 in the case i*
*n which p = 2,
and X is the 2-completion of the exceptional compact Lie group G2. In other wor*
*ds, we
will prove that ffXhas no nontrivial H*-clones.
Suppose that G is a compact Lie group (eventually G2). Let AG be the categor*
*y whose
objects are the non-trivial elementary abelian subgroups of G; a morphism V !- *
* V 0in
AG is a monomorphism f : V!- V 0of abelian groups with the property that there*
* exists
an element g 2 G such that f(x) = gxg-1 for all x 2 V . As in [12], there is a*
* functor
ffopG: AG!- Sp which sends V to the Borel construction EGxG (G=CG (V )), where*
* CG (V )
is the centralizer of V in G. (Note that this Borel construction has the homoto*
*py type of
the classifying space BCG (V ).)
5.1 Remark. The effect on a morphism f : V!- V 0of ffG is as follows. Let g 2 *
*G be an
element such that f(x) = gxg-1 for all x 2 V . The map h 7! g-1 hg gives a homo*
*morphism
CG (V 0)!- CG (V ), and so there is a G-equivariant map G=CG (V 0)!- G=CG (V *
*) given by
xCG (V 0) 7! xgCG (V ). The induced map of Borel constructions is ffG (f). It*
* does not
depend upon the choice of the element g.
The following proposition is well-known; it is a consequence of [8, 2.2], [1*
*1, pf. of 8.1],
and the fact that if G is a compact Lie group with ss0G a p-group, and V is an *
*elementary
abelian p-group, then there is a natural weak equivalence Map (B V; BG)^p-! Map*
* (B V; BG^p)
[6, 4.5].
5.2 Proposition. Suppose that G is a compact Lie group such that ss0G is a p-gr*
*oup, and
let X be the p-compact group G^p. Then there is an equivalence of categories e *
*: AG!- AX
such that the composite functor ffX . e is weakly equivalent to (ffG )^p.
Remark. A weak equivalence between two functors ff; ff0 : AG!- Sp is a natura*
*l trans-
formation which gives an ordinary weak equivalence of spaces for each object of*
* AG (cf.
DIAGRAMS 13
6.1). The functors ff and ff0 are weakly equivalent if they are connected by a*
* zigzag of
weak equivalences.
We will need to deal with certain elements and subgroups of SO (4). Let o1 d*
*enote the
central diagonal matrix diag(-1; -1; -1; -1) in SO (4), o2 the matrix diag(-1; *
*-1; 1; 1)
and o3 the block matrix
o3 = N0 0N with N = 01 10 :
These matrices generate a subgroup of SO (4) isomorphic to (Z=2)3. The symbol T*
* 2will
denote the maximal torus of SO (4) given by block matrices
M1 0 cosi - sini
0 M2 with Mi= sini cosi
and Te2xt T 2the subgroup of SO (4) generated by T 2and o3.
If A is a subgroup of G2 let NG2(A) denote its normalizer; note that the quo*
*tient
NG2(A)=CG2(A) acts faithfully on A by conjugation.
5.3 Proposition. For each integer i with 1 i 3, the group G2 contains up to c*
*onjugacy
a unique subgroup Vi isomorphic to (Z=2)i. The group CG2(V1) is isomorphic to S*
*O (4),
the group CG2(V2) is isomorphic to Te2xt, and the group CG2(V3) to V3 itself. I*
*n each case
the conjugation action gives an isomorphism
~=
NG2(Vi)=CG2(Vi) -! Aut (Vi) ~=GL (i; F2) :
Proof. This is stated for i = 1 and i = 3 in [13, 2.3]. For the case i = 2, no*
*te that
since SO (4) is the centralizer of a non-trivial involution oe in G2 and oe is *
*unique up
to conjugacy, the conjugacy classes of homomorphisms Z=2 x Z=2!- SO (4) corres*
*pond
bijectively to conjugacy classes in SO (4) of non-central involutions. Up to co*
*njugacy there
is only one such non-central involution in SO(4), which can be taken to be the *
*matrix
o2. The centralizer of o2 in SO (4), which is the same as the centralizer of in G2, is
Te2xt. Since there is only one conjugacy class of homomorphism Z=2 x Z=2!- G2,*
* every
automorphism of must be realized by conjugation with an element of G2; *
*this gives
the isomorphism NG2(V2)=CG2(V2) ~=GL (2; F2).
Choose specific subgroups V1 V2 V3 of G2 as in 5.3, together with an isomo*
*rphism
CG2(V1) ~= SO (4) G2; the groups Vi are then subgroups of SO (4), and the choi*
*ces
can be made in such a way that V1 = , V2 = and V3 = . *
* Let C
be the full subcategory of AG2 generated by the Vi (i = 1; 2; 3) and A3 the ca*
*tegory
of x4. The unique basis-preserving vector space isomorphisms Vi ~= Ai give a f*
*unctor
: C!- A3 which by 5.3 is an isomorphism of categories. Since C is a skeletal *
*subcategory
of AG2 (i.e., a full subcategory which contains one object of each isomorphism *
*type) the
composite of -1 : A3!- C with the inclusionC!- AG2 is an equivalence of categ*
*ories.
Let ff : Aop!- Sp be the restriction of ffG2 to A3 , ^ffthe 2-completion of ff*
*, and F , ^Fthe
diagrams in Ho underlying ff, ^ffrespectively.
By 5.2, the following proposition is equivalent to the result stated at the *
*beginning of
this section.
14 W. DWYER AND C. WILKERSON
5.4 Proposition. The functor ^F: Aop!- Ho has no H*-clones.
5.5 Notation. If G is a compact Lie group, we will let ^BG denote BG^2. The sym*
*bol Z2 will
stand for the ring of 2-adic integers, and i(n) for the subgroup of GL (n; Z2) *
*consisting of
matrices which are congruent to the identity matrix mod 2i.
Proof of 5.4. Since the functor F^is H *-centric up to homotopy (3.1), the prop*
*osition is
equivalent to the assertion that lim1OEF^is trivial.
The functor ^ffhas the following properties:
(1) ^ff(A1) ~ ^BSO(4), ^ff(A2) ~ ^BTe2xt, and ^ff(A3) ~ ^BA3.
(2) The map ^ff(e1) : ^ff(A2)!- ^ff(A1) is up to homotopy the map ^BTe2xt-*
*!B^SO (4)
induced by the inclusion Te2xt SO (4).
To study the functor OEF^it is necessary to calculate the groups Autiof homotop*
*y classes of
homotopy self-equivalences of ^ff(Ai), i = 1; 2; 3, and then determine the subg*
*roups OEF^(Ai)
of equivalences which induce the identity on H*. Picking out the groups OEF^(Ai*
*) turns out
to be relatively easy, because for i = 1; 2; 3 the restriction map H *^ff(Ai)!-*
* H *B V3 is
a monomorphism. The case i = 1 of this assertion is classical, the case i = 3 *
*is trivial,
and the case i = 2 is proved either by direct calculation or by combining the e*
*xactness of
Lannes' functor T with the case i = 1 [14, 2.1, p. 203].
Let L denote the Z2 module given by ss2^BT 2. The group V2 is the subgroup o*
*f elements
of exponent 2 in T 2, and so the long exact homotopy sequence associated to the*
* fibration
sequence
^B(t7!t2)
BV2!- ^BT 2-----! ^BT 2
gives a natural isomorphism
(5.6) V2 ~=Z=2 L :
Let W1 be the rank 2 elementary abelian 2-subgroup of GL (2; Z2) generated by t*
*he matrices
-1 -1 1 1
0 1 and 0 -1 :
(Note that the square of each matrix is the identity, and the product of these *
*matrices in
either order is the negative of the identity matrix.) The group SO (4) is isomo*
*rphic to the
quotient of Spin(4) ~=SU (2) x SU(2) by a diagonal central Z=2, and from this i*
*t follows
easily that it is possible to choose a basis {`1; `2} for L (equivalently, an i*
*somorphism
L ~= (Z2)2) in such a way that the conjugation image of the Weyl group of SO (4*
*) in
Aut (L) is exactly W1 (see [9,x3] for a similar calculation). After a possible*
* replacement
of `2 by `1 - `2 the basis can be adapted to the basis {o1; o2} of V2, in the s*
*ense that the
reduction mod 2 (5.6) of `i is oi.
Let N1 denote the normalizer of W1 in GL (2; Z2). By [9, x5] there is a nat*
*ural iso-
morphism Aut1 ~=N1=W1. The group N1=W1 is itself isomorphic to the wreath produ*
*ct
2 o 2(1) [9, 5.4, pf. of 5.5]. (Note in checking this reference that the obvi*
*ous map
2(1)!- GL (1; Z2)=< 1> is an isomorphism.) It follows easily that the subgroup*
* OEF^(A1)
DIAGRAMS 15
of Aut1 is the kernel 2(1) x 2(1) of the projection Aut1!- 2. The inclusion o*
*f this
product into N1=W1 is induced by the map 2(1) x 2(1)!- 1(2) given by the formu*
*la
(5.7 .) (u; v) 7! u0 (u -vv)=2
Let W2 denote the central subgroup of GL (2; Z2) generated by the negative o*
*f the
identity matrix and N2 = GL (2; Z2) its normalizer. An analysis along the lines*
* of [9, x5] but
substantially more elementary shows that there is a natural isomorphism Aut2 ~=*
*N2=W2.
Denote this quotient by GL (2; Z2). The group OEF^(A2) is the image 1 (2) in GL*
* (2; Z2) of
2(1).
It is clear that Aut3 is isomorphic to GL (3; F2) and that its subgroup OEF^*
*(A3) is trivial.
The next step is to obtain information about the maps in the diagram OEF^. A*
* naturality
argument shows that the map OEF^(e1) : 2(1) x 2(1)!- 1 (2) is induced by formu*
*la 5.7.
The action of Aut(A2) on ^F(A2) is induced by the conjugation action of NG2(V2)*
*=CG2(V2)
on BCG2(V2); homotopically this conjugation action is the one provided by the f*
*act that
B CG2(V2) is a regular covering space of BNG2(V2) with covering group NG2(V2)=C*
*G2(V2).
From this it follows that the map f : Aut(A2)!- Aut2 ~=GL (2; Z2) given by ^Fi*
*nduces
an isomorphism Aut(A2) ~=GL (2; F2) upon reducing mod 2. Since the basis {`1; `*
*2} used
in making the identification Aut2 ~=GL (2; Z2) is adapted to the basis {o1; o2}*
* for L, this
isomorphism Aut (A2) ~=GL (2; F2) is actually the identity map. The action of *
*Aut (A2)
on 1 (2) given by OEF^is obtained by letting Aut(A2) act on 1 (2) by conjugatio*
*n via the
homomorphism f.
Denote by gl(2; F2) the vector space of 2 x 2 matrices over F2, and by pgl(2*
*; F2) its
quotient by the subgroup generated by the identity matrix. Matrix conjugation *
*gives
actions of Aut(A2) = GL (2; F2) on both gl(2; F2) and pgl(2; F2). Let i (2) be *
*the image
in 1 (2) of i(2). By the discussion in the last paragraph there are Aut(A2)-equ*
*ivariant
isomorphisms ae
i (2)=i+1 (2) ~= pgl(2; F2) i = 1
i(2)=i+1(2) ~=gl(2; F2) i > 1
Consider now an element i 2 Z1sp(OEF^) (x4); i is a function GL (2; F2)!- 1 (*
*2) which
satisfies conditions (1) and (2) of 4.4. (Note that condition 4.4(3) is automat*
*ically satisfied,
because OEF^(A3) is the trivial group.) Consider the elements of GL (2; F2) gi*
*ven by the
matrices
s = 11 10 and t = 10 11 :
These generate GL (2; F2) subject to the relations s3 = 1, t2 = 1, tst = s2. Si*
*nce i(t) =
1, the cocycle i is determined by i(s). The idea of the argument is now to gra*
*dually
deform i toward the trivial cocycle, and use the completeness of Z2 to pass to *
*the limit.
More precisely, we will inductively construct elements wi2 i+1(1) x i+1(1) such*
* that if
xi = wi. .w.1and yi = e1#(xi), then yii(s)s# (yi)-1 2 i+1(2). The sequence {xi}*
* then
converges in the 2-adic topology on 2(1) x 2(1), and its limit x gives a trivia*
*lization of
the cocycle i.
16 W. DWYER AND C. WILKERSON
The construction of w1 is left to the reader since it is very similar to the*
* inductive
step we are about to describe. Assume that suitable wi, together with the asso*
*ciated
elements xi and yi, have been chosen for i < n (n > 1). Let 2 Z1sp(OEF^) be de*
*termined
by (s) = yn-1i(s)s# (yn-1)-1, and let : GL (2; F2)!- gl(2; F2) be the reduct*
*ion of
modulo i+1(2). With the group operation in gl(2; F2) written additively, sati*
*sfies the
cocycle condition
(5.8) (gh) = (g) + g# (h) g; h 2 GL (2; F2) :
Here, as noted above, the action of g on (h) is by conjugation. Expanding the *
*left hand
side of the equation (s3) = 0 with 5.8 gives (s) + s# (s) + s2# (s) = 0, whi*
*ch by explicit
calculation implies that (s) has the form
(5.9) b +cc b b+ c
Expanding both sides of (tst) = (s2) by the same technique gives
b b + c b c
c b = t# (s) = (s) + s# (s) = b + c b
which implies b = 0. If c = 0 let wn be the identity element, otherwise choose*
* wn =
(u; v) 2 2(1)n + 1x 2(1)n + 1 such that (u - v)=2 is not congruent to zero mod *
*2n. If
M is the reduction mod n(2) of e1#(wn), then (5.7)
M + (s) + s# (M) = 00 10 + 11 01 + 11 11 = 0 :
This shows that wn has the required inductive property.
x6. The cohomology category
For convenience of exposition, in this section "space" means "simplicial set*
*". Let CoHo
denote the category whose objects are spaces and whose morphisms are cohomology*
* classes
of maps; more formally, CoHo is the quotient category of Ho in which two map*
*s f; g :
X !- Y are considered equivalent if they induce the same map H *Y !- H *X. *
*Let
ss : Sp!- CoHo be the obvious functor.
6.1 Definition. Suppose that X : D!- CoHo is a functor. A realization of X *
* is a pair
(X; s), where X : D!- Sp is a functor and s : ssX!- X is a natural equivalenc*
*e. A weak
equivalence t : (X; s)!- (X0; s0) between two such realizations is a natural t*
*ransformation
from X to X0 such that
(1) for each object d of D, td : X(d)!- X0(d) is a weak equivalence of spa*
*ces, and
(2) the composite natural transformation s0. ss(t) is equal to s.
DIAGRAMS 17
The realization complex rX of X is defined to be the nerve of the category who*
*se objects
are the realizations of X and whose morphisms are the weak equivalences betwee*
*n these
realizations.
The aim of this section is to calculate the homotopy type of rX for diagram*
*s X which
satisfy a special condition. The main result, which requires some preparation t*
*o state, is
a combination of 2.4 and the results in [6].
Remark. The realization complex rX of 6.1 is the nerve of a category which is *
*not small.
Nevertheless this nerve is homotopically small [3] and it is possible to make s*
*tandard ho-
motopy theoretic constructions with it. The components of rX are in 1- 1 corre*
*spondence
with weak equivalence classes {(X; s)} of realizations of X , and the component*
* correspond-
ing to a particular (X; s) is equivalent to the classifying space of an appropr*
*iate complex
[3, 3.5] of homotopy automorphisms of X (in our situation this is the complex o*
*f homotopy
automorphisms of X which for each object d of D induce the identity automorphis*
*m of
H *X(d)).
6.2 Remark. Let X : D!- CoHo be a functor, D0 a small category, and F : D0-!*
* D a
functor. Let F *X : D0-! CoHo be the composite of X with F . It is easy to s*
*ee that F
induces a natural map rX !- r(F *X).
If f : A!- B is a map in Sp or CoHo , let Map (A; B){f} denote the subspac*
*e of the
mapping space Map (A; B) consisting of maps f0 such that H*(f0) = H*(f).
6.3 Definition. A map f : A!- B between fibrant simplicial sets is said to be *
*H*-centric if
(1) [A; A]{id}= ss0 Map (A; A){id}is a group under composition, and
(2) composition with f induces a weak equivalence Map (A; A){id}-! Map (A; *
*B){f}.
A map f in CoHo is said to be H*-centric if any representative of f is H*-cen*
*tric (equiv-
alently, if all representatives are H *-centric). A functor F : D!- CoHo is*
* said to be
H *-centric if F (g) is H*-centric for each morphism g of D.
Remark. It is clear that a map f : A!- B is H*-centric if and only if f is bot*
*h H*-centric
up to homotopy in the sense of x2 and centric in the sense of [6]. In particula*
*r, if X is a
p-compact group, then the diagram in CoHo underlying ffX is H*-centric (3.1, [*
*6, x4]).
6.4 Remark. If A is a fibrant simplicial set let h(A) denote Map (A; A){id}. Su*
*ppose that
f : A!- B is a map in CoHo , let M = Map (A; B){f}, G = h(A) and H = h(B). Le*
*t GM
denote the Borel construction of the right action of G on M, MH the Borel cons*
*truction
of the left action of H on M, and G MH the corresponding double Borel construc*
*tion
(which is the Borel construction of the right action of G on MH or equivalently*
* the Borel
construction of the left action of H on GM). There are fibration sequences
M!- GM!- BG
G M!- GMH!- B H
MH!- GMH!- B G
Suppose that f is H *-centric; this implies that GM is weakly contractible and *
*hence that
G MH!- B H is an equivalence, so that the third fibration sequence above deter*
*mines up
18 W. DWYER AND C. WILKERSON
to homotopy a map BH!- B G. We will denote this map f[. The construction of f[*
* is
parallel to what was done in x2 using 2.3 to construct OEF (g).
For any object d of D, let D#d denote the over category [1, XI, x2] [15, p. *
*46] of the
identity functor. The objects of this category are pairs (d0; g) 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*
*such that
gh = f. There is a forgetful functor Od : D#d!- D which sends a pair (d; f) to*
* d. Given
a diagram X : D!- CoHo , there is an induced diagram
O*dX = X . Od : D#d!- CoHo
for each object d of D, as well (6.2) as a functor rDopX : Dop!- Sp which map*
*s an object
d 2 Dop to the realization complex r(O*dX).
As in [6, p. 577], in order to form homotopy limits [1] which have homotopy *
*invariant
meaning we sometimes have to replace a given functor Y : D!- Sp by a fibrant o*
*ne Y 0,
i.e., by a weakly equivalent functor Y 0such that Y 0(d) is a Kan complex for e*
*ach object d
of D. We will write Y fto denote such a replacement. The following theorem is p*
*roved in
exactly the same way as [6, 2.1]; it is a derivative form of the basic diagram *
*classification
result in [2].
6.5 Theorem. Suppose that D is a small category and that X : D!- CoHo is a fu*
*nctor.
Then the natural map
rX = limrDopX !- holim(rDopX )f
is a weak equivalence.
We have now come to the main result of this section. In combination with 6.*
*5, it
expresses rX , for X a functor which is H*-centric, as the homotopy limit of a*
* diagram in
which the constituents are classifying spaces of self-equivalences of the space*
*s X (d).
6.6 Theorem. Suppose that D is a small category and that X : D!- CoHo is a fu*
*nctor
which is H *-centric. Then for each object d of D the space rDopX is weakly h*
*omotopy
equivalent in a natural way to Bh(X (d)). Under these equivalences, for each g *
*: d!- d0in
D the map rDopX (g) is homotopic to the map
X (g)[ : Bh(X d0)!- Bh(X d) :
described in 6.4.
Proof. This is essentially the same as the proof of [6, 2.2]. The main issue (c*
*f. the proof
of [6, 2.3]) is to show that if
f1 f2 fn
A0 -! A1 -! . .-.! An
is a chain of maps in CoHo such that each fi is H*-centric, then the natural p*
*rojection
B(h(A0); Map (A0; A1){f1}; h(A1); : :;:Map (An-1; An){fn}; h(An))
#
Bh(An)
DIAGRAMS 19
is an equivalence; the iterated bar construction on the left here is described *
*in [4, 9.1].
This is proved by an induction [4, 9.2(vii)] that depends only on the observati*
*on made in
6.4 about the contractibility of certain ordinary Borel constructions.
Let Grpd denote the category of groupoids, so that Grp is the full subcat*
*egory of
Grpd consisting of groupoids with a single object.
6.7 Definition. If H; H0 : D!- Grpd are functors, a natural transformation t *
*: H!- H0
is said to be a weak equivalence if td is an equivalence of categories for each*
* object d 2 D.
Let D and X be as in 6.6, let 1 be the composite of rDopX with the fundame*
*ntal
groupoid functor, and let B1 be the further composite with the classifying spac*
*e functor.
It is possible to check that finding a point in holim(B 1) is equivalent to lif*
*ting X to a
functor F : D!- Ho or, from another point of view, to finding a weak equivale*
*nce (6.7)
between 1 and a diagram of groups. If holim(B 1) is nonempty, i.e., such a lift*
* F exists,
then 1 is weakly equivalent to the diagram OEF and holim(B 1) is weakly equival*
*ent as a
space to holim(B OEF ). In particular ss0 holim(B 1) is isomorphic to lim1OEF [*
*1, p. 309] and
so corresponds bijectively (2.4) to the set of isomorphism classes of H*-clones*
* of F . There
is a natural map rDopX !- B1 and, given a component of holim(B 1) correspondin*
*g to
an H*-clone G, the problem of lifting this component to a component of holim(rD*
*opX )f is
equivalent to the problem of realizing G by a diagram of spaces.
In this way computing ss0 holim(rDopX )f can be broken down into two steps: *
* first,
compute the set of lifts of X to Ho (and observe that if any such lift exists,*
* the others are
exactly its H*-clones and so are enumerated by 2.4); second, determine which of*
* these lifts
of X to Ho lift further to Sp , and analyze in each case how many of these se*
*cond-level
lifts there are (this can be handled by [6]).
We end with a conjecture related to 1.1.
6.8 Conjecture. Let X be a p-compact group (perhaps connected) and ffHXthe diag*
*ram
in CoHo which underlies ffX . Then the realization space rffHXis connected.
Conjecture 6.8 asserts that the trivial clone of ffXis the only one which ca*
*n be realized
as a diagram of spaces, and adds that up to weak equivalence there is only one *
*such
realization. Technically, Conjecture 6.8 does not include 1.1; it would be poss*
*ible for 6.8 to
hold and 1.1 to fail if there existed nontrivial H*-clones of ffXwhich could no*
*t be realized
as diagrams of spaces. In fact we conjecture that this does not happen, and th*
*at more
generally the E2-term of the homotopy spectral sequence for ss* holim(rDopffHX)*
*f (see [6,
x3] and [1, XI, 6.2]) is trivial away from the y-axis.
Remark. Conjecture 6.8 can be proved for the special case in which X is the 2-c*
*ompletion
of G2 by combining the result of x5 with an argument along the lines of the one*
* in [9, x7].
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