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]. References [1]A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations,* * Lecture Notes in Mathematics vol. 304, Springer-Verlag, Berlin, 1972. [2]______, A classification theorem for diagrams of simplicial sets, Topology 2* *3 (1984), 139-155. [3]W. G. Dwyer and D. M. Kan, Realizing diagrams in the homotopy category by me* *ans of diagrams of simplicial sets, Proc. Amer. 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University of Notre Dame, Notre Dame, Indiana 46556 Purdue University, West Lafayette, Indiana 47907 Processed April 18, 1* *994