EQUIVARIANT p-ADIC HOMOTOPY THEORY MICHAEL A. MANDELL Abstract.Let G be a finite group. We show that the cochain functor with coefficients in Fp gives an equivalence between the p-adic equivariant h* *omo- topy category of finite type nilpotent G-spaces and a full subcategory o* *f the homotopy category of diagrams of E1 Fp-algebras indexed on the orbit cat* *e- gory of G. This turns out to be an easy consequence of Elmendorf's Theor* *em and Kan's work on diagrams in closed model categories plus the equivalen* *ce in the nonequivariant context. Introduction The paper [10] gave an algebraic description of the p-adic homotopy theory of finite type nilpotent spaces. This paper extends those results to the case when a finite group G acts. We will see that an algebraic description of the p-adic homotopy theory of finite type nilpotent G-spaces is an easy consequence of the algebraic description of nonequivariant spaces in [10], Elmendorf's description* * [6] of the homotopy theory of G-spaces, and Kan's unpublished but widely known work (q.v. [9, x14]) on diagram categories of Quillen closed model categories [14]. The algebraic models for G-spaces are diagrams of E1 Fp-algebras indexed on the orbit category of G. The orbit category OG is the full subcategory of the category of G-spaces consisting of the objects G=H for H G. In other words, there is one object G=H of OG for each subgroup H of G, and the set of maps OG (G=H; G=K) is the set of G-maps from G=H to G=K, or equivalently the set of H-fixed points of G=K. We define an OG -algebra to be a (covariant) functor from OG to the category of E1 Fp-algebras. A map of OG -algebras is a natural transformation of functors. We denote the category of OG -algebras as OG E. We have a singular cochain functor C_*from the category of G-spaces to the category of OG -algebras defined by C_*X(G=H) = C*(XH ): Since a weak equivalence of G-spaces is a G-map that is a (nonequivariant) weak equivalence on each fixed point space, it is clear that C_*preserves weak equiv* *a- lences. We say that a G-space X is G-connected, nilpotent, p-complete, or finite type when all of its fixed point spaces XH are connected, nilpotent, p-complet* *e, or finite type respectively. In [13, II.x4-5], it is explained why these defini* *tions for nilpotent and p-complete are good equivariant generalizations of the correspond- ing nonequivariant concepts. For example, just as in the nonequivariant case, t* *he homotopy category of the finite type p-complete nilpotent G-spaces is a full su* *b- category of the equivariant homotopy category and also a full subcategory of the p-adic equivariant homotopy category, the category obtained from the category of ____________ Date: February 29, 2000, 14:04. The author was supported in part by NSF postdoctoral fellowship DMS 9804421. 1 2 MICHAEL A. MANDELL G-spaces by formally inverting the maps that are Fp-homology isomorphisms on all fixed point spaces. We prove the following algebraization theorem for equivaria* *nt p-adic homotopy theory. Main Theorem. Let G be a finite group. The map Ho (GTop)(X; Y ) -! Ho(OG E)(C_*Y; C_*X) induced by the derived functor C_*is a bijection when Y is a finite type p-comp* *lete nilpotent G-space. In particular, C_*embeds the full subcategory of the equivar* *iant homotopy category consisting of the finite type p-complete nilpotent G-spaces a* *s a full subcategory of the homotopy category of OG -algebras. We have not restricted to the G-connected case, and so in the case of the tri* *vial group, the previous theorem is actually stronger than the Main Theorem of [10], which treated only the connected case. Section 3 presents a homotopy theoretic argument that extends the results of [10] to the nonconnected case. Theorem 3.2 gives the precise statement, the nonequivariant version of the Main Theorem abo* *ve. The argument in the nonequivariant case of [10] was to construct the functor U, a contravariant right adjoint to the derived functor C*, and to prove that t* *he unit of the adjunction Y -! UC*Y is an isomorphism in the homotopy category whenever Y is a (connected) finite type, p-complete, nilpotent space. We use a similar strategy. Here also the derived functor C_*has a contravariant right ad* *joint, UG ; we prove the following theorem in Sections 1-2. Theorem A. The derived functor C_*has a contravariant right adjoint, i.e. the* *re is a contravariant functor UG from the homotopy category of OG -algebras to the equivariant homotopy category and a bijection Ho(GTop)(X; UG A_) ~=Ho(OG E)(A_; C_*X) that is natural in the G-space X and the OG -algebra A_. The composite of the map in the Main Theorem with the adjunction bijection in Theorem A Ho (GTop)(X; Y ) -! Ho(OG E)(C_*Y; C_*X) ~=Ho(GTop)(X; UG C_*Y ) is induced by the unit of the adjunction Y -! UG C_*Y , by definition. Just as * *in the nonequivariant case, the Main Theorem follows immediately once we show that the unit of the derived adjunction Y -! UG C_*Y is a weak equivalence whenever Y is a finite type, p-complete, nilpotent G-space. In fact, we prove the follow* *ing theorem, which reduces the Main Theorem to the nonequivariant case. Theorem B. For all H G, there is a natural isomorphism in the nonequivariant homotopy category (UG C_*X)H ~=UC*XH . The composite XH - ! (UG C_*X)H ~=UC*XH is the unit of the nonequivariant adjunction. As additional corollaries of Theorem B, we get the equivariant analogues of t* *he observations [10, 5.1] and [10, B.1]. Namely, if X is a finite type G-connected G-space, then the map X -! UG C_*X performs Bousfield-Kan p-completion on each fixed point space. For an arbitrary G-connected G-space X, the map X -! UG C_*cX performs Sullivan p-pro-finite completion on each fixed point space, w* *here C_*cX(G=H) = C*cXH = C*(XH ; Fp) Fp. EQUIVARIANT p-ADIC HOMOTOPY THEORY 3 The proof of Theorem B in Sections 1-2 is an easy consequence of the con- struction of the adjoint UG , and we simultaneously prove the following symmetr* *ic result. Theorem C. For all H G, there is a natural isomorphism in the homotopy category of E1 Fp-algebras (C_*UG A_)(G=H) ~=C*U(A_(G=H)). The composite A(G=H) -! (C_*UG A_)(G=H) ~=C*U(A_(G=H)) is the unit of the nonequivariant adjunction. We obtain the following Characterization Theorem as an immediate consequence of Theorem C. We say that a G-space X is G-simply connected when each fixed point space has each of its components (nonequivariantly) simply connected. Characterization Theorem. An OG -space A is isomorphic in Ho(OG E) to the image under C_*of a finite type nilpotent or finite type G-simply connected G-s* *pace if and only if each A_(G=H) is isomorphic in Ho(E) to the image under C* of a finite type nilpotent or finite type simply connected space. Criteria for an E1 Fp-algebra to be isomorphic in Ho(E) to the image under C* of a connected finite type nilpotent and to a finite type 1-connected space are* * given in [10, 7.3] and in the Characterization Theorem of [10] respectively. The crit* *erion for simply connected spaces is extended to nonconnected spaces in Section 4 and* * can again be stated purely in terms of the cohomology of the E1 Fp -algebra as a mo* *dule with an action of the "Steenrod operation" P 0. We state this as Theorem 4.2. Rational Equivariant Homotopy Theory. The methods used in this paper are very general and extend to the rational case, where we are free to use the Thom- Sullivan PL De Rham complex [16] of a simplicial set or of the singular simplic* *ial set of a space in place of the cochain complex. In this case, Bousfield-Gugenhe* *im [2] gives the category of rational commutative differential graded algebras a c* *losed model structure and constructs an adjoint to the De Rham complex functor. Our arguments apply verbatim to this context. In this way, we obtain the following variant of [17]. In it, A_*denotes the functor that sends a G-space to the OG - diagram of rational commutative differential graded algebras obtained by applyi* *ng the Thom-Sullivan PL De Rham complex functor to the singular simplicial sets of the fixed point spaces, A_*X(G=H) = A*(XH ). Theorem. Let G be a finite group. (i)The map Ho(GTop)(X; Y ) -! Ho(OG CQ )(A_*Y; A_*X) induced by the derived functor A_*is a bijection when Y is a finite type r* *a- tional nilpotent G-space. In particular, A_*embeds the full subcategory of* * the equivariant homotopy category consisting of the finite type rational nilpo* *tent G-spaces as a full subcategory of the homotopy category of OG -diagrams of rational commutative differential graded algebras. (ii)The full subcategory of the G-simply connected rational G-spaces is equiva* *lent to the full subcategory of those diagrams that are objectwise finite type * *simply connected algebras. 4 MICHAEL A. MANDELL Compact Lie Groups. The methods of this paper do not extend directly to the case of a compact lie group of dimension greater than zero. Even in the rational context, the problem of describing equivariant homotopy theory is current, the * *case G = S1 being the recent work of L. Scull [15]. The difficulty is that when G is a lie group of dimension greater than zero, the orbit category is best viewed as a topological category, and diagram categories of E1 Fp-algebras or of rational commutative algebras do not generalize to this context in any reasonable way. Work of Dwyer and Kan [4] does give a homotopy theory of G-spaces in terms of* * a category of diagrams on a discrete category, or more precisely, the the over-ca* *tegory of a certain classifying space diagram; one can consider the analogous (opposit* *e) diagram category in algebra. Unfortunately, this theory does not mix well with rationalization or p-completion. Explaining why would require a long digressio* *n, but the basic reason is that rational homotopy theory cannot distinguish a space from a finite cover and p-adic homotopy theory cannot distinguish a space from * *an n-fold cover when n is relatively prime to p. This often forces the diagram cat* *egory to have too many rational and p-adic equivalences, and so it loses some rational and p-adic homotopy theory information. Adaptation of [4] to give an algebraic description of equivariant p-adic and * *ra- tional homotopy theory for compact lie groups requires separating out and then mixing back in the action of the finite groups of components of the Weyl group- s. This approach is work in progress of the author and L. Scull [12]. The case * *of an abelian compact lie group is significantly easier [11], and the diagram cate* *gory simplifies to look similar to the one appearing in [15]. 1. Elmendorf's Theorem and a Reduction of Theorems A, B, and C As explained in the introduction, the Main Theorem follows immediately from Theorems A and B together with the Main Theorem of [10] (for G-connected s- paces) or the nonequivariant case of the Main Theorem, Theorem 3.2 (for its full generality). Likewise, the Characterization Theorem follows immediately from Th* *e- orems A and C. We prove Theorems A, B, and C in this section and the next. In this section, we use Elmendorf's Theorem [6] to reduce Theorems A, B, and C to the more natural compound theorem, Theorem 1.2 below. In the next section, we prove Theorem 1.2. Let OG Topdenote the category of contravariant functors from OG to the catego* *ry of topological spaces, and let OG S denote the category of contravariant functo* *rs from OG to the category of simplicial sets. We call the objects of OG Top "OG - spaces" and the objects of OG S "OG -simplicial sets". We say that a map X_-! Y_ of OG -spaces or of OG -simplicial sets is a weak equivalence when it is an obj* *ectwise (nonequivariant) weak equivalence, in other words, when it is a nonequivariant weak equivalence X_(G=H) -! Y_(G=H) for every object G=H in OG . We denote by Ho(OG Top) and Ho(OG S ) the categories obtained from OG Topand OG S by formally inverting the weak equivalences (see also Proposition 2.2 below). The geometric realization and singular simplicial set functors applied objectwise g* *ive us adjoint functors OG S -! OG Topand OG Top- ! OG S . Clearly these functors preserve all weak equivalences, and so pass to derived functors Ho (OG S ) -! Ho(OG Top) and Ho(OG Top) -! Ho(OG S ); it is easy to see the latter functors a* *re inverse equivalences. EQUIVARIANT p-ADIC HOMOTOPY THEORY 5 We have a functor from the category of G-spaces to the category of OG -spaces defined as follows. For a G-space X, let X(G=H) be the space of G-maps from G=H to X, X(G=H) = GTop(G=H; X) = XH : A map of G-spaces G=H -! G=K then induces a map X(G=K) -! X(G=H) in a natural way, making X an OG -space, functorial in X. Since an equivariant weak equivalence is a map that is a (nonequivariant) weak equivalence on XH for* * all H G, we see that a map of G-spaces X -! Y is an equivariant weak equivalence if and only if the induced map X -! Y is a weak equivalence in OG Top. Thus, in particular, passes to a derived functor Ho (GTop) -! Ho (OG Top) that by abuse we also denote as . Elmendorf's Theorem [6] is that the derived functor is an equivalence of categories. Proposition 1.1.[6] The categories Ho(GTop), Ho(OG Top), and Ho(OG S ) are equivalent. The functor C_*from G-spaces to OG -algebras is the composite of the functor : GTop -! OG Top, the singular simplicial set functor OG Top -! OG S , and the normalized cochain functor (applied objectwise) C_*:OG S - ! OG E. Since all of these functors preserve all weak equivalences, the composite of the deri* *ved functors is the derived functor of the composite. Since the derived functor of * * and the derived singular simplicial set functor are equivalences of categories, to * *prove Theorems A, B, and C, it suffices to prove the following compound theorem. Theorem 1.2. The derived functor C_*:Ho(OG S ) -! Ho(OG E) has an adjoint U_:Ho(OG E) -! Ho(OG S ), satisfying Ho (OG S )(X_; U_A_) ~=Ho(OG E)(A_; C_*X_): There is a natural isomorphism (U_A_)(G=H) ~=U(A_(G=H)) in Ho E, and under this isomorphism, the unit maps X_-! U_C_*X_ and A_-! C_*U_A_ induce on each object G=H of OG the units of the nonequivariant adjunction of [* *10] X_(G=H) -! UC*(X_(G=H)) and A_(G=H) -! C*U(A_(G=H)) for every OG -algebra A_and every OG -simplicial set X_. 2. Kan's Work on Diagram Categories and the Proof of Theorem 1.2 Theorem 1.2 is about the derived functor of an adjoint functor applied object- wise on diagram categories. When the categories are Quillen model categories of a certain type and the functor is a Quillen adjoint functor, this is exactly the situation studied by Kan and presented for example in [9, x14]. Kan defines a "cofibrantly generated" Quillen model category and shows that diagram categories of cofibrantly generated Quillen model categories are cofibrantly generated Qui* *llen model categories with weak equivalences and fibrations defined objectwise. We do not review this theory in general but only in the concrete cases we need. In t* *he case of E1 algebras, this theory provides the following result. 6 MICHAEL A. MANDELL Proposition 2.1.Let k be a commutative ring, let E be a cofibrant E1 operad of differential graded k-modules, and let D be a small category. The category * *of covariant functors from D to E1 k-algebras (over E) is a closed model category* * with weak equivalences the objectwise quasi-isomorphisms and fibrations the objectwi* *se surjections. All cofibrations are objectwise cofibrations but generally not vic* *e-versa. For brevity, we call a functor D -! E a D-algebra and denote the category of D-algebras as DE. We explain the proof of Proposition 2.1 below. The proof uses the observation of Goerss-Hopkins (unpublished) and Hinich [8] that the category of algebras over a cofibrant operad is a closed model category with cofibration* *s, fi- brations, and weak equivalences as defined in [10, 2.2]. For an arbitrary E1 op* *erad, the diagram category has exactly the same kind of "almost" closed model structu* *re that the category of E-algebras has, as explained in [10, x2]. However, since * *the axioms for this kind of structure were not spelled out in [10, x2], we have cho* *sen precision over generality and written Proposition 2.1 in terms of genuine closed model structures. In the case of simplicial sets, the result we need was already given in [3] a* *nd implicitly in [14]. As it is well known, we do not repeat the proof below. Proposition 2.2.Let D be a small (discrete) category. The category of contravar* *i- ant functors from D to simplicial sets is a closed model category with weak equ* *iva- lences the objectwise weak equivalences and fibrations the objectwise Kan fibra* *tions. All cofibrations are objectwise injections but generally not vice-versa. These two propositions are all that is needed for the proof of Theorem 1.2. Proof of Theorem 1.2.The functor C_*is the normalized cochain functor C* applied objectwise. As shown in [10, 4.2], C* has a contravariant right adjoint U. Let * *U_ be the functor from OG -algebras to OG -simplicial sets obtained by applying U objectwise. The functor C_*converts all weak equivalences to quasi-isomorphisms and converts objectwise injections to fibrations, and so the pair C_*; U_satisf* *ies the contravariant right adjoint form of [14, Theorem 4-3]. In particular the ri* *ght derived functor U_of U_exists and gives an adjunction on homotopy categories Ho (OG S )(X_; U_A_) ~=Ho(OG E)(A_; C_*X_): This proves the first part of Theorem 1.2. The derived functor U_is constructed by taking a cofibrant approximation and then applying the functor U_. It follows from Proposition 2.1 that if QA_-! A_i* *s a cofibrant approximation in OG E, then in particular for each object G=H of OG ,* * the map QA_(G=H) -! A_(G=H) is a cofibrant approximation in E. Since UA_(G=H) is constructed by taking a cofibrant approximation of A_(G=H) in E and applying U, we have (U_A_)(G=H) = (U_QA_)(G=H) = U(QA_(G=H)) = U(A_(G=H)): The remainder of Theorem 1.2 now follows. |___| We now return to the category of D-algebras and the explanation of the closed model structure. We begin by describing the cofibrations. These begin with the following "free" functors. Definition 2.3.For each object d in D, let F_d:E -! DE be the functor ` (F_dA)(e) = A: D(d;e) EQUIVARIANT p-ADIC HOMOTOPY THEORY 7 In other words, F_dA is the diagram that at an object e of D is the coproduct of copies of A indexed on the set D(d; e) of maps from d to e in D. A map fl :e -!* * f in D sends the copy of A corresponding to ff: d -! e to the copy of A corresponding to fl O ff via the identity map of A. Write Evd for the functor from DE to E that takes a D-algebra B_ to the E1 algebra B_(d). The following variant of Yoneda's Lemma is elementary. Proposition 2.4.The functor F_dis left adjoint to the functor Evd. We define cofibrations of D-algebras as follows. Definition 2.5.We say that a map A_-! B_of D-algebras is a relative cell inclu- sion, if B_is the colimit of a sequence of maps A_= A_0-! A_1-! . .-.! A_n-! . . . such that A_0= A_and A_n+1is a pushout A_n+1= A_nqX_nY_n; where ` ` X_n = F_dEMd;n; and Y_n = F_dECMd;n; d2D d2D for some differential graded k-modules Md;nthat are degreewise free with zero differential. Here E denotes the free E1 algebra functor. We say that A_is a c* *ell D-algebra if the initial map k_-! A_is a relative cell inclusion. We say that a* * map of D-algebras A_-! B_is a cofibration if it is the retract of a relative cell i* *nclusion. For fixed e in D, X_n(e) = EMn and Y_n(e) = ECMn for a differential graded k-module Mn that is degreewise free with zero differential, namely L Mn = Md;n: D(d;e) It follows that when A_-! B_is a relative cell inclusion or cofibration of D-al* *gebras, each A_(e) -! B_(e) is a relative cell inclusion or cofibration of E1 algebras.* * Propo- sition 2.4, the small objects argument, and the usual lifting argument give the following result. Proposition 2.6.Every map of D-algebras can be factored as a cofibration fol- lowed by an acyclic fibration. A map of D-algebras is a cofibration if and only* * if it has the left lifting property with respect to the acyclic fibrations. A map * *of D- algebras is an acyclic fibration if and only if it has the right lifting proper* *ty with respect to the cofibrations. We can also factor a map of D-algebras functorially as a cofibration that has* * the left lifting property with respect to all fibrations followed by a fibration. E* *xplicitly, given f :A_- ! B_, let Xd be the free graded k-module with one generator in each degree n for each homogeneous degree n element of B_(d), and let Yd be the differential graded k-module obtained giving Xd a free differential. So for eac* *h d, we have a map of differential graded k-modules Yd -! B_(d) obtained by sending each generator to the element indexing it. We obtain a map of E1 algebras EYd -! B_(d) and a map of D-algebras F_dEYd -! B_. The map f factors as ` A_-! A_q ( F_dEYd) -! B_: d2D 8 MICHAEL A. MANDELL By construction, the righthand map is surjective at each object d of D and is therefore a fibration. The lefthand map is clearly a relative cell inclusion, a* *nd we need to identify when it is a weak equivalence. On each object d the inclusion of A_in the coproduct above is isomorphic to A_(d) -! A_(d) q ECZd for some differential graded k-module Zd that is degreewise free with zero diff* *er- ential. Now if A_ is a cofibrant D-algebra or even just an objectwise cofibrant D-algebra, it follows from [10, 13.2] that the inclusion of A_in the coproduct * *above is an objectwise quasi-isomorphism. On the other hand, if the underlying oper- ad is cofibrant, the main lemma of Hinich [8] implies that for arbitrary A_, the inclusion of A_in the coproduct above is an objectwise quasi-isomorphism. Using Proposition 2.4 and the usual lifting argument, we obtain the following proposi* *tion. Proposition 2.7.Let E be a cofibrant E1 operad or let A_be a cofibrant D-algeb* *ra. A map A_-! B_can be factored as an acyclic cofibration followed by a fibration.* * A map A_-! B_is an acyclic cofibration if and only if it has the left lifting pro* *perty with respect to the fibrations. A map is a fibration if and only if it has the * *right lifting property with respect to the acyclic cofibrations. Proof of Theorem 2.1.The category of D-algebras has all limits and colimits; th* *ese are formed objectwise from limits and colimits of E1 algebras. The classes of * *cofi- brations, fibrations, and weak equivalences are closed under retracts, and the * *weak equivalences satisfy the two-out-of-three property. The previous two propositio* *ns provide the factorization and lifting properties. We have already observed abov* *e __ that a cofibration is in particular an objectwise cofibration. * * |__| 3.Extension to Nonconnected Spaces This section expands the Main Theorem of [10] to the case of nonconnected spaces, proving in full generality the nonequivariant version of the Main Theor* *em of this paper. The essential step of the proof is to understand the effect of * *the composite functor UC* on nonconnected spaces. We know that C* sends a disjoint union of spaces or simplicial sets to the cartesian product of E1 algebras; we* * show here that the functor U sends finite cartesian products of E1 algebras to disj* *oint unions of spaces. The major part of the section is devoted to the proof of the following theorem. Because this theorem may be interesting in other contexts, we state it in slightly more generality than we need here. Theorem 3.1. Let k be an integral domain, and let A1 and A2 be E1 k-algebras. The map UkA1 q UkA2 -! U(A1 x A2) induced by the projections A1 x A2 -! Ai is an isomorphism in the homotopy category. Here Uk denotes the derived functor adjoint to C*(-; k), the cochain functor with coefficients in k [10, App A]. When k is the ring of rational numbers, the analogous theorem holds when Uk is replaced with the derived functor adjoint to the Thom-Sullivan PL De Rham functor [2]. EQUIVARIANT p-ADIC HOMOTOPY THEORY 9 It follows from Theorem 3.1 that if X = X1 q . .q.Xn, then the map (induced by the inclusions) UC*X1 q . .q.UC*Xn -! UC*(X1 q . .q.Xn) is an isomorphism in the homotopy category. In particular, if X is a (not neces* *sarily connected) finite type nilpotent p-complete space, then the unit of the derived adjunction X -! UC*X is an isomorphism in the homotopy category. As in [10, x1], the following theorem is a formal consequence. Theorem 3.2. The map HoTop (X; Y ) -! HoE(C*Y; C*X) induced by the derived functor C* is a bijection when Y is a finite type p-comp* *lete nilpotent space (and not necessarily connected). In particular, C* embeds the f* *ull subcategory of the homotopy category consisting of the finite type p-complete n* *ilpo- tent spaces as a full subcategory of the homotopy category of E1 Fp-algebras. This extension of the Main Theorem of [10] was first (to our knowledge) proved by Paul Goerss in a course on p-adic homotopy theory at Northwestern University* * in Spring 1999. Goerss' approach was to compare the Bousfield-Kan unstable Adams resolution of the space (the p-completion cosimplicial space) with the mapping * *space of the bar resolution of the E1 algebra. We give a different argument here. Let A1, A2 be E1 k-algebras, and consider the map k x k -! A1 x A2. Let R -! kxk be a cofibrant approximation of kxk, and factor the map R -! A1xA2 as a cofibration R -! A followed by an acyclic fibration A -! A1xA2. Let ss1 and ss2 be the composite maps R -! k induced by the two projections k x k -! k, and write Css1 and Css2 for the pushouts k qR A over these two maps. We have maps Css1 -! A1 and Css2 -! A2 induced by the universal property of the pushout, and the first observation we need is the following proposition. Proposition 3.3.The E1 k-algebras Css1 and Css2 are cofibrant, and the maps Css1 -! A1 and Css2 -! A2 are quasi-isomorphisms. Proof.Since the map R -! A is a cofibration, the pushout maps k -! Css1 and k -! Css2 are cofibrations, but k is the initial object, so Css1 and Css2 are c* *ofibrant. For the second statement, we use the Eilenberg-Moore spectral sequence of [10, * *3.6] to calculate H*Cssi. The E2 term is Ep;q2= Torp;qH*R(H*k; H*A) ~=Torp;qkxk(k; H*A1 x H*A2): Since this is concentrated in homological degree p = 0, the spectral sequence d* *e- generates at E2 and we get H*Cssi= k kxk (H*A1 x H*A2) = H*Ai: The map H*A -! H*Cssi is the quotient map, and so it follows that the map__ Cssi- ! Ai is a quasi-isomorphism. |__| The other observation we need is the following special case of Theorem 3.1. Proposition 3.4.Uk(k x k) is isomorphic to S0 in the homotopy category. Proof.Since R is cofibrant, we have that Uk(kxk) is represented by UkR. Note th* *at by definition [10, 4.2 and App A], vertices in UkR are in one-to-one correspond* *ence with E1 k-algebra maps from R to k. Let ffl: R -! k be any map of E1 k-algebras, and factor ffl as the composite of a cofibration e: R -! K and an acyclic fibra* *tion 10 MICHAEL A. MANDELL K -! k. Let Di be the pushout of e along ssi; then Di is cofibrant. Since k is * *an integral domain, the only k-algebra maps k x k -! k are the two projections, and so either H*ffl = H*ss1 or H*ffl = H*ss2. Applying the Eilenberg-Moore spectral sequence to calculate H*Di, the E2 term is ( * * Ep;q2= Torp;qH*R(H*k; H*K) ~=Torkxk(k; k) = k if H ffl = H ssi 0 if H*ffl 6= H*ssi Thus, for one of D1, D2, the initial map k -! Di is a quasi-isomorphism and for the other, the final map D3-i- ! 0 is a quasi-isomorphism. Applying the functor Uk, we get the following pullback square of simplicial s* *ets. UkDi _____//UkK | | | | fflffl|fflffl|fflffl|fflffl| Ukk _____//_UkR Note that Ukk = * and its image in UkR is the point corresponding to the map ssi:R -! k. Since Uk converts cofibrations into Kan fibrations, we have that UkDi is the fiber of a Kan fibration of Kan complexes. We also have that UkK is contractible and one of UkD1, UkD2 is contractible and the other is empty. This shows that the point of UkR corresponding to ffl must either be in the componen* *t of the point corresponding to ss1 or be in the component of the the point correspo* *nding to ss2, and that that component is contractible. Since ffl was arbitrary, we co* *nclude_ that UkR has exactly two components, both of which are contractible. |_* *_| We can now prove Theorem 3.1. Proof of Theorem 3.1.Applying Uk to the pushout square defining Cssi, we obtain the following pullback square of simplicial sets. UkCssi_____//UkA | | | | fflffl|fflffl|fflffl|fflffl| Ukk ______//UkR Again, we have that Ukk = *, and Uk converts cofibrations to Kan fibrations. From the previous proposition, we have that the two maps * = Ukk -! UkR corresponding to ss1 and ss2 induce a homotopy equivalence S0 -! UkR. It follows that the maps UkCssi- ! UkA induce a homotopy equivalence UkCss1 q UkCss2 -! UkA: The theorem now follows from Proposition 3.3. |___| 4. The Characterization Theorem for Nonconnected Spaces The purpose of this section is to extend to the case of nonconnected spaces t* *he results of [10, xx7-8] characterizing the subcategory of E1 algebras that are * *quasi- isomorphic to the cochain complexes of finite-type nilpotent or simply connected spaces. Of course, an E1 Fp-algebra A is quasi-isomorphic to C* of a nilpotent finite type nonconnected space if and only if A is quasi-isomorphic to a cartes* *ian product A1x . .x.An of some E1 Fp-algebras A1, : :,:An, each of which is quasi- isomorphic to C* of a nilpotent finite type connected space. In order to make this helpful for identification, we need a way of identifying when an E1 algeb* *ra EQUIVARIANT p-ADIC HOMOTOPY THEORY 11 can be decomposed as a cartesian product. We show that for an E1 algebra A whose cohomology is concentrated in non-negative degrees, decompositions of A as a cartesian product up to quasi-isomorphism are in one-to-one correspondence wi* *th decompositions of H*A as a cartesian product up to isomorphism. We state the following theorem for a general commutative ring k since we find the general ca* *se interesting and since restriction to k = Fp provides no simplification. Theorem 4.1. Let A be an E1 k-algebra with HqA = 0 for q < 0. Suppose H*A ~=A*1xA*2for some graded k-algebras A*1, A*2. Then there exist E1 k-algebr* *as A1, A2 with H*A1 ~=A*1, H*A2 ~=A*2, and an isomorphism in the homotopy cate- gory of E1 k-algebras between A and A1 x A2 that realizes the given isomorphism on H*. As a consequence, we obtain the following cohomological criterion for an E1 F* *p - algebra to be quasi-isomorphic to C* of a simply connected space. Theorem 4.2. An E1 Fp-algebra A is quasi-isomorphic to the singular cochains on a simply connected (but not necessarily connected) finite type space if and * *only if H*A satisfies the following conditions. (i)HqA = 0 for q < 0 and q = 1. (ii)Each HqA has a finite Fp-module basis of P 0fixed points. Proof.By Theorem 4.1 and the Characterization Theorem of [10], it suffices to show that H0A is isomorphic as an Fp-algebra to the cartesian product of copies of Fp. Let R be the subset of H0A fixed by P 0. Since P 0is additive, it is cle* *ar that R is an Fp-submodule of H0A. On the other hand, P 0is the Frobenius (the p-th power operation) in dimension 0, so it is multiplicative, and R is actuall* *y an Fp-subalgebra of H0A; in fact, we must have that H0A is the Fp-extension of the Fp-algebra R. Now we show by induction on the Fp-module dimension of R that a finite Fp-algebra fixed by the Frobenius is a cartesian product of copies of * *Fp. The base case is when R is one dimensional and is immediate. The induction step is to show that when R has dimension more than one, R must contain a nontrivial idempotent. Let x 2 R be any element that is not a multiple of the unit, and le* *t Q be the subalgebra of R generated by x; it suffices to find a nontrivial idempot* *ent in Q. We regard Q as the quotient of the PID Fp[X]. Since x satisfies the polynomi* *al q(X) = Xp - X = (X - 1) . .(.X - p), but does not satisfy any of the individual factors X - 1, : :,:X - p, the minimal polynomial for x decomposes as the produ* *ct of more than one linear factor. The Chinese Remainder Theorem then splits Q, as a ring, as a cartesian product of more than one copy of Fp. This finds_in Q a nontrivial idempotent. |__| We now return to the proof of Theorem 4.1, which begins with the following "discrete" case. It is the heart of the homotopy theory portion of the argument. Proposition 4.3.Let R be a cofibrant E1 k-algebra with HqR = 0 for q 6= 0. If H*R ~=R*1x R*2as a graded k-algebra, then there exists an E1 k-algebra S with H*S ~=R*1, and a map R -! S that realizes the projection map on H*. Proof.Choose a cocycle x in R representing (0; 1) 2 R*1x R*2~=H0R. Let X0 = k regarded as a differential graded k-module concentrated in degree 0, and consid* *er the map of differential graded k-modules X0 -! R sending the generator to x. Let S1 be the pushout S1 = R qEX0 ECX0, where E denotes the free E1 k-algebra 12 MICHAEL A. MANDELL functor. Since H*EX0 is concentrated in non-positive degrees, the Eilenberg-Moo* *re spectral sequence of [10, 3.6] implies that S1 is concentrated in non-positive * *degrees: note that the homological degree subtracts from the total degree with our cohom* *o- logical grading. A quick calculation shows that H0S1 ~=R01and the map R -! S1 induces the projection map on H0. We construct S by killing the negative cohomology modules of S1. In detail, starting with R -! S1, assume we have inductively constructed a sequence of cofibrations of E1 k-algebras R -! S1 -! . .-.! Sn such that Sn is concentrated in non-positive degrees, HqSn = 0 for -n < q < 0, and H0S1 -! H0Sn is an isomorphism. We construct a map of E1 k-algebra Sn -! Sn+1 as follows. Let Xn be the differential graded k-module concentrated in degree -n, where it is the free k-module with one basis element for each ele* *ment in H-n Sn. For each element of H-n Sn, choose a cocycle representing it and let Xn - ! Sn be the map of differential graded k-modules that sends each basis element of Xn to the cocycle of the class indexing it. Let Sn+1 = Sn qEXn ECXn, and let Sn -! Sn+1 be the canonical inclusion. Now consider the Eilenberg-Moore spectral sequence that calculates H*Sn+1. Since Xn is concentrated in degree -n, the inclusion of k Xn in EXn is an isomorphism on Hq for q -n, and it is easy to see from the E2-term of the spectral sequence that the map Sn -! Sn+1 induces an isomorphism on Hq for q > -n. Since the map H-n Xn -! H-n Sn is surjective, we also see from the E2-term that H-n Sn+1 = 0. __ Let S = ColimSn. The inclusion R -! S is as desired. |__| We will use the following observation to extend the previous result to more general E1 k-algebras. Proposition 4.4.Let A be an E1 k-algebra with HqA = 0 for q < 0. Then there exists an E1 k-algebra R with HqR = 0 for q 6= 0, H0R ~=H0A and a map of E1 k-algebras R -! A realizing the isomorphism on H0. Proof.Let R be the differential graded submodule of A which is zero in positive degrees, the kernel of the differential in degree zero, and all of A in negativ* *e de- grees. Since an E1 operad is concentrated in non-positive degrees, the E1 algeb* *ra structure maps for A restrict to R, making R an E1 k-algebra and the inclusion_ R -! A a map of E1 k-algebras. |__| We can now prove Theorem 4.1. Proof of Theorem 4.1.Let A, A*1, A*2be as in the statement. Let R -! A be as in Proposition 4.4, and let R*1be A01considered as a graded k-algebra (concentrate* *d in degree zero) and similarly for R*2. Then the isomorphism H*A ~=A*1x A*2induces an isomorphism H*R ~=R*1x R*2. By choosing a cofibrant approximation of R and factoring the map R -! A, we can assume without loss of generality that R is cofibrant and R -! A is a cofi- bration. Then Proposition 4.3 produces E1 k-algebras R1, R2, with H*R1 ~=R*1, H*R2 ~=R*2and E1 k-algebra maps R -! R1, R -! R2 realizing the projection maps H*R -! R*1, H*R -! R*2. Let A1 = R1 qR A, A2 = R2 qR A. Then the Eilenberg-Moore spectral sequence to calculate H*Ai has E2-term Ep;q2= Torp;qH*R(H*Ri; H*A) ~=Torp;qR01xR02(R0i; A*1x A*2): EQUIVARIANT p-ADIC HOMOTOPY THEORY 13 Since this is concentrated in homological degree p = 0, the spectral sequence d* *e- generates at E2 and we get H*Ai~=R0iR01xR02(A*1x A*2) ~=A*i: Under this isomorphism, the map A -! Ai is the projection. 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Department of Mathematics, University of Chicago, Chicago, IL E-mail address: mandell@math.uchicago.edu February 29, 2000, 14:04