A Cohomology Decomposition Theorem W. G. Dwyer and C. W. Wilkerson University of Notre Dame Purdue University x1. Introduction In [9] Jackowski and McClure gave a homotopy decomposition theo- rem for the classifying space of a compact Lie group G; their theorem states that for any prime p the space BG can be constructed at p as the homotopy direct limit of a specific diagram involving the classifying spaces of centralizers of elementary abelian p-subgroups of G. In this paper we will prove a parallel algebraic decomposition theorem for cer- tain kinds of unstable algebras over the mod p Steenrod algebra. This algebraic result gives a new proof of the theorem of Jackowski and Mc- Clure and has the potential to lead to homotopy decompositon theorems for many spaces which are not of the form BG (see x6). Before stating our results we will recall some material from [9]. Choose a prime p. Let G be a compact Lie group, and let AG be the cate- gory whose objects are the non-trivial elementary abelian p-subgroups of G; a morphism A ! A0 in AG is a homomorphism f : A ! A0 of abelian groups with the property that there exists an element g 2 G such that f (x) = gxg-1 for all x 2 A. There is a functor from AopGto the category of topological spaces which sends A to the Borel construc- tion EG xG (G=C(A)), where C(A) denotes the centralizer of A in G. (Notice that this Borel construction has the homotopy type of the clas- sifying space BC(A).) Jackowski and McClure prove that the natural map from the homotopy direct limit of this functor to EG xG * = BG is an isomorphism on mod p cohomology. They derive this from a spectral sequence argument [2, XII, 5.8] and the following calculation with the inverse limit functor lim and its right derived functors lim i. Let H* denote mod p cohomology and ffG the functor on AG which sends A to H* (EG xG (G=C(A)). Theorem 1.1 [9, Prop. 3-4]. The natural map H* BG ! lim ffG is an isomorphism and the groups lim iffG vanish for i > 0. The proof of Theorem 1.1 in [9] uses the Feshbach double coset formula and so depends heavily on the presence of a genuine compact Lie group. ______________________________________ This work was supported in part by the National Science Foundation. 2 Dwyer and Wilkerson What we do is much more algebraic. Let K be the category of unstable algebras over the mod p Steenrod algebra Ap . Given an object R of K we will build an index category AR together with a functor ffR : AR ! K and natural map R ! lim ffR ; if R = H* BG then AR is equivalent to AG in such a way that ffR corresponds to ffG . This construction depends heavily on work of Lannes [11]. Using [11] again, we will define what it means for an object R of K to "have a non-trivial center"; if R = H* BG this condition holds if G has a non-trivial central element of order p. Our main result is the following one. Theorem 1.2. Suppose that i : R ! S is a map of K such that: (1) Both R and S are finitely generated as algebras, and the map i makes S into a finitely generated module over R. (2) The map i has an additive left inverse S ! R which is both a map of R modules and a map of unstable modules over the Steenrod algebra. (3) The algebra S has a non-trivial center. Then the natural map R ! lim ffR is an isomorphism and the groups lim iffR vanish for i > 0. Remark: For a functor such as ffR which takes values in the category K, we write lim iffR (i > 0) for the ordinary higher limits [2, p. 305] of the composite of ffR with the forgetful functor from K to the category of graded Fp vector spaces. The connection between Theorem 1.2 and Theorem 1.1 is provided by the following proposition, which lists some standard properties of compact Lie groups. Proposition 1.3. Let G be a compact Lie group, T a maximal torus in G, N (T ) the normalizer of T , Np(T ) the inverse image in N (T ) of a p- Sylow subgroup of N (T )=T , and i* the natural restriction map H* BG ! H* BNp(T ). Then the following assertions hold. (1) Both H* BG and H* BNp(T ) are finitely generated algebras [17] [15, 2.2]. The map i* makes H* BNp(T ) into a finitely generated module over H* BG [15, 2.4]. (2) The map i* has an additive left inverse H* BNp(T ) ! H* BG which is both a H* BG module map and a map of unstable mod- ules over the mod p Steenrod algebra [1]. (3) The group Np(T ) has a non-trivial central element of order p. Remark 1.4: Note that 1.3(3) follows from the fact that the conjuga- tion action of Np(T )=T on the elements of order p in T must pointwise fix a non-trivial subgroup [8, p. 47]. Decomposition Theorem 3 There is another basic example of the situation described in Theorem 1.2, an example which is purely algebraic. Suppose that o is either a torus or an elementary abelian p-group. Let W be a finite group of K-automorphisms of H* Bo and Wp a p-Sylow subgroup of W . Then the rings of invariants R = H* (Bo )W and S = H* (Bo )Wp are finitely generated as algebras and the natural inclusion R ! S has a left inverse given by averaging over the coset space W=Wp. It is not hard to see that S has a non-trivial center (cf. 1.4, 3.4 and the definition in x4 of "having a non-trivial center"). This leads to Proposition 1.5. Let o be either a torus or an elementary abelian p- group. Suppose that W is a finite group of K-automorphisms of H* Bo and that R is the ring of invariants H* (Bo )W . Then the natural map R ! lim ffR is an isomorphism and the groups lim iffR vanish for i > 0. Organization of the paper: Section 2 constructs the decomposition functor ffR and compares it with ffG if R = H* BG. Section 3 describes some properties of Lannes's functor T . In x4 there is a definition of what it means for an object R of K to "have a non-trivial center" and a proof of (a slight generalization of) the special case of 1.2 in which R = S. Section 5 completes the proof of 1.2 itself, and x6 concludes the paper with a topological application of 1.5. Notation and terminology: The prime p will be fixed for the rest of the paper. If V is an elementary abelian p-group (ie. a finite dimen- sional Fp vector space) then HV will stand for the mod p cohomology algebra H* BV . At several points in the paper we will speak for con- venience of inverse limits and related constructions over certain large categories. In each case the large category in question is equivalent to a small subcategory of itself (ie. the category has a small skeleton [13, p. 91]) and the constructions can be performed in the usual way on these small subcategories. The authors would like to thank the referee for his suggestions. x2. The decomposition diagram Given an object R of K, define VR to be the category whose objects consist of pairs (V; f ), where V is an elementary abelian p-group and f : R ! HV is a K-map. A map (V; f ) ! (W; g) in VR is an abelian group map h : V ! W such that the composite of the induced map h* : HW ! HV with g is f . We will usually write an object (V; f ) of VR as f : V R or just V R; in this notation V is identified with the object HV of K and " " denotes a morphism of Kop . A morphism 4 Dwyer and Wilkerson from f : V R to g : W R is then a commutative diagram V f R h # k W g R in which "commutativity", of course, means h*g = f ; sometimes we will even refer to f as the "composite" of g and h. The category AR is a subcategory of VR . An object f : V R is in AR if V 6= {0} andf makes HV into a finitely generated module over R; a map h as above is in AR if the abelian group map h : V ! W is a monomorphism. Recall from [10, 3.4] that for any elementary abelian p-group V there is a functor T V : K ! K left adjoint to the functor given by tensor product over Fp with HV ; moreover [4], for any fixed f : V R there is a distinguished quotient TfV(R) of T VR. We will alter the standard notation slightly and write T (V; R) instead of T VR and T (V; R)f instead of TfVR. A homomorphism h : V ! W of elementary abelian p-groups induces a natural transformation T (h; - ) : T (V; - ) ! T (W; - ); given f : V R and g : W R with with h*g = f , the map T (h; R) : T (V; R) ! T (W; R) passes to a quotient map T (V; R)f ! T (W; R)g (see 3.1). Define the functor ffR : AR ! K by setting ffR (f : V R) equal to T (V; R)f . For any V the inclusion 0 V induces a composite map ffl : R = T (0; R) ! T (V; R) ! T (V; R)f (cf. [4, x2]); these combine to give a map R ! lim ffR . Remark 2.1: If R = H* X for some space X, then under mild as- sumptions [10] T (V; R) is naturally isomorphic to the cohomology of the mapping space Map (BV; X) and T (V; R)f to the cohomology of the sub- space Map (BV; X)f consisting of maps which induce f on cohomology. If R = H* BG for a compact Lie group G, then f : V R corresponds to a homomorphism f : V ! G (unique up to conjugacy) and T (V; R)f is naturally isomorphic to H* BC(im (f )) (see [11]). The connection im- plied here between mapping spaces and centralizers is made explicit in [6]. Note that [15, 2.4] a homomorphism V ! G is a monomorphism if and only if the induced map V H* BG makes HV into a finitely generated module over H* BG. Proposition 2.2. If R = H* BG for a compact Lie group G, then there is an equivalence of categories e : AG ! AR such that the composite functor ffR . e is naturally equivalent to ffG . Proof: Define e to be the functor which sends a non-trivial elementary abelian p-subgroup A of G to the pair (A; jA ), where iA : BA ! BG is Decomposition Theorem 5 the map induced by inclusion and jA = i*A; as above the map jA makes HA into a finitely generated module over H* BG. If h : A ! B is a map between two elementary abelian subgroups of G which is realized by conjugation with an element of G, then jA = h*jB because inner automorphisms of G induce the identity map on cohomology. Since homomorphisms between elementary abelian p-groups are detected in cohomology, the functor e is faithful. The fact that e is full and has image intersecting every isomorphism class in AR follows from the result of [11] referred to in 2.1. By [13, p. 91], then, the functor e is an equivalence of categories. A map h : A ! B in AG can be represented by an element xh 2 G which conjugates A in a specific way to a subgroup of B. It is clear that xh is unique up to right multiplication by elements of C(A). Conjugation with x-1h then carries C(B) to C(A) in a way which is well-defined up to inner automorphisms of C(A). Since any inner automorphism of a group induces a self-map of the classifying space which is homotopic to the identity [15, p. 551], we can construct a functor F from AopGto the homotopy category of spaces by setting F (A) = BC(A); it is easy to check that up to homotopy this is the same as the functor [9] which assigns to A the space EG xG (G=C(A)) = EG=C(A). Consequently, the functor ffG is naturally equivalent to H* F . The multiplication map A : BA x BC(A) ! BG induces a K-map H* BG ! HA H* BC(A) which has as adjoint an isomorphism [11] ffR (e(A)) = T (A; H* BG )jA ! H* BC(A) = H* F (A). The fact that these isomorphisms combine to give a natural equivalence ffR . e ! H* F follows from the fact that for any h : A1 ! A2 in AG the diagram 1xx-1h(:)xh BA1 x BC(A2) --- - - - - ! BA1 x BC(A1) ? ? hx1 ?y ?yA1 BA2 x BC(A2) --- - ! BG A2 commutes up to homotopy. It will be convenient below to work with a functor fiR which is closely related to ffR . Let A denote the category whose objects are non-trivial elementary abelian p-groups and whose morphisms are monic group ho- momorphisms. For any object R ofQK, let fiR denote the functor A ! K which assigns to V the product f T (V; R)f , where f runs through all V R which make HV into a finitely generated R module. If j : V ! W is a map of A (ie. a monomorphism), there is for each such 6 Dwyer and Wilkerson Q f : V R a natural map T (V; R)f ! gT (W; R)g, where this latter product is indexed by g : W R such that g makes HW into a finitely generated R module and j* g = f . Combining these maps for all such f gives a map fiR (V ) ! fiR (W ). The natural maps ffl : R ! T (V; R)f combine to give a natural map R ! lim fiR . Proposition 2.3. There are natural isomorphisms lim iffR ! lim ifiR for all i 0. Under these isomorphisms the natural map R ! lim ffR corresponds to the natural map R ! lim fiR . Proof: There is a forgetful functor : AR ! A which assigns to an object (V; f ) of AR the underlying vector space V . Associated to this is a composition of functors spectral sequence E2p;q= lim pq(ffR ) =) lim p+q ffR in which the functors q(ffR ) are defined as follows. For V 2 A, let V # be the under category [13, p. 46] and ffVR the functor obtained by composing ffR with the forgetful functor V # ! AR . Then q(ffR )(V ) is lim qffVR. (For a dual (direct limit) description of this spectral sequence see [7, pp. 155-157].) To complete the proof it is enough to show that 0(ffR ) is isomorphic to fiR and that q(ffR ) vanishes for q > 0. For V 2 A let -1 V denote the subcategory of AR consisting of objects A such that (A) = V ; a morphism in -1 V is a morphism in AR which projects under to the identity map of V . A short calculation shows that the evident inclusion map j : -1 V ! V # is left cofinal [2, XI, x9] so that the induced maps lim qffVR ! lim qffVR O j are isomorphisms (cf. [2, XI, 7.2 and 9.2]). The desired result now follows from the fact that -1 V is equivalent to a discrete category, that is, a category with no non-identity morphisms Remark: The functor fiR is the right Kan extension [13, p. 232] of ffR along the forgetful functor which appears in the above proof. Proposition 2.3 can also be deduced from [9, 3.1]. x 3. Properties of T In this section we will describe some properties of the functor T which are used in x4 and x5. Most of these properties are algebraic analogs of simple properties of function spaces (cf. 2.1). Throughout the section R will stand for an object of K and V for an elementary abelian p-group. Recall [4, x2] that T (V; R)f is the tensor product Fp T (V;R)0T (V; R), where the composite map T (V; R) ! T (V; R)0 ! Fp is adjoint to f : R ! HV . Let I(R) R denote the ideal of positive dimensional elements. Decomposition Theorem 7 Lemma 3.1. Let f : V R be an object of VR , and S an object of K with unit inclusion j : Fp ! S0 and projection ss : S ! S=I(S) = S0 . Then the set Hom K(T (V; R)f ; S) is naturally isomorphic to the set of maps g : R ! HV S such that (1 ss) . g is equal to the composite (1 j) . f : R ! HV = HV Fp ! HV (S0 ). Proof: This follows immediately from the tensor product formula for T (V; R)f and the defining adjointness property of T (V; - ). Lemma 3.2. Given f : V R, the natural map ffl : R ! T (V; R)f [4,x2] induces via 3.1 the map which assigns to R ! HV S the composite R ! HV S ! S obtained using the projection HV ! Fp . Proof: This follows immediately from [4, x2] and could in fact be used to define ffl. Proposition 3.3. Given f : V W R, let a : V R and b : W R be the composites of f with the appropriate direct summand inclusions, and let g : W T (V; R)a be the map which corresponds via 3.1 to f : R ! HV HW ~= HV W . Then there is a natural isomorphism T (W; T (V; R)a )g ! T (V W ; R)f . Under this isomorphism the map T (W; R)b ! T (V W ; R)f induced by W ! V W corresponds to the map T (W; R)b ! T (W; T (V; R)a )g obtained by applying T (W; - ) to the natural map ffl : R ! T (V; R)a. Proof: This is a routine application of Yoneda's lemma [13, p. 62, Ex. 2] that involves using 3.1 and 3.2 to identify the functors corepre- sented by the objects T (W; R)b, T (V W ; R)f , and T (W; T (V; R)a )g. Recall [12] that a module M over the mod p Steenrod algebra Ap is said to be locally finite if every element of M is contained in a finite Ap submodule. Let Q(R) denote the quotient I(R)=I(R)2. Proposition 3.4. Let f : V R be an object of VR , and assume that Q(R) is locally finite as a module over Ap . Then the natural map R ! T (V; R)f is an isomorphism iff (1) R is connected (ie. R0 ~= Fp ) and (2) there exists a K-map R ! HV R which, composed with the evident projections HV R ! R and HV R ! HV , gives, respectively, the identity map of R and the map f . Proof: This follows from [5, 4.7, proof of 4.1]. x4. Centers In this section we will define what it means for an object R of K to have a non-trivial center and then prove a special case (Proposition 4.10) 8 Dwyer and Wilkerson of Theorem 1.2. Throughout the section R will stand for an object of K and V for an elementary abelian p-group. We will develop a formal analogy (cf. 2.1) between properties of the objects V R for R 2 K and properties of homomorphisms up to conjugacy V ! G for a compact Lie group G. It will become clear below that this analogy works well only if Q(R) is locally finite as a module over Ap . Once the analogy is set up, the proof of 4.10 can be carried out by mimicing a group theoretic argument. Definition 4.1: An object f : V R of VR is said to be (1) monic, if f makes HV into a finitely generated module over R, and (2) central, if the natural map R ! T (V; R)f is an isomorphism. . The algebra R is said to have a non-trivial center if there exists a monic central map V R with V 6= {0}. Recall from 2.1 that if R = H* BG for a compact Lie group G then an object f : V R of VR corresponds to a conjugacy class f of group homomorphisms V ! G. The object f is monic in the above sense iff (any representative of) f is a group monomorphism, and central iff the image of f lies in the center of G. In particular, the following is true. Proposition 4.2. If G is a compact Lie group with a non-trivial central element of order p, then H* BG has a non-trivial center. In 4.3-4.9 below we will prove in a more general setting statements which are obvious from the above remarks in the special case R = H* BG. Let f : V R be an object of VR . Choosing an element v of V amounts to giving a homomorphism Ov : Z=p ! V with Ov(1) = v; we will let fv : Z=p R stand for the composite Z=p Ov!V R. Definition 4.3: An object f : V R of VR is said to be null if the K- map f : R ! HV is trivial above dimension 0. The kernel of f , denoted ker (f ), is the set consisting of all elements v 2 V with the property that fv : Z=p R is null. Proposition 4.4. An object f : V R of VR is monic iff ker (f ) = {0}. Proof: If ker(f ) is not trivial, there is a surjective map g : HV ! HZ=p such that the composite gf is trivial above dimension 0; since HZ=p is not of finite rank as an Fp vector space, this implies that HV is not finitely generated as an R module, ie. that f is not monic. Conversely, if HV is not finitely generated as an R module then the quotient ring Fp R HV is not finite-dimensional over Fp . This quotient ring is generated as an Decomposition Theorem 9 algebra by exterior generators {yi} of dimension 1 together with their Bockstein images {fiyi} in dimension 2. If the quotient ring is not finite dimensional, there must be some y 2 {yi} such that fiy is not nilpotent. The subring S of Fp R HV generated by y and fiy is clearly closed under the action of Ap and isomorphic as an Ap algebra to HZ=p . By the injectivity of HZ=p as an Ap algebra [10, 3.6], any chosen isomorphism S ! HZ=p extends to a K-map Fp R HV ! HZ=p ; the composite of such an extension with the quotient map HV ! Fp R HV is a K map HV ! HZ=p representing [10, 4.2] a non-trivial element of V in ker (f ). Lemma 4.5. Suppose that R 2 K is connected and that Q(R) is locally finite as a module over Ap . Then any null object f : V R is central. If g : C R is central and C0 is a subgroup of C then the composite object C0 ! C R is also central. Proof: This is a consequence of the characterization of central objects in Proposition 3.4. Lemma 4.6. Let f : V R and g : C R be objects of VR , and assume that g is central. Then there is a unique object f g : V C R which restricts to f (resp. g) along the summand inclusion V ! V C (resp. C ! V C). Proof: The desired object V C R corresponds to a map h : R ! HV C ~= HV HC which agrees with f when composed with HV HC ! HV and with g when composed with HV HC ! HC . According to 3.1 and 3.2, such an h amounts to a map h : T (C; R)g ! HV which agrees f when composed with ffl : R ! T (C; R)g. The lemma follows from the fact that, since g is central, the map ffl : R ! T (C; R)g is an isomorphism. Lemma 4.7. Suppose that R 2 K is connected and that Q(R) is locally finite as a module over Ap . Let f : V R be an object of VR , v 2 V an element of ker(f ), and < v> the subgroup of V generated by v. Then f extends to a unique map g : V =< v> R. Proof: Let W V be a complement to < v> and h : W R the composite of f with the inclusion W ! V . There is an isomorphism V ~=W < v> and by 4.5 the composite < v> ! V R is central; by 4.6, then, f is the unique object V R which is null on < v> and agrees with h on W . Clearly, then, f is the composite of h with the projection map V ! V =< v> ~= W . Remark: The following example shows that in Lemma 4.7 some re- striction on R is necessary. Let V be an elementary abelian p-groupQof rank greater than 1, J HV the ideal generated by the product y fiy 10 Dwyer and Wilkerson as y runs through the non zero one-dimensional elements of HV , R the ring Fp J, and f : R ! HV the evident inclusion. Every element of V is in the kernel of f : V R but f does not extend over V =W for any non-trivial subgroup W of V . Proposition 4.8. Suppose that R 2 K is connected and that Q(R) is locally finite as module over Ap . Let f : V R be an object of VR . Then ker(f ) is a subgroup of V , and f extends uniquely to a monic map g : V = ker(f ) R. Proof: Let W be a maximal subgroup of V such that f extends to an object g : V =W R. It is clear that W ker(f ). However, g must be monic by 4.7, so it follows easily that ker (f ) W . Proposition 4.9. Suppose that R 2 K is connected and that Q(R) is locally finite as a module over Ap . Let f : V R, g : C R, and h : W R be objects of VR with g central and h monic, and let f g : V C R be the unique object (4.6) which extends f and g. Then, given a map u : f ! h in VR , there is at most one way of extending u to a map ^u: f g ! h. Proof: The map ^uamounts to an abelian group map V C ! W which is prescribed on V ; to show that ^uis unique (if it exists) it is enough to treat the special case V = {0} and (see 4.5) C ~= Z=p . Suppose that ^u1 and ^u2are two maps Z=p ! W which give the same object g : Z=p R when composed with h : W R, and let ^u1+ ^u2: Z=p Z=p ! W be their sum. The composite of ^u1+ ^u2with h is an object Z=p Z=p R which by 4.6 agrees with the composite Z=p Z=p !+ Z=p R and therefore has each element of the form (x; -x) in Z=p Z=p in its kernel. Since h is monic, it follows easily that ker (^u1 + ^u2) also contains all elements of the form (x; -x). This shows that u^1(x) = u^2(x) for all x 2 Z=p . Proposition 4.10. Suppose that R is an object of K with the property that Q(R) is locally finite as a module over Ap . Assume that R has a non-trivial center. Then the natural map R ! lim ffR is an isomorphism and the groups lim iffR vanish for i > 0. Remark: The proof of 4.10 shows that if R has a non-trivial center then the nerve of the category AR is contractible. In fact, the main argument in this proof is very similar to the argument in Quillen's proof that the poset of elementary abelian subgroups of a finite group is contractible if the group has a non-trivial normal p-subgroup [16, 2.4]. Proof of 4.10: Observe by 3.4 that R is connected. Choose a monic central object g : C R of VR such that C 6= {0}, so that g also Decomposition Theorem 11 represents an object of AR . Let g # AR be the under category [13, p. 46] and j : g # AR ! AR the forgetful functor. The category g # AR has the identity map of g as an initial object and by assumption the natural map R ! ffR (g) = ffR (j(g !1 g)) is an isomorphism; it follows [2, XI, 7.2 and 9.2] that the natural map R ! lim (ffR . j) is an isomorphism and that lim i(ffR . j) vanishes for i > 0. Given an arbitrary object f : V R of AR there is (4.6) a unique map f g : V C R of VR which restricts to f (resp. g) along the summand inclusion V ! V C (resp. C ! V C); let oe(f ) : (V C)= ker(f g) R be the corresponding (4.8) object of AR . It is clear that the construction f 7! oe(f ) produces a functor oe : AR ! g # AR and that the natural map V ! (V C)= ker(f g) induces a natural transformation o from the identity functor of AR to the composite j . oe. Now the group (V C)= ker(f g) can be written as a direct sum V C0, where C0 C is complementary to the kernel of the composite map C ! V C ! (V C)=(V + ker(f g)); by (4.5) the composite C0 ! C R is central, and so by (3.3) the map T (V; R)f ! T ((V C)= ker(f g) ; R)oe(f)induced by V ! (V C)= ker(f g) is an isomorphism. It follows that the natural transformation from ffR to ffR . j . oe induced by o is a natural equivalence. Now let x = (C ! W R) be an object of g # AR , let f : V R be an object of AR , and let h : W R in AR be the image j(x) of x under the forgetful functor. Suppose that oe(f ) ! x is a map in g # AR . According to (4.9) and the above remarks there is a unique map w : f ! h in AR such that the given map oe(f ) ! x is the composite of oe(w) with the evident natural isomorphism oe(j(x)) = oe(h) ~= x. This shows that the over category oe # x has a terminal object and therefore a contractible nerve [2, XI, x2]; consequently, the functor oe is left cofinal [2, XI, x9]. The proposition now follows directly from the fact [2, XI, 7.2 and 9.2] that the natural maps lim i(ffR . j) ! lim i(ffR . j . oe) ~= lim iffR are isomorphisms for all i. x5. Completion of the proof In this section we will complete the proof of Theorem 1.2 by showing that the functor fiR of 2.3 is a retract of the acyclic (4.10) functor fiS . Note that if R is an object of K which is finitely generated as an algebra and V is an elementary abelian p-group, there are only a finite number of K-maps R ! HV (ie.Qobjects V R). In this case T (V; R) splits as a direct product fT (V; R)f indexed by f : V R (see [10, 3.5]). Recall from x2 that there is a functor fiRQ : A ! K which assigns to each elementary abelian p-group V the product fT (V; R)f where f runs through monic (x4) objects V R, (that is, maps f : R ! HV which make HV into a finitely generated module over R). 12 Dwyer and Wilkerson Lemma 5.1. Suppose that i : R ! S is a map of K such that R and S are finitely generated as algebras and i makes S into a finitely gen- erated module over R. Then for any elementary abelian p-group V the quotient fiS (V ) of T (V; S) is naturally isomorphic to the tensor product fiR (V ) T (V;R) T (V; S). Proof: For fixed f : R ! HVQ, the tensor product T (V; R)f T (V;R) T (V; S) is a direct product gT (V; S)g where g runs through all maps S ! HV which extend f . The lemma follows from the fact that such a g makes HV into a finitely generated module over S iff f makes HV into a finitely generated module over R. Remark 5.2: Let U be the category of unstable modules over Ap and : K ! U the forgetful functor [10, x1]. There is a functor T U(V; - ) : U ! U defined like T (V; - ) such that for R 2 K the module T U(V; (R) ) is naturally isomorphic to (T (V; R)) (see [10, 3.4]). If M in U is a module over R in such a way that the Cartan formula holds for mod- ule multiplication (write M 2 U (R) [4, x1]) then T U(V; M ) is a module over T (V; R) [10, 3.2.1]. Finally, if i : R ! S is a map of K which is used to make (S) into a module over R, then the T (V; R) module structure on T U(V; (S) ) described above agrees via the natural isomor- phism T U(V; (S) ) ~= (T (V; S)) with the module structure on T (V; S) derived from the induced ring homomorphism T (V; R) ! T (V; S). Proof of 1.2: Let flR : A ! K assign to the elementary abelian p- group V the ring T (V; R) and let flS : A ! K assign the ring T (V; S). The functors fiR , fiS and flS are modules over flR (ie., for any V in A the object flR (V ) is a ring and the objects fiR (V ), fiS (V ) and flS (V ) a* *re modules over flR (V ) in a way that varies in an evident functorial way with V ). By 5.1 there is a natural equivalence fiS ~= fiR flRflS . By 5.2 the U (R)-map S ! R, left inverse to R ! S, induces a map fiS ~= fiR flRflS ! fiR flRflR ~=fiR left inverse to the natural map fiR ! fiS . By 4.10 and 2.3 the natural map S ! lim fiS is an isomorphism and lim ifiS = 0 for i > 0. It follows easily that R ! lim fiR is an isomorphism (any retract of an isomorphism is an isomorphism) and that lim ifiR = 0 for i > 0. The conclusion of the theorem results from another application of 2.3. x6. A homotopy decomposition theorem In this section we will show that our main algebraic results give rise to homotopy-theoretic decompositions in some situations which are not covered by [9]. Decomposition Theorem 13 The first step is to give a topological construction of the decomposition diagram (x2). Given a space X, define VX to be the category whose objects consist of pairs (V; f ), where V is an elementary abelian p-group and f is a homotopy class of maps from BV to X. A map (V; f ) ! (W; g) in VX is an abelian group map h : V ! W such that f = g O h, where h : BV ! BW is the homotopy class of maps induced by h. The category AX is defined to be a subcategory of VX . An object (V; f ) of VX is in AX if V 6= {0} and the cohomology map f * makes HV into a finitely generated module over H* X; a map h as above is in AX if h : V ! W is a monomorphism of abelian groups. Let Map (BV; X) denote the space of maps BV ! X. There is a functor ^ffX from AopXto the category of spaces which assigns to (V; f ) the path component Map (BV; X)f of Map (BV; X) determined by f . (To construct f^fX it is necessary to settle on some strictly functorial way of constructing BV from V - see, for instance, [14, 23.2, Ch. 3]). Evaluating at the basepoint of BV gives a map f^fX(V; f ) ! X; these maps combine to give a natural map holim!^ffX ! X. Define ffX : AX ! K to be the functor determined by the formula ffX (V; f ) = H* (f^fX(V; f )). Let R denote H* X. Taking mod p cohomology gives a functor HX : AX ! AR , and it is clear from the universal property of T (V; -)f (cf. 3.1, [10]) that there is a natural transformation X : ffR O HX ! ffX . The following proposition is proved by combining the spectral sequence of [2, XII, 5.8] with a naturality argument. Proposition 6.1. Let X be a space and let R = H* X. Suppose that (1) HX : AX ! AR is an equivalence of categories, (2) X : ffR O HX ! ffX is a natural equivalence, and (3) the natural map R ! lim ffR is an isomorphism and the groups lim iffR vanish for i > 0. Then the natural map ho lim!^ffX! X induces an isomorphism on mod p cohomology. Lannes has provided some mild conditions under which assumptions (1) and (2) of Proposition 6.1 are satisfied. Say that an object R of K is of finite type if each Rk is finite dimensional as an Fp vector space. Proposition 6.2 [10]. If X is a p-complete [2] space such that H* X is of finite type, then HX is an equivalence of categories. Proposition 6.3 [10]. Let X be a p-complete space such that R = H* X is of finite type. Assume that for each object f : V R of VR the graded algebra T (V; R)f is of finite type and is trivial in dimension 1. Then X is a natural equivalence. 14 Dwyer and Wilkerson Remark: It is also true that assumptions (1) and (2) of 6.1 are satisfied if G is a compact Lie group and X = BG (see 2.2, [11], [6]). Now let o be a torus, W a finite group of K-automorphisms of H* Bo , and R the ring of invariants H* (Bo )W . Suppose that V is an elementary abelian p-group. By the exactness property of the functor T (V; -) [10] there is an isomorphism T (V; R) ~= T (V; H* Bo )W , and by [11] the alge- bra T (V; H* Bo ) is isomorphic to a direct product of copies of H* Bo , one for each homomorphism V ! o . It follows that for any f : V R the algebra T (V; R)f is of finite type and is concentrated in even dimensions. In light of 6.1-6.3 and 1.5, this proves the following theorem. Theorem 6.4. Let X be a p-complete space with the property that H* X is isomorphic to the ring of invariants H* (Bo )W for some torus o and some finite group W of K-automorphisms of H* Bo (see for instance [3]). Then the natural map ho lim!^ffX! X induces an isomorphism on mod p cohomology. Remark: Sometimes conditions (1) and (2) of 6.1 are satisfied but the map ho lim!^ffX! X does not induce an isomorphism on mod p cohomol- ogy. By more or less explicit calculation this happens if p is 2 and X is the one-point`union BZ=2_BZ=2 _ in this case ho lim!^ffXis the disjoint union BZ=2 BZ=2. It is not yet clear to us whether or not there is a reasonable characterization of spaces X for which the conclusion of 6.1 holds. References [1] J. C. Becker and D. H. Gottlieb, The transfer map and fiber bundles, Top* *ology 14 (1975), 1-12. [2] A.K. Bousfield and D.M. Kan, "Homotopy Limits, Completions and Localiza- tions," Lecture Notes in Mathematics vol. 304, Springer-Verlag, Berlin, 19* *72. [3] A. Clark and J. Ewing, The realization of polynomial algebras as cohomol* *ogy rings, Pacific J. Math. 50 (1974), 425-434. [4] W. G. Dwyer and C. W. Wilkerson, Smith theory and the functor T , prepri* *nt (University of Notre Dame) 1988. [5] W. G. Dwyer and C. W. Wilkerson, Spaces of null homotopic maps, preprint (University of Notre Dame) 1989. [6] W. G. Dwyer and A. Zabrodsky, Maps between classifying spaces, in "Algeb* *raic Topology, Barcelona 1986," Lecture Notes in Math. 1298, Springer, Berlin, * *1987, pp. 106-119. [7] P. Gabriel and M. Zisman, "Calculus of Fractions and Homotopy Theory," Ergebnisse der Mathematik vol. 35, Springer, Berlin, 1967. [8] M. Hall, "The theory of groups," Macmillan, New York, 1959. [9] S. Jackowski and J. E. McClure, A homotopy decomposition theorem for cla* *s- sifying spaces of compact Lie groups. Decomposition Theorem 15 [10] J. Lannes, Sur la cohomologie modulo p des p-groupes Abelienselementair* *es, in "Homotopy Theory, Proc. Durham Symp. 1985," edited by E. Rees and J.D.S. Jones, Cambridge Univ. Press, Cambridge, 1987. [11] __________, Cohomology of groups and function spaces, preprint (Ecole P* *oly- technique) 1987. [12] J. Lannes and L. Schwartz, Sur la structure des A-modules instables inj* *ectifs, preprint (Ecole polytechnique). [13] S. Mac Lane, "Categories for the Working Mathematician," Graduate Texts in Mathematics vol. 5, Springer, Berlin, 1971. [14] J. P. May, "Simplicial Objects in Algebraic Topology," Math. Studies No* *. 11, Van Nostrand, Princeton, 1967. [15] D. G. Quillen, The spectrum of an equivariant cohomology ring: I, Annal* *s of Math. 94 (1971), 549-572. [16] D. G. Quillen, Homotopy properties of the poset of nontrivial p-subgrou* *ps of a group, Advances in Math. 28 (1978), 101-128. [17] B. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Ak* *ad. Nauk SSSR 127 (1959), 943-944. University of Notre Dame, Notre Dame, Indiana 46556 Purdue University, West Lafayette, Indiana 47907