Topology 40 (2001) 993}1016
Realizing coalgebras over the Steenrod algebra
David Blanc* Department of Mathematics, University of Haifa, 31905 Haifa, Israel Received 9 September 1998; accepted 13 January 2000
Abstract
We describe algebraic obstruction theories for realizing an abstract (co)algebra KH over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the p-homotopy types of di!erent realizations. The theories are expressed in terms of the Quillen cohomology of KH. 2001 Elsevier Science Ltd. All rights reserved.
MSC: primary 55S10; secondary 18G55
Keywords: Steenrod algebra; Realization; Unstable coalgebra; (Co)homology operations; Cohomology ring; Cosimplicial resolution; Quillen cohomology
1. Introduction
The question of which graded R-algebras can occur as the cohomology ring of a space X with coe$cients in R was "rst raised explicitly by Steenrod [59], but it goes back to Hopf, for R"0 "% H % * see [37]. When R N, the cohomology H (X; N) also has a compatible action of the Steenrod algebra, so it is natural to ask: Which algebras over the Steenrod algebra can be realized as the cohomology of a space, and in how many di!erent ways? This question has been addressed repeatedly in the past * see, for example, [1}3,5,23,29,30,56,57] and the survey in [4]. Two related algebraic questions have also been often % considered: which N-algebras can be provided with a compatible action of the Steenrod algebra (see, e.g., [26,60,63]), and conversely, which unstable modules over the Steenrod algebra can be provided with a compatible algebra structure, or directly: which unstable modules are realizable
* Fax: #972-4-824-0024. E-mail address: [email protected] (D. Blanc).
0040-9383/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 0 - 9 383(00)00007-0 994 D. Blanc / Topology 40 (2001) 993}1016
* see, e.g. [24,39,54]. The analogous stable question of whether a given module over the Steenrod algebra can be realized by a spectrum, which has also been extensively studied (e.g. [8,21,22]), can be answered in terms of suitable Ext groups (see [45, Chapters 16, 3]), and an unstable version of this for the Massey}Peterson case was developed by Harper (see [36]). However, we shall not be concerned with these variants here: our goal is to describe a general obstruction theory for the original realization problem, which can be stated purely algebraically, in terms of the Quillen cohomology of the given algebra * analogous to the stable theory. This answers a question of Lannes, Miller and others in the 1980s, asking for an unstable analogue of the stable obstruction theory, which was also (independently) one of the motivations for the project begun by Dwyer et al. [27,28] (see also [14]). It turns out to be more natural to consider the dual question, that of realizing a coalgebra over the Steenrod algebra as the homology of a space. This is because the cohomology of a space, in general, has the structure of a proxnite unstable algebra; it is only when the space is of "nite type that it is an unstable algebra, and the realizability of an unstable algebra of "nite type is, of course, strictly equivalent to that of the corresponding coalgebra (its vector space dual). Part of the theory we describe here actually works over a more general ground ring R, but to obtain its full force we restrict attention to the case when R is a "eld. Thus, if we de"ne an unstable R-coalgebra to be a graded coalgebra over R equipped with a compatible action of the unstable R-homology operations, we have:
"% 0 " Theorem A. For R N or , let KH be a connectedunstable R-coalgebra , such that either K 0, or KH has xnite projective dimension. Then there is a sequence of cohomology classes 3 L> L "2" " L H (KH; KH) such that L is dexnedwhene ver L\ 0, andall the classes vanish if andonly if KH HH(X; R) for some space X.
L (See Theorems 5.10 and 6.3 below.) The coaction of KH on KH here is essentially a re-indexing of the coalgebra structure of KH * see Proposition 4.4 w and Section 5.1. The proof involves showing that, for any space X, any (algebraic) cosimplicial resolution of the unstable coalgebra "% HH(X; R) can be realized as a cosimplicial space, and conversely. As a side bene"t, when R N, this provides a way of constructing minimal `unstable Adams resolutionsa (see Remark 6.6 below). There is a similar theory for distinguishing between di!erent realizations:
Theorem B. For any two simply connectedspaces X and Y of xnite type such that % % HH(X; N) HH(Y; N) KH as unstable coalgebras over the Steenrodalgebra , there is a sequence of 3 L> L cohomology classes L H (KH; KH) whose vanishing implies that X and Y are p-equivalent.
(See Theorems 7.3 and 7.6 below.) This result is less satisfactory, in that X and Y may be p-equivalent even if the cohomology classes do not all vanish (which simply expresses the fact that the homotopy theory of cosimplicial spaces is richer than that of topological spaces.) Note that in this the classes L resemble the usual k-invariants, and indeed Theorem Bcan be thought of as providing a system of algebraic `invariantsa for the p-type of a space, dual to those of [13] or [14] (see Section 7.8). D. Blanc / Topology 40 (2001) 993}1016 995
Theorem Balso holds for R"0; in this case we simply recover the homology version of the obstruction theory of [35,32].
1.1. Notation and conventions. T will denote the category of topological spaces, and TH that of pointed connected topological spaces with base-point preserving maps. We denote objects in T or TH by boldface letters: X,Y, and so on, to help distinguish them from the various algebraic objects we consider. 0 % denotes the rationals, and for p prime, N denotes the "eld with p elements. For a ring R (always assumed to be commutative with unit), R-Mod denotes the category of R-modules, and 1 2 R XH the free R-module on a (possibly graded) set of generators XH. Tensor products of R-modules will always be over the ground ring R, unless otherwise stated, and the dual module of 3 ଙ " A R-Mod is denoted by A : Hom0U+(A, R). 3 HH(X; R) R-Mod is the homology of a topological space (or simplicial set) X with coe$cients in P P R. We write fI : HH(X; R) HH(Y; R) for the graded homomorphism induced by f : X Y. RX is the Bous"eld}Kan R-completion of X (cf. [19, I, 4.2]). For an abelian category M, we let cHM denote the category of chain complexes over M (in non-negative degrees); similarly, cHM denotes the category of cochain complexes. For any category C, we denote by gr C the category of non-negatively graded objects over C, with " "" 0 3 3 3 ! x n x XL for XH gr C. Given a ("xed) object B C, we denote by C B the category of objects under B (cf. [43, II, Section 6]).
1.2. De5nition. A cosimplicial object Xⅷ over any category C is a sequence of objects X, X,2, XL,2 in C equipped with coface and codegeneracy maps dG : XLP XL>, sH : XL>PXL (0)i, j)n) satisfying the cosimplicial identities
dHdG"dG>dH if i*j,
dGsH\ if i(j, sHdG"Id if i"j, j#1, (1.3) dG\sH if i*j#2,
sHsG"sGsH> if i)j
(cf. [19, X, Section 2.1]). We denote by cC the category of cosimplicial objects over C. If we restrict attention to X, X,2, XL, with their coface and codegeneracy maps, we have an n-cosimplicial object; the category of such will be denoted by c6L7C.
Dually, we denote by sC the category of simplicial objects over C (cf. [44, Section 2]). The category of simplicial sets, however, will be denoted simply by S (rather than sSet), and that of pointed simplicial sets by SH. Objects in these two categories will again be denoted by boldface 3 I3 letters. The standard n simplex in S is denoted by [n], generated by L [n]L, and L S is the O sub-simplicial set of [n] generated by dG L for i k. 996 D. Blanc / Topology 40 (2001) 993}1016
Since we shall be dealing for the most part with simplicial sets as our model for the homotopy category of topological spaces, we shall call cosimplicial pointed simplicial sets * i.e., objects in cSH * simply cosimplicial spaces.
1.4. Example. The cosimplicial space ⅷ3cS has the standard simplicial n-simplex [n]in cosimplicial dimension n, with coface and codegeneracy maps being the standard inclusions and projections (cf. [19, I, 3.2]).
P 6L7 1.5. De5nition. If C has enough limits, the obvious truncation functor trL : cC c C has a right L" P adjoint L, and the composite cosk : L trL : cC cC is called the n-coskeleton functor. (This is P dual to n-skeleton functor skL : sC sC.)
1.6. Organization In Section 2 we recall some basic facts about coalgebras over the Steenrod algebra, and in Section 3 we show how certain convenient CW resolutions for such coalgebras may be constructed. Section 4 deals with the coaction of the fundamental group of a cosimplicial coalgebra, and the Quillen cohomology of unstable coalgebras. In Section 5 we describe the cohomology classes which determine whether a given algebraic resolution may be realized topologically (Theorem 5.10), and in Section 6 we apply this to the original question of realizing an abstract coalgebra (Theorem 6.3). Finally, in Section 7 a similar theory is developed for distinguish- ing between di!erent possible realizations of a given algebraic resolution (Theorem 7.3), and thus for determining the p-type of a space (Theorem 7.6).
2. Unstable coalgebras
We "rst recall some basic facts about the category of the coalgebras over the Steenrod algebra:
2.1. De5nition. For a "eld R, let coAlg0 denote the category of graded coalgebras over R: an object in coAlg0, which we shall call simply an R-coalgebra, is thus a (non-negatively) graded R-module 3 2.2. Unstable coalgebras. As usual, the homology HH(X; R) of a space X with coe$cients in "eld R is an R-coalgebra, with induced by the diagonal XPX;X (composed of inverse of the KuK nneth D. Blanc / Topology 40 (2001) 993}1016 997 ; isomorphism HH(X; R) HH(X; R) HH(X X; R)). However, HH(X; R) also comes equipped with an action of the primary R-homology operations: these are natural transformations ! P ! ' HG( ; R) HI( ; R), dual to the corresponding cohomology operations, and they vanish if k i (see [58, Section 9]). For any "eld R,anunstable coalgebra (over R) is a non-negatively graded R-module equipped with an action of primary R-homology operations (which include the coalgebra structure), satisfying the universal identities for these operations. We denote the category of such unstable R-coalgebras by CA0. The simplest case is when R"0: CA0+coAlg0 , since there are no non-trivial primary 0 "% -homology operations besides the coproduct (see [50, I, Section 1]). The next simplest is R N: 2.3. De5nition. For any prime p,anunstable module over the mod p Steenrodalgebra , AN,is % a non-negatively graded N-vector space KH, equipped with a right action of AN * i.e., a graded homomorphism P : KH AN KH, (2.4) where "x SqG"""x"!i if p"2, and "xPG"""x"!2(p!1)i, "x """x"!1ifp'2 (where we write x SqG for (x SqG), etc.). The action is required to be unstable in the sense that x SqG"0if2i'"x" (for p"2) and xPG"0if2pi'"x" (for p'2). The category of such unstable modules will be denoted by Mod-AN. ` a The category Mod-AN is dual to the more familiar category U of unstable cohomology-like modules over the Steenrod algebra (see, e.g. [53, Section 1.3]). 2.5. De5nition. For any prime p, the category CA% of graded coalgebras over the mod p Steenrod N % algebra AN, has as objects non-negatively graded coalgebras CH over N, which are at the same time unstable AN-modules. The two structures are related by the Cartan formula, which says that the of (2.4) is a homomorphism of coalgebras (see [47, Section 4]) * dual to L ) " I ) L\I " L ) " I ) L\I ) " ) # Sq (a b) ISq a Sq b for p 2, and P (a b) IP a P b, (a b) ( a) b (!1) ? a ) b for p'2. There is also a Verschiebung formula, due to the fact that the top Steenrod operation equals the Frobenius * i.e., for "a""n, we have SqL a"a if p"2, and PL a"aN if p'2 and n is even. See [18, Section 5]. In particular, we can think of H (X; % ) as an object in either od- or A% for any space X. H N M AN C N % " % Again, CA N is dual to the more familiar category K K N of unstable algebras over the Steenrod algebra (cf. [53, Section 1.4]). However, taking vector space duals yield a strict equivalence of categories only when dealing with (co)algebras of xnite type, and our approach is more naturally presented in terms of coalgebras, as noted above. 2.6. (Co)abelian (co)algebras. Recall that any abelian R-algebra (i.e., abelian group object in Alg0) "0 must have a trivial multiplication (for any ring R). When R , the subcategory (K0 ) of abelian objects in 0 is actually equivalent to the category of graded vector spaces over 0.( % ) is K K N equivalent to a subcategory of Mod-AN (viewed as algebras with a trivial product): for p"2, ( % ) " is the category of -modules with SqGx"0 for "x")i; for p'2, K U A 998 D. Blanc / Topology 40 (2001) 993}1016 ( % ) " is the category of -modules with Gx"0 for "x")2i (cf. [46, Section 1]). In all K N V AN P P H3 these cases the abelianization functor ( ) : K0 (K0) assigns to any algebra A K0 its `module of indecomposablesa, Q(AH). Dualizing, we see that the coabelian objects (i.e., abelian cogroup objects) in CA0 must have trivial comultiplication, so (CA0) U is equivalent to the appropriate sub-category of Mod-AN; the 3 coabelianization functor is just the R-module of primitives P(AH) 6AH, for any AH CA0 (see [17, Section 8.6], and cf. [40, Section 2]). P 2.7. Functors and limits of coalgebras. The underlying-set functor coAlg0 grSet factors through K P K P K ; : coAlg0 gr R-Mod, with right adjoint G :grR-Mod coAlg0, where G( G " G CH colim colim ( CH )?, (2.8) GZ' (ZF ?Z ( GZ( G 3 G where ( GZ(CH )? ( A() runs over all xnite sub-coalgebras of the "nite tensor product GZ(CH (see [33, 1.2]). " 3 2.9. Remark. In fact, any cofree unstable R-coalgebra GH G( 3. Resolutions of coalgebras We now prove some basic facts about cofree resolutions for coalgebras: 3.1. De5nition. For any concrete cocomplete category C, the co-matching object functor M : SMN;cCPC, written M)Xⅷ for a "nite simplicial set K3S and any Xⅷ3cC,isde"ned by ⅷ ⅷ ⅷ L " L " ) " )G requiring that M X : X , and if K colimG KG then M X colimG M X (cf. [16, Section 6; 28, Section 2.1]. ⅷ ⅷ In particular, write MLX for M L\ L X . Note that each coface map dI : XL\PXL factors L L ⅷP L through the map : M X X induced by the inclusion skL\ [n]6 [n]. A cosimplicial space ⅷ3 L X cSH will be called coxbrant if each of these maps is a co"bration. This concept refers to the resolution model category structure on cSH (see Section 3.13 below). D. Blanc / Topology 40 (2001) 993}1016 999 Fig. 1. 3.2. De5nition. Given a cosimplicial object Xⅷ over a concrete complete category C, the analogous construction for the codegeneracies yields ¸LXⅷ which is de"ned (in the cases of interest to us) by ¸L ⅷ "+ 3 L\ "L " G " H\ ) ( ) , X : (x,2, xL\) (X ) s xH s xG for all 0 i j n . Again, each codegeneracy map sG : XLPXL\ equals the natural map L : XLP¸LXⅷ, composed of the projection onto the ith factor. ¸LXⅷ has been called the nth `co-latching objecta for Xⅷ * cf. [27, Section 2.3]. In [19, X, Section 4.5] it was denoted by MLXⅷ; the notation we have here was chosen to be consistent with that of [27,28,13], since ML is used here to denote yet another version of the original matching objects. ⅷ3 L ⅷ 3.3. De5nition. If X cSH is co"brant, its n-cochains object, written C X ,isde"ned to be the ⅷ ⅷ ⅷ L L P L L " L L G L\ L co"ber of : M X X (Section 1.2), so C X X /( Gd X ). Similarly, the object B X is L L ⅷP L L ⅷ" L L G L\ de"ned to be the co"ber of : M X X , so that B X X /( Gd X ). L LZ L ⅷ L L ⅷZ L ⅷ These all "t into the commutative diagram of Fig. 1, in which p6 : X C X and q6 : C X B X are quotient maps, and the bottom row is a co"bration sequence (with the co"bration " L ⅷP L> ⅷ L L ⅷP L d d6L : B X C X induced from : M X X by cobase change). We will call d the nth principal face map for Xⅷ. 3.4. De5nition. The same de"nitions may be applied to a cosimplicial object Aⅷ3cC over any suitable category C (e.g., if C has a faithful `underlying objecta functor into an abelian category). In " L ⅷP L> ⅷ ⅷ this case the kernel of d B A C A is de"ned as usual to be the nth cohomotopy object of A , and denoted by LAⅷ (see [19, X, Section 7.1]). When C is an abelian category, Aⅷ3cC is equivalent under the Dold}Kan equivalence (cf. [25, Theorem 1.9]) to a cochain complex AH, and LAⅷ HLAH. However, in most cases LAⅷ will have additional structure * e.g., it will be a (coabelian) object in C (see [15, Section 5]). L 3 3.5. Remark. When R is a "eld, let K(R, k) cSH denote the cosimplicial space consisting of the usual simplicial Eilenberg}Mac Lane space K(R, k) in cosimplicial dimension n, a single point in lower dimensions, and ¸NLK(R, k) (de"ned inductively) in dimension p'n; similarly CLK(R, k)is obtained from LK(R, k) by attaching a single copy of K(R, k) in cosimplicial dimension n#1 (see Section 3.9 and cf. [28, Section 3.6]). 1000 D. Blanc / Topology 40 (2001) 993}1016 L " 1L 2 We now de"ne the analogous cosimplicial unstable coalgebras: EM (R, k)C : G(R K(R, k) ) L " 1 L 2 L L> and CEM (R, k)C : G(R C K(R, k) ). These represent the functors B and C , respectively, in that for any cosimplicial coalgebra Aⅷ 3c A we have Hom (Aⅷ , ML(R, k) ) (BLAⅷ )ଙ (the H C 0 AC 0 H E C I vector space dual), and Hom (Aⅷ , C ML(R, k) ) (CL>Aⅷ )ଙ. Therefore, it makes sense to AC 0 H E C I L ⅷ ଙ ⅷ L denote ( AI) by [AH,EM (R, k)C ] * and these are in fact the homotopy classes of maps into L EM (R, k)C in the model category structure on cCA0 described in [10, Section 7]. The following statement is dual to [13, Lemma 2.29]: ⅷ3 L ⅷ L 3.6. Proposition. For coxbrant X cSH, the inclusion n : C X 6X induces an isomorphism L ⅷ L ⅷ * nଙ : HH(C X ; R) C (HH(X ; R)) for each n 0. P Proof. The free R-module functor F : SH sR-Mod is a left adjoint, so preserves all colimits, and ⅷ L ⅷ ⅷ L ⅷ thus CL(FX ) F(C X ) (we need X to be co"brant in order for C X to be meaningful). If P" 3 K : sR-Mod cHR-Mod is the Dold}Kan equivalence functor, then for any X SH, HH(X; R)is the homology of the chain complex KFX,soitsu$ces to show that for any cosimplicial chain ⅷ " ⅷ L ⅷ L ⅷ complex AH( KFX ) the map nଙ : HI(C AH) C (HIAH) is an isomorphism for all k, n. Now for a cosimplicial object Aⅷ over any abelian category, we can de"ne the Moore cochain L ⅷ "L\ H - L "L ! G G complex by N A : HKer(s ) A (with di!erential : G( 1) d ). We claim that the composite NLAⅷ6ALZCLAⅷ is an isomorphism : NLAⅷ P" CLAⅷ. L ⅷ5L G " "L G First note that N A GIm(d ) 0, since if Gl d (xG) (which we may assume by induc- ) ) H" ) ) " l\" #L G\ l\ tion on 0 l n) and s 0 for 0 j n, then 0 s xl Gl>d s (xG), so l "!L G l l\ L G " d xl Gl>d d s (xG). Thus is in fact of the form Gl>d (x G) * which implies that 0 (for l"n). This shows that is one-to-one. Next, given 3AL with sH()"0 for 0)j(l (which we again assume by induction on 0)l(n), then there is an 3AL with []"[ ]3CLAⅷ such that sH( )"0 for 0)j)l * name- ly, ": !dl>sl. This shows that is onto. L ⅷ L ⅷ Thus the proposition will follow if we show nଙ : HI(N AH) N (HIAH) is an isomorphism: given 123 L ⅷ 3 L * " * N (HIAH) represented by AI with I( ) 0, (where is the boundary map of the chain L H " ) ( 3 L\ complex AH), we assume that s ( ) 0 for 0 j l, and that there are bl,2, bL\ AI> such that H "* ) ( "!* l> s ( ) I>(bH) for l j n. Replacing by : I>(d (bl)), we see by induction on 0)l)n!1 that we may choose a representative for 12 with sH()"0 for all j. Thus nଙ is 12" 3 L ⅷ 3 L ⅷ 3 L * " surjective. Finally, if 0 N (HIAH) for N AI, there is a AI> such that I>( ) , H " ) ) " ! l> l> 3 L with s ( ) 0 for 0 j l; setting : d s ( ), we see that we can assume N BI>,so nଙ is one-to-one. ᮀ ⅷ * 3.7. De5nition. A cosimplicial coalgebra AH is called cofree if for each n 0 there is a graded set ¹L L L " ¹L H of elements in AH such that AH G( H) (cf. Section 2.7), and H L P L\ ¹L ¹L\ each codegeneracy map s : AH AH takes H to H . (3.8) D. Blanc / Topology 40 (2001) 993}1016 1001 3.9. De5nition. A CW complex over a pointed category C is a cosimplicial object Cⅷ3cC, together with a sequence of objects CML3C (n"0,1,2) * called a CW basis for Cⅷ * such that L" ML;¸L ⅷ * LP ML G" C C C for each n 0, with projection proj!ML : C C , such that proj!ML d 0 for 1)i)n (cf. [38 Section 3; 9 Section 5]). M" L\P ML ML The coface map d L : proj!ML d : C C is called the attaching map for C , and it is readily M " veri"ed that the attaching maps d L (n 0,1,2), together with the cosimplicial identities (1.3), ⅷ M determine all the face and degeneracy maps of C . Note that dL factors through a map BL\CⅷPCML. ⅷ 3 In particular, we require that a CW basis for a free cosimplicial algebra AH CA0 be a sequence ML of cofree colagebras (AH)L. On the other hand, for a cosimplicial object over SH, it will be convenient to require only that CLKCML;¸LCⅷ for all n. For any "eld R, the category cCA0 of cosimplicial unstable R-coalgebras has a model category structure in the sense of Quillen (cf. [49, I, Section 1]), induced from that of cR-Mod+cHR-Mod by the obvious pair of adjoint functors (see [10, Section 7]). All we shall need from this are the following: 3.10. De5nition. A cofree cosimplicial resolution of an unstable R-coalgebra KH is de"ned to be ⅷ P a cofree cosimplicial coalgebra AH, equipped with a coaugmentation KH AH, such that in each * ⅷ degree k 1, the cohomotopy groups of the cosimplicial R-module (A )I (i.e., the cohomology groups of the corresponding cochain complex over R-Mod) vanish in dimensions n*1, and the ⅷ coaugmentation induces an isomorphism KH AH. 3.11. Constructing CW resolutions. As usual, such a resolution is simply a "brant (and co"brant) ⅷ object in cCA0 which is weakly equivalent to the constant cosimplicial object c(KH) , in the model 3 category structure mentioned above. In particular, such resolutions always exist, for any KH CA0; there are a number of ways to construct them, including the (very large) canonical monad resolution described in [20, Section 11.4] when R is a "eld (see [10, Section 7.8]). We shall be interested in a particular type, namely, those equipped with a CW basis, which will be called CW resolutions, since these can be chosen to be small (e.g., minimal). Their construction is \" \ ⅷ " \ straightforward: starting with AH : KH and B AH : AH , we assume that we have constructed ⅷ ! AH through cosimplicial dimension n 1; then we simply choose some cofree unstable coalgebra ML 3 M L\ ⅷ ML AH CA0 with a one-to-one map dL : B AH6AH. These always exist, by the universal property of ML " L\ ⅷ L " ML ;¸L ⅷ cofree coalgebras * e.g., one could take AH : G;(B AH). Setting AH : AH AH (the product ML ¸L ⅷ in CA0 * i.e., AH 0 AH) completes the inductive stage. The dual construction, for simplicial groups, algebras, and so on, is classical: see [38,62] and [7, I, Section 6]. Note, however, the following analogue of [12, Proposition 3.18]: ⅷ 3 ML 3.12. Proposition. Any cofree cosimplicial unstable coalgebra AH cCA0 has a CW basis (AH)L. M " Proof. Start with AH : AH, and note that (3.8) of De"nition 3.7 implies (by induction on n) that ¸L ⅷ 1 2 3 AH G(R >H ) for some >H grSet. 1002 D. Blanc / Topology 40 (2001) 993}1016 ` a Now because CA0 has the underlying structure of an abelian category, we may de"ne L L P L a homomorphism of the underlying abelian groups : AH AH by L> L():" (!1) '> ( >d's(, L I ' ( ZLI L"+ 3-I;-I " ' ' ) ) , " where LI (I, J) jI\R iR jI>\R for all 1 t k (and we let j : n). It then follows from the cosimplicial identities (1.3) that sHL()"sH for 0)j)n!1. Since the L L\ 3¸L ⅷ L L P¸L ⅷ de"nition of depends only on (s ,2, s ) AH, we see that : AH AH * and in fact even ML ": L"L G * are epimorphisms. G ' B "L I 3L G H" ) ) ! Moreover, given Gd aI GIm(d ) such that s 0 for 0 j n 1, identities (1.3) imply that L N\ N\ O\ L\ " N O ! N\G\ G # ! N\H H # 3 8 G d d ( 1) s aO ( 1) s (aO> aN) Im(d ), N O GO H G so by induction on n we see "0 * and thus ML is one-to-one, so in fact it is an isomorphism of ⅷ " 1 Z 3 Z unstable coalgebras. We thus have AH G(R XH >H) for some XH grSet, where denotes L G " 1 2 the disjoint union, and we may assume GIm(d ) G(R >H ). 3 " " ! I> I Finally, for each x XH, set x : x, and de"ne xI inductively by xI> : xI d s xI ) ( H " ) ( " H " L " (0 k n). We see that s xI 0 for 0 j k,sox( : xL has s x( 0 for all j, i.e., (x( ) 0. Moreover, x!x( 3G(R1> 2), so if we set AML ": G(R1+x( , 2), we get the required CW basis. ᮀ H H VZ6H 3.13. The resolution model category cSH. In [27, Section 5.8], Dwyer et al. de"ne a model category structure on the category cSH of cosimplicial spaces (for each choice of R), which they called the `E-model categorya, though the term resolution model category (cf. [34]) may perhaps be more appropriate: Amapf : XⅷPYⅷ of cosimplicial spaces is L * (i) a weak equivalence if HH(f; R) is an isomorphism (of graded R-modules) for each n 0; * L3 LP L (ii) a cofree map if for each n 0 there is a "brant R-GEM K SH and a map X K which ⅷ LP L; ⅷ ¸L ; L induces a trivial "bration X (X *LX Y ) K ; (iii) a xbration if it is a retract of a cofree map; ⅷ LNL LZ ⅷ L P>L * (iv) a coxbration if f : X +LY M X (cf. Section 3.1) is a co"bration for each n 0, and Lf is a levelwise co"bration of graded R-modules. The advantage of such a model category is that it provides a way to de"ne a cosimplical 3 resolution of a simplicial set (or topological space) X SH,asa"brant replacement for the constant cosimplicial space c(X)ⅷ * where a special case (in fact, the motivating example) is the R-resolution presented in [19, I, Section 4.1]. See also [13, Section 2] for a slight generalization of the original construction. Compare also [31]. 4. The fundamental group and cohomology As noted in Section 3.5 above, the category cCA0 of cosimplicial unstable R-coalgebras has L * * a model category structure in which the objects EM (R, k)C (n 0, k 1) play the role of D. Blanc / Topology 40 (2001) 993}1016 1003 cosimplicial Eilenberg}Mac Lane objects, in the sense of representing the cohomotopy groups. Thus, if we take homotopy classes of maps between (products of) these Eilenberg}Mac Lane objects as the primary cohomotopy operations (see [64, V, Section 8]), we can endow the H ⅷ " G ⅷ ⅷ 3 cohomotopy groups AH ( AH)G of any AH cCA0 with an additional structure: that of a CA0-coalgebra, that is, a graded object over CA0 (coabelian, in positive dimensions), endowed with an action of these primary cohomotopy operations. This concept is dual to that of a K0- - algebra, in the terminology of [15, Section 3.2]. By de"nition, this structure is a homotopy ⅷ invariant of AH. 4.1. The coaction of the fundamental group. In our case we shall only need the very simplest part of ⅷ this structure * namely, the coaction of the fundamental group AH on each of the higher L ⅷ cohomotopy groups AH. This is described in terms of homotopy classes of maps L P ; L EM (R, p)C EM (R, k)C EM (R, l)C ; but since these are cosimplicial coalgebras of "nite type, it may be easier to follow the dual description, in sK0, of homotopy classes of maps between simplicial suspensions ML(R, k) 3s of the free unstable algebras E K0 K0 M(R, k) ": HH(K(R, k)); R)3 . E K0 K0 First, if >ⅷH is any simplicial graded-commutative algebra over a "eld R, we can de"ne the `ଙ a 3 I 3 l ଙ " ) 3 I>l " 3 I -action of any a > on b >L by a b : a( b >L , where a( : sL\2sa >L.Ifwede"ne >HL >H L ⅷZ L ⅷ the n-cycles and n-chains algebras ZL ⅷ CL ⅷ dually to C X B X of Section 3.3 (see [44, ଙ ` a H Section 17]), then since commutes with the face maps, it de"nes a (bilinear) action of > on >H >H >H >H * CL ⅷ and ZL ⅷ and thus an action of ⅷ on L ⅷ for any n 1. H ; L Now let Xⅷ denote the simplicial unstable algebra M (R, k) M (R, ) 3s , for E K0 E l K0 K0 k,l, n'0. Note that we have a short exact sequence of unstable algebras: I P H H ; P H P 0 ZLXⅷ 6H (K(R, k) K(R, l);R) H (K(R, k); R) 0. (4.2) H ଙ 3 H O 3 H Evidently, ZLXⅷ consists of elements of the form GaG bG where aG X and 0 bG XL . However, " ) H " ) ) ! ) 3 H if b b b is non-trivially decomposable in XL , then : (sLa( ) (sb sb sb sb ) XL> " ) ) " ) ) # H H satis"es d a( b b and dH 0 for 1 j n 1, so that LXⅷ is just the free Xⅷ -module generated by ( M(R, ) ) (see Section 2.6, and cf. [15, Section 5]), where XHⅷ E l K0 ?@ M(R, k) "HH(K(R, k); R)3 . E K0 K0 For the dual category of cosimplicial coalgebras, we need the following 3 4.3. De5nition. For a given coalgebra JH CA0, an unstable coalgebra CH equipped with bilinear ` a P ` a co-operations CH JH CH (satisfying the universal identitites for the dual of action ଙ >H >HP >H : ⅷ L ⅷ L ⅷ de"ned above) will be called a JH-coalgebra. The category of such will be denoted by A . C (H On the other hand, a coabelian unstable coalgebra KH equipped with a (left) coaction map of P coalgebras : KH JH KH, satisfying the usual identities (see [61, Section 2.1]) is called a JH- comodule. We denote the category of such by JH-coMod. This is an abelian category. 3 " We say that a JH-comodule KH is cofree, with basis The above discussion for the case of unstable algebras may now be summarized in: ⅷ 3 ⅷ 3 4.4. Proposition. Any cosimplicial unstable coalgebra AH cCA0 has a coaction of AH CA0 on L ⅷ 3 L AH (CA0) U , induced by an AH-coalgebra structure on AH. This coaction of AH commutes with the Mod-AN-structure (i.e., the action of the Steenrodalgebra ), andrespects the coface maps , andthus L ⅷ P L> ⅷ * d : B AH C AH is a map of AH-coalgebras for each n 0. ⅷ L ⅷ Note that the AH-comodule structure on the coabelian coalgebra AH is just part of H "H ⅷ a bigraded cocommutative coalgebra structure on CH : AH, in which the diagonal respects the unstable R-operations. This is the cosimplicial analogue of the C--structure on the homotopy objects of a simplicial object over a category of universal algebras C (see [15, Section 3.2]). 3 3 4.5. Quillen cohomology. Given an unstable coalgebra JH CA0, and KH JH-coMod * that is, 3 a coabelian object KH (CA0) U , with a coaction of JH * one may de"ne its Quillen cohomology by dualizing [15, Section 2], as follows: ⅷ 3 Choose some cofree cosimplicial resolution AH cCA0 of JH, and note that the JH-comodule L KH is in particular an AH-comodule, and each AH is a coalgebra over AH. Moreover, by the usual ⅷ ⅷ universal property we have a natural equivalence Hom (K , A ) Hom (K ,(A ) ), C H H H HU + H H U Y where M U Y denotes the coabelianization of the AH-coalgebra M (see Section 2.6). Thus Cⅷ ": Hom (K , Aⅷ ) is a cosimplicial abelian group, and LCⅷ is called the nth Quillen cohomol- C H H H L ogy group of JH with coezcients in KH, and denoted by H (JH;KH) (cf. [40, Section 3]). If JH, ⅷ KH and AH are of "nite type, this is the vector space dual of the usual Quillen cohomology of ଙ3 (JH) K0 (see [17, Section 8], and cf. [13, Section 4]). ⅷ 4.6. Remark. As for any abelian category, given a cosimplicial object (AH) U Y over AH-coMod, in * L " L ⅷ each dimension n 1 there is a direct product decomposition (AH) U Y C ((AH) U Y) ¸L ⅷ ML ⅷ ((AH) U Y) (cf. [44, Corollary 22.2]). If we choose a CW basis (AH)L for AH (Section 3.9), we L ⅷ M ML have: C (AH) U Y (AHn) U AH as unstable modules (where (AH) U is the usual coabelianiz- L ation of Section 2.6). Moreover, each (AH) U Y is a cofree AH-comodule (with a basis which may be ⅷ described explicitly in terms of the C=-basis for AH * see [10, (6.3)]), and the coface maps are L ⅷ " L ⅷ cofree (Section 4.3), so we have C ((AH) U Y) (C AH) U Y. We may therefore use the cochain complex L ⅷ B ⅷ 2PHom (K ,(CLA ) )P Hom (K ,(CL>A ) )P2 HU + H H U Y H U + H H U Y L L L ⅷ P L> ⅷ H (where is induced by d L q : C AH C AH) to calculate H (JH;KH) (compare [19, X, Section 7.1]). 4.7. An alternative description. Quillen's original description of the cohomology of a (simplicial) algebra included several variant approaches (cf. [51, Section 3]), and by dualizing one of these, Bous"eld obtained an alternative description of the cohomology of a coalgebra, as follows: 3 3 For any "eld R, given an unstable coalgebra JH CA0 and a JH-comodule KH JH-coMod, with P P¸ P¸ coaction : KH JH KH, and a map of coalgebras : JH H,de"ne a derivation : KH H to D. Blanc / Topology 40 (2001) 993}1016 1005 be a od- -morphism such that " # ( ) (see Section 2.1). Write M AN (H Der (K , ¸ ) for the R-vector space of all such derivations. C 0 H H 3 For every comodule KH JH-coMod, we can think of JH KH as a coalgebra under JH, with diagonal de"ned by (H=)H ## #0:J K P(J J ) (J K ) (K J ) (K K ). (H H H H H H H H H H H P¸ Thus, given a map : JH H, we have a natural identi"cation Hom ! (J K , ¸ ) Der (K , ¸ ). C 0 (H H H H C 0 H H C In fact, the functor KH JH KH is left adjoint to the coabelianization functor ! P () U : CA0 JH JH-coMod (cf. Section 2.6), and it induces an equivalence of categories between ! JH-coMod and (CA0 JH) U . Moreover, as in [51, Section 4], one has an explicit description of ¸ the functor ( ) U in terms of a cotensor product of JH with H (the coalgebraic analogue of the usual KaK hler module of di!erentials). See [17, Section 8.5; 53, Section 7.7}8] for more details on this approach. P ⅷ Now if JH AH is a resolution, we can show that there is a natural map ⅷ Hom (K , CLA )PHom ! (J K , AL ), C H H H C 0 (H H H H which induces an isomorphism in cohomology, so Remark 4.6 above implies that the Quillen cohomology groups HL(J ; K ) coincide with the derived functors of Der (K ,!) applied to J , H H C 0 H H which were considered by Bous"eld (who showed in [17, Section 9] that the groups Q R ¸ " Q I R % ¸ Der (JH, H): H (JH; HH(S ; N) H) serve as the E-term of a certain unstable Adams spectral sequence). To show this, use the fact that a morphism in either Hom-set must take values in a coabelian P L unstable coalgebra, and that the unique iterated coface map : JH AH vanishes when projected L ⅷ to C AH. We omit the details, since we shall not require this result in what follows. However, it may " be observed that in the Massey}Peterson case JH ;(MH), Bous"eld's approach allows us to identify HL(J ; K ) with the usual ExtL (K , M ,), so we can recover Harper's results in [36, H H MMBUAN H H Proposition 4.2] as a special case of Theorem 6.3 below. It is possible that one could prove the results of the following sections, using Bous"eld's description of cohomology as the derived functors of derivations, and dualizing Quillen's identi- "cation of these derived functors (for algebras) with suitable groups of extensions (see [51, Section 3]). However, there may be computational advantages to the explicit approach we have taken here. 5. Realizing resolutions The key to our approach to the realization question for an unstable coalgebra KH lies in the realization of a suitable cofree resolution of KH * by analogy with the method used in [13] for "0 % -algebras. In what follows R or N. 3 5.1. Trying to realize a resolution. Given an unstable coalgebra KH CA0, choose some cosimplicial P ⅷ + ML , resolution KH AH, with CW basis AH L, as in Section 3.11. We would like to realize this ⅷ3 algebraic resolution at the space level, i.e., "nd a co"brant cosimplicial space Q cSH, with a CW 1006 D. Blanc / Topology 40 (2001) 993}1016 + ML, L ML * M MLP ML> basis Q L, such that HH(Q ; R) AH for all n 0, and the attaching maps d/L : Q Q ⅷ ⅷ ⅷ realize those for AH, so that HH(Q ; R) AH. We attempt to construct such a Qⅷ (with its CW basis) by induction on the cosimplicial dimension: M 3 The "rst two steps are always possible: because AH CA0 is cofree by assumption (Section 3.9), M MP M M M P M M M P M we can "nd a map d/ : Q Q in SH such that (d/)I : HH(Q ; R) HH(Q ; R)isd : AH AH (Section 2.9). We then set Q ": QM and QI ": QM;¸Qⅷ"QM;Q, with dI ": dM?Id and dI ": 0?Id. In order to end up with a co"brant cosimplicial space (Section 3.1), we now change the resulting IL : MQⅷPQI into a co"bration: L : MQⅷPQ. It will be convenient to denote P ⅷ " ⅷP ⅷ d p : Q C Q by d d/ : B Q C Q , to conform to the notation of Fig. 1. ⅷ If we let B Q denote the co"ber of d/ , from the long exact sequence in homology: / d O/ I 2 ⅷ P ⅷ P ⅷ P ⅷ 2 HG>(B Q ; R) HG(B Q ; R) HG(C Q ; R) HG(B Q ; R) and from Proposition 3.6 we obtain a short exact sequence of unstable coalgebras: P ⅷ PG ⅷ ZR P 0 B (AH) HH(B Q ; R) KH 0, where KH is the graded R-module, shifted one degree up, with the trivial coalgebra structure (and "% the suspended action of the Steenrod algebra, if R N). The given coalgebra structure on KH makes KH into a KH-comodule, in an obvious way, and this is compatible with the AH-coalgebra structure. Now assume Qⅷ as required has been constructed through cosimplicial dimension n, so we have ⅷ3 6L7 an n-cosimplicial object which (by a slight abuse of notation) we denote by trL Q c SH (cf. ⅷ ⅷ Section 1.5), with HH(trL Q ; R) trL AH. By considering the cosimplicial chain complex corresponding to R1Qⅷ2, one can verify (as in the L proof of Proposition 3.6) that there always exists a factorization of (q/)I as in Fig. 2: and since ⅷ L 3 L\ ⅷ AH is a resolution, d L, and thus q , must be monic. Moreover, for any a HG(B Q ; R) we have L " 3 L ⅷ " L ⅷ ⅷ (q/)I((d/L\ )I(a)) 0; thus (d/L\ )I(a) HG(C Q ; R) C AH is a cocycle for the resolution AH,soit must be in Im(d L\ ). Thus, we must have L # " L ⅷ Im(q/)I Ker(d/L)I HH(B Q ; R). (5.2) We therefore assume by induction that we have a short exact sequence of unstable coalgebras: L L P L ⅷ PO L ⅷ ZR L P 0 B AH HH(B Q ; R) Coker (q ) 0, (5.3) where Coker(q L) LK , and in fact L" is an isomorphism onto Coker(q L), with * again H ' /L> L> the connecting homomorphism in the long exact sequence /L> d O I /L> 2 L> ⅷP L ⅷP L ⅷ P L ⅷP2 HG>B Q HGB Q HGC Q HGB Q . (5.4) D. Blanc / Topology 40 (2001) 993}1016 1007 Fig. 2. L " L Since Im(q/)I Im q in Fig. 2, in fact we have a direct sum of R-modules in (5.2), and this is by de"nition a semi-split extension of coalgebras. This implies that * " is one-to-one, and surjects onto Im(* ). (5.5) L\ ' /L L\ 5.6. Aside. Observe that if we continue our construction `naivelya by choosing some GEM ML>3 M L>P ML> Q SH with an attaching map dL> : Q Q which induces a monomorphism L ⅷ ML> HH(B Q ; R) 6 HH(Q ; R) in homology, we can easily continue this process to obtain a cosim- ⅷ ⅷ plicial space Y , such that HH(Y ; R) is a free cosimplicial coalgebra satisfying: " KH for i 0, G ⅷ L " # HH(Y ; R) KH for i n 1, (5.7) 0 otherwise. Such a Yⅷ should be thought of as the (n!1)th Postnikov section for the resolution Qⅷ.We denote a co"brant version of it by PL\Qⅷ, and observe that it is unique up to homotopy equivalence (in the model category structure of Section 3.13). This provides a convenient homotopy-invariant version of the n-coskeleton of a cosimplicial space. See [13, Section 3.4] for an explanation of the indexing. In [14], we present an alternative approach to the dual problem of realizing (simplicial) -algebras, via Postnikov systems (including k-invariants) for simplicial -algebras and simplicial spaces. This was in fact the original program of [27,28]. However, because there is no satisfactory homotopy theory for cosimplicial sets, an analogous approach here would require developing additional machinery not presently available. ⅷ 5.8. Continuing the construction. If we can extend our n-truncated object trL Q one more dimension, L ⅷP L> ⅷ we will have a principal face map dL : B Q C Q , which induces a map " L ⅷ P L> ⅷ L> ⅷ (dL)I : HH(B Q ; R) HH(C Q ; R) C AH (by Proposition 3.6). It turns out that such a is essentially all we need in order to proceed. ML> First note that since AH is a cofree unstable coalgebra by assumption, and L L ⅷ P L ⅷ M L ⅷ P ML> q : B AH HH(B Q ; R) in Fig. 2 above is monic, d L : B AH AH extends to a map L ⅷ P ML> : HH(B Q ; R) AH as in Fig. 3. We can realize by a map l : BLQⅷPQML>, for a suitable GEM QML>, and thus a map M LP ML> ⅷ ⅷ d : Q Q (see Fig. 1), which in turn determines an extension of trLQ to trL> Q ;wemay modify this to be co"brant. Moreover, from the exactness of the bottom row, by (5.2), we have L P L> ⅷ " # a unique lifting : Coker(q ) B AH as in Fig. 4, and if 0, then (5.2) will hold for n 1. 1008 D. Blanc / Topology 40 (2001) 993}1016 Fig. 3. Fig. 4. 3 L> L 5.9. De5nition. The cohomology class L H (KH; KH) represented by the cocycle ⅷ ": dL> 3Hom (LK , A ) (see Section 4.6) is called the characteristic class of the extension C H H H (5.3) (cf. [42, IV, Section 5; 13, Section 4.5]). 3 L> L 5.10. Theorem. The cohomology class L H (KH; KH) is independent of the choice of lifting , " L\ ⅷ L ⅷ and L 0 if andonly if one can extend P Q to an nth Postnikov approximation P Q of a resolution of KH. L ⅷ P ML> Proof. Assume that we want to replace by a di!erent lifting : HH(B Q ; R) AH , and choose maps l,l : BLQⅷPQML> realizing , , respectively; their respective extensions to d and LP L> ¸L> ⅷ L P L> ⅷ " (d ) : Q Q agree on Q . We correspondingly have : KH B AH and : d L> in Fig. 4. Since QL> can be any "brant GEM realizing AL>, we may assume it is a simplicial R-module, L L> " ! LP L> and thus HomSH (Q ,Q ) has a natural R-module structure. Set h : ((d ) d ):Q Q . L ⅷ P L> ⅷ ML> ! Then h induces a map : HH(B Q ; R) C AH whose projection onto AH is . Moreover, L ⅷ L" because d and (d ) agree with d L when pulled back to HH(C Q ; R), we have q 0, and thus L P L> ⅷ L factors through : KH C AH, and this is a map of AH-coalgebras because KH is a coabelian AH-comodule (actually, a KH-comodule), and is induced by group operations from the AH-coalgebra maps d and (d ) . See Fig. 5 below. D. Blanc / Topology 40 (2001) 993}1016 1009 Moreover, in the abelian group structure on Hom (LK ,!) we have )H U + H !" ! "L> d L> ( ) ( ) (see Section 4.6), so this is a coboundary, which proves indepen- dence of the choice of . ⅷK L ⅷ ⅷ ⅷ Now assume that there exists Y P Q (Section 5.6) with trL Y trL Q . By the discussion in # ML> Section 5.1, we know that (5.2) is a direct sum for n 1, and since AH is cofree, we can choose L ⅷ P ML> " " : HH(B Q ; R) AH in Fig. 3 to extend d by zero, so 0 in Fig. 4, and thus 0. " " Conversely, if L 0, we can represent it by a coboundary d L> for some AH-coalgebra ⅷ ⅷ map : K PCL>A , and thus get proj ML> L : H (BLQ ; R)PAML>, for H H H H H proj ML> : AL>PAML> the projection. If we set ": !proj ML> L (we can subtract maps, H H H H ML> L # " L ⅷ because AH is a graded R-module), we have Im(q/)I Ker HH(B Q ;R). We can therefore M L ⅷP ML> " ⅷ choose d/L> : B Q Q realizing , and then 0, so that trL> Q so constructed yields PLQⅷ, as required. ᮀ 5.11. Notation. If we wish to emphasize the dependence on the choice of , we shall write PLQⅷ[ ] L\ ⅷ 3 L> L> for the extension of P Q so constructed, and write L>( ) H (KH; KH) for the next cohomology class (which does depend on , in principle). " Remark 5.12. Note that if L 0, the choice of determines the AH-comodule structure L> # * on KH via (5.3) for n 1. Moreover, for each n 1, the resulting coaction L P L L : KH KH KH in fact agrees with the obvious KH-comodule structure, de"ned via P " PL the original comultiplication : KH KH KH: that is, if (a) Ga G a G, and L : KH KH ! " is the re-indexing isomorphism (in Mod AN), then L( L(a)) Ga G L(a G). This follows from the description in Section 4.1, and the fact that the exact sequences (5.4) (and thus also (5.3)), respect the AH-coalgebra structure (Proposition 4.4). 5.13. De5nition. Note also that by standard homotopical algebra arguments the elements 3 L> L P ⅷ L H (KH; KH) do not depend on the choice of resolution KH AH. If for some (and thus P ⅷ any) cosimplicial cofree resolution KH AH, there are successive choices of liftings ( L)L in " " Fig. 3 such that L>( L) 0 for n 0,1,2, we say that we have a coherently vanishing sequence of characteristic classes. Thus we may encapsulate our results so far in ⅷ 3 5.14. Corollary. Any cofree cosimplicial resolution AH of an unstable coalgebra KH CA0 is realiz- ⅷ3 ⅷ ⅷ able by a cosimplicial space Q cSH (with AH HH(Q ; R)) if andonly if KH has a coherently vanishing sequence of characteristic classes. 6. Realizing coalgebras We now apply Theorem 5.10, on the realization of cosimplicial resolutions of coalgebras, to the original question, namely, that of realizing a given abstract coalgebra as the cohomology of a space. 1010 D. Blanc / Topology 40 (2001) 993}1016 It turns out that the obstructions described in the previous section are all that is needed, at least in the simply connected case. 6.1. The homology spectral sequence. In [52, Section 3] and [6], Rector and Anderson de"ned the homology spectral sequence of a cosimplicial space Yⅷ (see also [16, Section 2]). This is a second quadrant spectral sequence with N ⅷ EN O HO(Y ; R) (6.2) ⅷ ⅷ3 ⅷ3 abutting to HH(Tot Y ; R), where the total space Tot Y S of a cosimplicial space Y cS is ¹ 3 ¹ " ;ⅷ ⅷ de"ned (cf. [19, I, Section 3]) to be the simplicial set ⅷ S with L HomAS( [q] ,Y ) (see Section 1.4). In general, this spectral sequence need not converge. However, under rather special conditions one does have strong convergence (see [16,55]), and this yields the following: "0 % 6.3. Theorem. For R or N, a simply-connectedunstable R-coalgebra KH is realizable as the 3 homology of some simply-connectedspace X TH if andonly if KH has a coherently vanishing sequence of characteristic classes. Proof. First note that any simply connected coalgebra over 0 is realizable by [50, Theorem I] and its corollary, so in this case the theorem merely states that one always has a coherently vanishing sequence of characteristic classes. 3 ⅷ L" ML> Given any connected space X SH, the cosimplicial space Y de"ned by Y R X (where M P M "+ 3 1 2" " , R : SH SH is the Bous"eld}Kan monad (R X)I : GrGxG R XI GrG 1 * cf. [19, I, ⅷ ⅷ Section 2]), is a cosimplicial resolution of c(X) in the sense of Section 3.13 * i.e., HH(Y ; R)is a cosimplicial cofree resolution of HH(X; R) (see [20, 11.5]). But then by Corollary 5.14, HH(X;R) has a coherently vanishing sequence of characteristic classes. Conversely, assume that KH is a simply connected unstable R-coalgebra with a coherently vanishing sequence of characteristic classes. By Corollary 5.14, any cosimplicial cofree resolution P ⅷ ⅷ3 " KH AH may be realized by a cosimplicial space Q cSH. In particular, since K R and " L L K 0, we may assume that the same holds for each AH, so that each R-GEM Q is simply ⅷ L I ⅷ " " ) * connected. Because AH is a resolution, HL>Q(Q ; R) 0 for n 0 and s 1orn 1, and thus by ⅷ ⅷ [16, Theorem 3.4] the homology spectral sequence for Q converges strongly to HH(Tot Q ; R). Since the E-term of (6.2) is concentrated along the 0-line, we get ⅷ " P" ⅷ KH AH E H HH(Tot Q ; R) and this is an isomorphism of unstable coalgebras, since the edge homomorphism is induced by ⅷP ⅷ" ᮀ a topological map Tot Q Tot Q Q . 6.4. The non-simply connected case. The simple connectivity of KH only was needed to guarantee convergence of the homology spectral sequence, using [16, Theorem 3.4]; the (algebraic) obstruc- tion theory described in the previous section is of course also valid in the non-simply connected case. In particular, by dualizing [9, Proposition 5.1.4] we can construct a resolution with CW basis ML ML" ( ) (AH)L with strictly increasing connectivity * so that in particular AQ 0 for 0 s n * and D. Blanc / Topology 40 (2001) 993}1016 1011 ⅷ ⅷ then realize AH by a cosimplicial space Q , assuming that KH has a coherently vanishing sequence of characteristic classes. ¹" ¹ P Now consider the functor ( G)G : CA0 gr R-Mod, de"ned on cofree coalgebras by ¹ " " (GH) However, it is not clear on the face of it whether unstable coalgebras can ever have non-trivial "nite projective dimension (cf. [41, Theorem 4.3]). "% 6.6. Remark. As noted in the introduction, when R N, Theorem 6.3 (and perhaps also Proposi- tion 6.5) provides a way of constructing small, even minimal, `unstable Adams resolutionsa Qⅷ of a given (simply connected) space X, which could be used in computing the Bous"eld}Kan spectral ⅷ sequence of [20] for H(RX). In particular, when X is of "nite type, one can choose Q so that each L % space Q is a "nite-type product of copies of K( N, k) (for various k). 7. Distinguishing between realizations Another interesting question is how one can distinguish between non-homotopy equivalent 3 realizations X,Y SH of a given unstable coalgebra KH; we shall try to do this in terms of di!erent ⅷ ⅷ3 P ⅷ realizations Q , T cSH of a "xed cosimplicial cofree resolution KH AH, where we assume to begin with that KH is in fact realizable. Our goal is to "nd necessary conditions in order for two realizations Qⅷ and Tⅷ to yield homotopy equivalent total spaces (cf. [13, Theorem 4.21]). Again the key lies in the extension of coalgebras (5.3). Of course, we may assume that the 3 L> L characteristic class L H (KH; KH) vanishes, so that it is possible to "nd various splittings of (5.3) as a `semi-direct producta, given by di!erent choices of lifting in Fig. 3. The di!erence between two such semi-direct products is represented by a suitable cohomology class (cf. [40, Section 6]; IV, [42, Section 2]), constructed as follows: L ⅷ P ML> 7.1. De5nition. Given two liftings , : HH(B Q ;R) AH in Fig. 3 above * determining exten- ⅷ ⅷ sions of trL Q to trL> Q ) * as in the proof of Theorem 5.10, we may assume that the L P L> ⅷ corresponding maps , : KH B AH vanish. We extend , as in Section 5.8 to coface maps LP L> L ⅷ P L> ⅷ L" d , d : Q Q ,de"ne : HH(B Q ;R) C AH with q 0, and extend to a map of AH- L P L> ⅷ L> " algebras : KH C AH (again, as in the proof of Theorem 5.10). Again (q L> )I 0, so is a cocycle in Hom (K , CHAⅷ ), representing a cohomology class 3HL>(K ,LJ ), which C H H H H HY H H we call the diwerence obstruction for the corresponding Postnikov sections PLQⅷ[ ] and PLQⅷ[ ] (in the notation of Section 5.6). 1012 D. Blanc / Topology 40 (2001) 993}1016 7.2. Remark. Again, by standard arguments this cohomology class is independent of the P ⅷ 3 speci"c algebraic resolution KH AH in cCA0. Now assume that X,Y SH are two (di!erent) ⅷ ⅷ ⅷ realizations of KH, with Q and T respectively cosimplicial spaces realizing AH (so that ⅷK ⅷ ⅷK ⅷ Q c(X) and T c(>) in the resolution model category cSH of Section 3.13), with the ! L\ ⅷK L\ ⅷ ⅷ" ⅷ same (n 1)-type (that is, P Q P T , so in particular we can assume that trL Q trL T ). L ⅷ " L ⅷ L ⅷ " L ⅷ Then we can choose and so that P Q [ ] P Q , P Q [ ] P T * and thus H HY depends only on the n-type of Qⅷ and Tⅷ, respectively, so in particular only on the homotopy types of X and Y. " L> L L ⅷ K L ⅷ 7.3. Theorem. If L 0 in H (KH; KH), then P Q [ ] P Q [ ] in the resolution model category structure. " L P L ⅷ L " Proof. If L 0, there is a map of AH-comodules : KH C AH such that d q , and by L P L the discussion in Section 4.6 can be lifted to a map : KH AH (actually factoring through L L " L L L ⅷ P (AH) [email protected]"ne a map of AH-comodules: : (q/) , then : HH(B Q ; R) ML> AH is just M" " ! d proj M (7.4) in Fig. 5. L L L ⅷP L Note that because AH is cofree, we can realize by a map f : B Q Q in SH,so is realized L L L ⅷP L ⅷ by q/ p/ f : B Q B Q . We may take the simplicial GEM QL to be a simplicial R-module, with : QL;QLPQL the addition map, and de"ne g : QLPQL to be the composite (Id?( fpLqL)). For every 0)i)n we have gdG"dG : QL\PQL,sopLqLg induces a map h : BLQⅷPBLQⅷ, with "# 3 L ⅷ hI( ) ( ) for HH(B Q ;R), (7.5) and thus by (7.4) the following diagram in CA0 commutes: # ⅷ P ⅷ which yields a map of (n 1)-truncated objects trL> Q [ ] trL> Q [ ], or equivalently, L ⅷ P L ⅷ I ! amap : P Q [ ] P Q [ ], which clearly induces an isomorphism in HH( ;R) for k)n#1. D. Blanc / Topology 40 (2001) 993}1016 1013 Fig. 5. L> L ⅷ * L> Note that for any choice of we have P Q [ ] Im( L>) KH, by (5.7). Since L L " L" L "L (q/)I 0 by (5.3) and Fig. 2, we have 0, so by (7.5) hI , and since the following square commutes: and h " is an isomorphism, so is h " , by (5.5). Thus L>( ) is an isomorphism, too, so I ' /L> I ' /L> I is a weak equivalence by (5.7) for PLQⅷ. ᮀ "% 0 3 7.6 Theorem. For R N or , assume X,Y SH are two R-goodrealizations of a gi ven unstable coalgebra KH of xnite type, which is either simply connectedor of xnite projective dimension. If the diwerence obstructions for X and Y (Section 7.2) all vanish, then X and Y are R-equivalent K (i.e., RX RY). ⅷ 3 Proof. When KH is of "nite type, we can choose a cosimplicial resolution AH cCA0, with a CW ML L ⅷ ⅷ basis in which each AH (and thus each AH)isof"nite type. Let Q and T be cosimplicial spaces ⅷ realizing AH, which are resolutions (in the sense of Section 3.13) of X and Y, respectively, as in ⅷ ⅷ Section 7.2. By Theorem 6.3 (resp., Proposition 6.5), we have HH(Tot Q ; R) KH HH(Tot T ; R). ⅷ3 ⅷⅷ Let R W cSH denote the diagonal of the bicosimplicial space RW obtained from a given ⅷ3 cosimplicial space W cSH by applying the Bous"eld}Kan R-resolution functor [19, I, Section 4.1] dimensionwise to Wⅷ. By Theorem 7.3 there is a map of cosimplicial spaces : QⅷPTⅷ which is a weak equivalence in cSH, so induces an isomorphism in the E -terms of the homology spectral ⅷ ⅷ L L * sequences for Q and T . Since HH(Q ; R) and HH(T ; R) are of "nite type for each n 0, by Shipley [55, Theorem 9.1], induces a homotopy equivalence Tot RQⅷP! Tot RTⅷ (and similarly ⅷP ⅷP Tot Q RX and Tot T RY). 1014 D. Blanc / Topology 40 (2001) 993}1016 However, for each n*0, the spaces QL and TL are R-GEMs, so they are R-complete (cf. [19, V, 3.3]), and thus Tot RQⅷKTot Qⅷ by [55, Theorem 10.2], and similarly for Tⅷ,sowe"nd that X and Y are indeed R-equivalent. ᮀ "% "0 7.7. Remark. Shipley's theorems, in [55], were originally stated for R N; when R it is no longer true that all relevant homotopy groups are "nite. However, they are "nite dimensional vector spaces over 0, so [18, IX, Section 3], and the rest of Shipley's arguments, still apply. As noted in the proof of Theorem 6.3, when R"0 the only question of interest is distinguishing between di!erent realizations of a given R-(co)algebra; the obstruction theory described here is just the vector-space dual of the theory for graded algebras over 0 de"ned by Halperin and Stashe! in [35] (see also [32]). "% 7.8. Remark. When R N, Theorem 7.6 can be thought of as providing a collection of algebraic % invariants * starting with the homology coalgebra HH(X; N) * for distinguishing between p-types of spaces. As with the ordinary Postnikov systems and their k-invariants, these are not actually invariant, in the sense that distinct values (i.e., non-vanishing di!erence obstructions) do not guarantee distinct p-types. This approach is the Hilton}Eckmann dual of the theory described in [13] or [14] for distinguishing (integral) homotopy types, starting with the homotopy -algebra HX,in terms of an analogous collection of cohomology classes. It is reasonable to expect a more general version of Theorem 7.6 to hold, without the assumption of "nite type, and for any R-0; but this would require a stronger convergence result than that provided by Shipley [55, Sections 9}10]. Perhaps it should be observed that many non-realization results proven in the past (see the Introduction) have used higher order cohomology operations; these are implicit in the Quillen cohomology cohomology classes of Theorems 5.10 and 7.3, and were made explicit in the -algebra analogue in [11]. We hope to return to this point in the future. 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