ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES DAVID BLANC Abstract.We define a sequence of purely algebraic invariants - namely, * *classes in the Quillen cohomology of the -algebra ss*X - for distinguishing between diff* *erent homotopy types of spaces. Another sequence of such cohomology classes allows one t* *o decide whether a given abstract -algebra can be realized as the homotopy -algebra of a spa* *ce. 1.Introduction The usual Postnikov system for a (simply-connected) CW complex X serves to det* *ermine its homotopy type. One begins with purely algebraic data, consisting of the hom* *otopy groups (ssnX)1n=2. However, in order to construct the succesive approximations X(n)(* *n 2), with X ' holimX(n), one must specify a sequence of cohomology classes kn 2 Hn+2(X(* *n); ssn+1X) (see [W , IX, x2]). These can hardly qualify as algebraic invariants, since th* *eir description involves the cohomology groups of topological spaces. In this paper we show th* *at if one is willing to invest the graded group ss*X := (ssnX)1n=1with some further algebra* *ic structure, the additional information needed to determine the homotopy type of X can be de* *scribed in purely algebraic terms. The structure needed on ss*X is that of a -algebra - i.e., a graded group equi* *pped with an action of the primary homotopy operations (Whitehead products and compositions)* *. In this context, the additional data needed consists of cohomology classes in the Quill* *en cohomology of this -algebra - which can be defined as usual in algebraic terms (see x4.1* * below). We show: Theorem A. Given two realizations X and X0 of a -algebra J*, there is a s* *uccessively defined sequence of "difference obstructions" ffin 2 Hn+1(J*; nJ*), taking valu* *e in the Quillen cohomology groups of J*, with coefficients in the J*-module nJ*, whose van* *ishing implies that X ' X0. (See Theorems 4.18 and 4.21 below). The (n + 1)-st cohomology class is defined * *whenever the n-th Postnikov section of the simplicial space resolutions of the spaces X and * *X0, respectively, agree, up to homotopy. Even though the obstructions are defined in terms of a s* *pecific choice of -algebra resolution of J*, in fact they depend only on the homotopy type of * *the Postnikov sections. Moreover, these cohomology groups can also be used to determine the realizabil* *ity of an abstract -algebra as the homotopy groups of some space: Theorem B. Given a -algebra J*, there is a successively defined sequence of "c* *haracteristic classes" 2 Hn+2(J*; nJ*), which vanish if and only if J* is realizable by a to* *pological space. (See Theorems 4.8 and 4.15 below). The vanishing requirement should be understo* *od in the sense of an obstruction theory: if any such sequence of cohomology classes vani* *shes, the - algebra is realizable; if one reaches a non-trivial obstruction, one must back* *-track, and try to ____________ Date: (Revised version) September 18, 1998. 1991 Mathematics Subject Classification. Primary 55S45; Secondary 55Q35, 55P1* *5, 18G10, 18G55. Key words and phrases. homotopy invariants, simplicial resolution, homotopy t* *ype, ;-algebra Quillen cohomology. 1 2 DAVID BLANC vary the choices involved in order to obtain a realization. These choices again* * depend only the homotopy type of a suitable Postnikov section - this time, of a simplicial re* *solution we are trying to construct for the putative topological space X realizing J*. See Pr* *oposition 4.10 below. The theory is greatly simplified if we are only interested in the rational hom* *otopy type of a simply-connected space X. In that case, a rational -algebra is simply a graded * *Lie algebra over Q, and the cohomology theory in question reduces to the usual cohomology o* *f Lie algebras. Theorem A thus provides an integral version of (the dual to) the Halperin-Stash* *eff obstruction theory for rational homotopy types (see [HS ] and x4.22 below). It is in order to be able to deal with this case, too (and other possible vari* *ants - see x2.14 below), that we have stated our results for a general model catgory C (su* *bject to certain somewhat restrictive simplifying assumptions on C - not all of which are really* * necessary). For technical convenience we have chosen to describe the ordinary topological versi* *on of our theory within the framework of simplicial groups, rather than topological spaces (see * *x4.12 below). 1.1. notation and conventions. T will denote the category of topological space* *s, and T* that of pointed connected topological spaces with base-point preserving maps. T* *he base-point will be written * 2 X. The category of groups is denoted by Gp, that of graded groups by grGp, th* *at of (left) R-modules by R-Mod , and that of sets by Set. 1.2. Definition. is the category of ordered sequences n = <0; 1; : :;:n> (n * *2 N), with order-preserving maps. opis the opposite category. As usual, a simplicial ob* *ject over any category C is a functor X : op! C; more explicitly, it is a sequence of objec* *ts {Xn}1n=0in C, equipped with face maps di: Xn ! Xn-1 and degeneracies sj : Xn ! Xn+1 (0 i; * *j n), satisfying the usual simplicial identities ([May , x1.1]). We usually denote * *such a simplical object by Xo. The category of simplicial objects over C is denoted by sC. T* *he standard embedding of categories c(- )o : C ! sC is defined by letting c(X)o 2 sC denote* * the constant simplicial object on any X 2 C (with c(X)n = X, di= sj= idX ). The category of simplical sets will be denoted by S, rather than sSet, tha* *t of pointed connected simplicial sets by S*, and that of simplicial groups by G. If we con* *sider a simplicial object Xo over G, say, we shall sometimes call n in X1; : :;:Xn; : :t:he exte* *rnal simplicial dimension, written (-)extn, in distinction from the internal simplicial dimen* *sion k, inside G, denoted by (-)intk. In this case we shall sometimes write (Xo)intk2 sGp, in* * contrast with Xn 2 G, to emphasize the distinction. The standard n simplex in S is denoted by [n], generated by oen 2 [n]n, wi* *th k[n] the subobject generated by dioen for i 6= k. If we denote by the category obtained from by omitting the objects {* *k}1k=n+1, the category of functors ( )op! C is called the category of n-simplicial o* *bjects over C - written sC. If C has enough colimits, the obvious truncation functor trn: s* *C ! sC has a left adjoint aen : sC ! sC, and the composite skn:= aen O trn: sC ! sC * * is called the n-skeleton functor. 1.3. organization. In section 2 we review some background material on closed mo* *del category structures for categories of simplicial objects and show how certain convenient* * CW resolutions may be constructed therein. In section 3 we construct Postnikov systems for suc* *h resolutions, and define the action of the fundamental group on them; and in section 4 we exp* *lain how these resolutions are determined in terms of appropriate cohomology classes, which ma* *y also be used to determine the realizability of a (generalized) -algebra (Theorems 4.8 and 4.* *15), as well as to distinguish between different possible realizations (Theorems 4.18 and 4.21). ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 3 1.4. Acknowledgements.I would like to thank Dan Kan for suggesting that I conti* *nue the project begun in [DKS1 ] and [DKS2 ], and Bill Dwyer, Phil Hirschhorn and Emman* *uel Dror- Farjoun for several useful conversations. I am especially grateful to Hans Baue* *s and Paul Goerss for pointing out the necessity of taking into account the action of the fundame* *ntal group in describing the coefficients of the cohomology groups. I would also like to than* *k the referee for his comments. It should be noted that Baues had previously constructed the first difference * *obstruction of Theorem A, lying in H2(J*; J*), by different methods, and has since extended hi* *s construction to the full range of invariants we define here: see [Ba3]. Yet a third descri* *ption of these invariants, more in the spirit of the original approach of Dwyer, Kan, and Stov* *er, is planned in [BG ]. 2. model categories of simplicial objects We first review some background material on model category structures for cate* *gories of simplicial objects, in particular a slightly expanded version of structure defi* *ned in [DKS2 ], and show how one can construct CW resolutions in such a context. 2.1. model categories. A model category in the sense of Quillen (see [Q1 ]) is * *a category C equipped with three distinguished classes of morphisms: W (weak equivalences),* * C, and F, satisfying the following assumptions: (1) C has all small limits and colimits. (2) W is a class of quasi-isomorphisms (i.e., there is some functor F : C ! * *D such that f 2 W , F (f) is an isomorphism). p (3) Any morphism f : A ! B in C has a factorization A -i!C -! B (f = p O i* *) with i 2 C \ W and p 2 F; moreover, this factorization is unique up to weak e* *quivalence, in 0 p0 the sense that if A i-!C0-! B is another such factorization of f (i02 C * *\ W, p02 F), then there is a map h : C ! C0 such that h O i = i0 and p0O h = p. p (4) Similarly, any morphism f : A ! B in C has a factorization A i-!C -!B * *(f = p O i) with i 2 C and p 2 F \ W - again unique up to weak equivalence. (5) We will assume here that the factorizations above may be chosen functoriall* *y (though this is not included in the original definition in [Q1 , I, x1]). We call the closures under retracts of C and F the classes of cofibrations and* * fibrations, respectively. The definition given here is then equivalent to the original on* *e of Quillen in [Q1 , Q3] (see [Bl5, x2]). An object X 2 C is called fibrant if X ! *f is a fibration, where *f is th* *e final object of C; similarly X is cofibrant if *i! X is a cofibration (*i= initial object* *). If X 2 C is cofibrant and Y 2 C is fibrant, we denote by [X; Y ]C (or simply [X; Y ]) the s* *et of homotopy equivalence classes [f] of maps f : X ! Y . For this to be defined we in fa* *ct need only require X to be cofibrant or Y to be fibrant (cf. [Q1 , I,x1]). A map in W \ F * *is called a trivial fibration, and one in W \ C a trivial cofibration. Given a model category , one can "invert the weak equivalences" to* * obtain the associated homotopy category hoC, in which the set of morphisms from X to Y is* * just [X; Y ] (at least when X and Y are both fibrant and cofibrant). See [Q1 , I], [Q3 , I* *I,x1], or [Hi, ch. IX-XI] for some basic properties of model categories. 2.2. pointed model categories. In a pointed model category - i.* *e., one with a zero object, denoted by 0 or * (= *f = *i) - we may define the fiber of a m* *ap (usually: a f fibration) f : X ! Y to be the pullback of X -!Y *, and the cofiber of a map* * (usually: a cofibration) i : A ! B to be the pushout of * A -i!B. The suspension A* * of a 4 DAVID BLANC (cofibrant) object A 2 C is then defined to be the cofiber of A q A ! A x I,* * where A x I is any cylinder object for A (cf. [Q1 , I,1,Def. 4]); it is unique up to homo* *topy equivalence. Similarly, the loops X of a fibrant object X is the fiber of XI ! X x X, wh* *ere XI is a path object for X (ibid.). Finally, the cone CA of a (cofibrant) object A 2 C* * is the cofiber of either map A ,! A x I. See [Q1 , I,2.8-9]. 2.3. simplicial objects. For any category C with coproducts, one has a simplic* *ial structure (cf. [Q1 , II, x1]) on the category sC of simplicial objects over C, defined a* *s usual by: ` (i)For any simplicial set A 2 S and X 2 C, we define X ^A 2 sC by (X ^A)n := * * a2AnX, with the face and degeneracy maps induced from those of A. We denote the c* *ofiber of A ^*! A ^X by A ^ X. ` Now for Xo 2 sC we define Xo A 2 sC by (Xo A)n := a2AnXn (the diagonal of the bisimplicial object Xo^A). (ii)For any Xo; Yo 2 sC we define the function complex map (Xo; Yo) by map (Xo; Yo)n := Hom sC(Xo [n]; Yo); where [n] 2 S denotes the standard simplicial n-simplex. 2.4. Definition.For any complete category C, the matching object functor M : So* *pxsC ! C, written MAXo for a (finite) simplicial set A 2 S and any Xo 2 sC, is defined b* *y requiring that M[n]Xo := Xn, and if A = colimiAi then MAXo = limiMAiXo (see [DKS2 , x* *2.1]). This may be defined by adjointness, via: HomsC(Z A; Xo) ~=Hom C(Z; MAXo) for Xo 2 sC and Z 2 C. In particular, we write MknXo for MAXo where A is the subcomplex of skn-1[n] g* *enerated by the last (n - k + 1) faces (dkoen; : :;:dnoen). When C = Set or Gp, f* *or example, this reduces to: (2.5) MknXo = {(xk; : :;:xn) 2 (Xn-1)n+1| dixj= dj-1xi for allk i < j n}; and the map ffikn: Xn ! MknXo induced by the inclusion A ,! [n] is defined * * ffin(x) = (dkx; : :;:dnx). The original matching object of [BK , X,x4.5] was M0nXo = M@[* *n]Xo, which we shall further abbreviate to MnXo; note that each face map dk : Xn+1 ! Xn * * factors through ffin := ffi0n. See also x3.1 below and [Hi, XVII, 87.17]. The dual construction yields the colimit LnXo, sometimes called the "n-th`latc* *hing object" of Xo - see [DKS1 , x2.3(i)]. For Xo 2 x, for example, we have LnXo := 0in-1* * Xn-1= ~, where for any x 2 Xn-k-1 and 0 i j n - 1 we set sj1sj2: :s:jkx in the i* *-th copy of Xn-1 equivalent to si1si2: :s:ikx in the j-th copy of Xn-1 whenever the simplic* *ial identity sisj1sj2: :s:jk= sjsi1si2: :s:ik holds (so in particular sjx 2 (Xn-1)i is equivalent to six 2 (Xn-1)j+1 for al* *l 0 i j n - 1). The map oen : LnXo ! Xn is defined oen(x)i= six, where (x)i2 (Xn-1)i. There are (at least) two ways to extend a given model category structure on C * *to sC: 2.6. Definition.In the Reedy model structure on sC (see [R ] or [Hi, XVII, x88]* *), a simplicial map f : Xo ! Yo is (i)a weak equivalence if fn : Xn ! Yn is a weak equivalence in C for each n * *0; (ii)a (trivial) cofibration if fn q oen : Xn qLnXoLnYo ! Yn is a (trivial) cofi* *bration in C for each n 0; ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 5 (iii)a (trivial) fibration if fn x ffin : Xn ! Yn xMnYoMnXo is a (trivial) fib* *ration in C for each n 0. Note that these definitions imply that Xo 2 sC is fibrant if and only if the m* *aps ffin : Xn ! MnXo are fibrations (in C) for all n. We shall require another structure, originally called the "E2-model category" * *(see [DKS1 , x3] and x4.20 below), defined under the following 2.7. Assumption.Assume that is a pointed cofibrantly generated mod* *el category, in which every object is fibrant (this holds, for example, if C = T* or C = G* *). Let F = FC be a small full subcategory of C with the following properties: (i)There is a subset {M}ff2F^ ObjF consisting of cogroup objects for C -* * so there is a natural group structure on Hom C(M; Y ) for any Y 2 C. (ii)F is closed under coproducts, and`every object Z 2 F is weakly equivalent* * to some (possibly infinite) coproduct iM with ffi 2 F^ - so Z is a homot* *opy cogroup object (i.e., [Z; Y ]C has a natural group structure). However, we do no* *t require the morphisms in F to respect the cogroup structure, even up to homotopy. (iii)F is closed under suspensions - that is, for each X 2 F, there is a mo* *del for X in F. We also assume CM 2 F for every ff 2 ^F (x2.2). We now wish to define an algebraic model for the collection of sets of homotop* *y classes of maps {[X; Y ]C}X2F , for a given object Y 2 C. This is provided by the followi* *ng 2.8. Definition.Given F C as in x2.7, we define a F-algebra to be a functor ho* *(F)op! Set, which takes coproducts in F to products in Set (compare [Dr]). The category of all F-algebras will be denoted by F -Alg, and the functor [* *hoF; -] : C ! F -Alg defined ([B; Y ])B2hoF will be denoted by ssF. F -Alg is a category* * of universal graded algebras, or CUGA, in the sense of [BS , x2.1]. In particular,`the`free* * F-algebras are those isomorphic to ssFX for some X 2 F. If we assume that X ' ff2F^t2TffM<* *ff>t for some F^-graded set T*, we say that ssFX is the free F-algebra generated by T*. If f : X ! Y is a morphism in C, the induced morphism of F-algebras, ssFf* * : ssFX ! ssFY , will be denoted simply by f#. 2.9. Remark.Since all objects in F are homotopy equivalent to coproducts of obj* *ects from the set ^F, a F-algebra may be thought of more concretely as an ^F-graded group * * - i.e., a collection of groups (Gff)ff2F^- equipped with a (contravariant) action of the * *homotopy classes of morphisms in F on them, modeled on the action of such homotopy classes on {[* *M; Y ]}ff2F^ by precomposition (cf. [W , XI, x1]). We shall write ssffX for (ssFX)ff:= [M; X], and ssff+kX for [kM; X]. 2.10. Definition.As usual, a F-algebra X is called abelian if Hom F -Alg(X; A) * *has a natural abelian group structure for any A 2 F -Alg (see [BS , x5.1] for an explicit de* *scription.). In particular, for any X 2 F -Alg, its abelianization Xab may be defined as in [B* *S , x5.1.4] as a suitable quotient of X. Another abelian F-algebra which may be defined for any * *X is its loop algebra X, defined by X(B) := X(B) (cf. [DKS2 , x9.4]; recall that F is clo* *sed under suspension). The fact that it is abelian follows as in [Gr, Prop. 9.9]. The (a* *belian) category of abelian F-algebras will be denoted by F -Algab. 2.11. Example. In C = T*, let F denote the subcategory whose objects are we* *dges of spheres of various dimensions; then for any space X 2 T*, the functor ssFX is d* *etermined up to isomorphism by ss*X, the homotopy -algebra of X - that is, its homotopy grou* *ps, together with the action of the primary homotopy operations (Whitehead products and comp* *ositions) 6 DAVID BLANC on them. See [Bl2, x2] or [St, x4]. In particular, the abelian -algebras are th* *ose for which all Whitehead products are trivial (cf. [Bl2, x3]). 2.12. Remark.This example does not quite fit our assumptions (x2.7), since the* * spheres are only co-H-spaces, i.e., homotopy cogroup objects in T*. This does not affect * *the arguments at this stage - in fact, this is the original example of an "E2-model catego* *ry" in [DKS1 ]. However, for our purposes G appears to be more convenient than T* as a model * *for the homotopy category of (connected) spaces (see [K2 ]; also, e.g., [Bl6, x5]). In fact, in all the examples we have in mind the objects in C will have an (un* *derlying) group structure, so it will be convenient to add to x2.7 the following additional 2.13. Assumption.C is equipped with a faithful forgetful functor ^U: C ! D - * * where D is one of the "categories of groups" D = Gp, grGp, G, R-Mod , or sR-Mod , for some* * ring R - and the cogroup objects M 2 ^Fof x2.7(i) are in the image of its adjoint ^F* *, with the group structure on Hom C(M; X) induced from that of U^(X). When D = G or D = sR* *-Mod , the objects M must actually lie in the image of the composite ^FO F 0: S* * ! C, where F 0: S ! D is adjoint to the forgetful functor U0: D ! S. We also assume that the adjoint pair (U^; ^F) create the model category stru* *cture on C in the sense of [Bl5, x4.13] - so in particular ^Ucreates all limits in C (cf. [Mc* *1 , V,x1]). 2.14. Remark.In fact, the categories C in which shall be interested are the fol* *lowing: o C = G, so sC, the category of bisimplicial groups, is a model for simplicia* *l spaces; o C = Gp, so sC = G is a model for the homotopy category of connected topolog* *ical spaces of the homotopy type of a CW complex; o C = dL, the category of differential graded Lie algebras (or equivalently,* * C = sLie), so sC is a model for simplicial rational spaces; o C = Lie, the category of Lie algebras, so sLie is a model for (simply conne* *cted) rational spaces (cf. [Q3 , II,x4-5]); o C = R-Mod , the category of (left) modules over a not-necessarily commutat* *ive, possibly graded, ring R, so sC is a model for chain complexes over R. and it is the desire to give a unified treatment for these five cases that forc* *es upon us the somewhat unnatural set of assumptions we have made in x2.7 and here. 2.15. Definition.A map f : Vo ! Yo in sC is called F-free if for each n 0, th* *ere is a) a cofibrant object Wn which is weakly equivalent to an object in F; b) a map 'n : Wn ! Yn in C which induces a trivial cofibration (VnqLnVoLnYo)qW* *n ! Yn. 2.16. The resolution model category. Given a model category C and a subcategory* * F as in x2.7, we define the resolution model category structure on sC, with respect * *to F by setting a simplicial map f : Xo ! Yo to be (i)a weak equivalence if ssFf is a weak equivalence of ^F-graded simplicial g* *roups (x2.9). (ii)a cofibration if it is a retract of an F-free map; (iii)a fibration if it is a Reedy fibration (Def. 2.6(iii)) and ssFf is a (le* *velwise) fibration of simplicial groups (that is, for each B 2 F and each n 0, the group hom* *omorphism [B;fn] ext ext [B; Xn] ---! [B; Yn] is an epimorphism (where for Go := [B; Yo] 2 G, Go * *denotes the connected component of the identity) - see [Q1 , II,3.8]. This was originally called the "E2-model category structure" on sC. See [DKS* *1 , x5] for further details. 2.17. Example. Let C = Gp with the trivial model category structure: i.e., on* *ly isomor- phisms are weak equivalences, and every map is both a fibration and a cofibrati* *on. Let FGp be the category of all free groups (which are the cogroup objects in Gp - cf.* * [K1 ]). The ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 7 resulting resolution model category structure on G := sGp is the usual one (c* *f. [Q1 , II, x3]). This observation is due to Pete Bousfield. We can then iterate the process by l* *etting FG be the category of (coproducts of) the G-spheres, defined: Sn := F Sn-12 G - see* * [Mi] - (with S0 := GS0), and obtain a resolution model category structure on sG (bisimplici* *al groups). Note that if we tried to do the same for C = Set, there are no nontrivial co* *group objects, while in S not all objects are fibrant (see x2.7). The category T* of pointed * *topological spaces, which is the main example we actually have in mind, does not quite fit our assu* *mptions (but see x2.12 above). Motivation for the name of "resolution model category" is provided by the foll* *owing 2.18. Definition.A resolution of an object Xo 2 sC (relative to F) is a cofibra* *nt replacement for Xo in the resolution model category on sC determined by F: that is, it i* *s any cofibrant object Qo, equipped with a weak equivalence to Xo, which may be obtained fr* *om the factorization of * ! Xo as * cof-!Qo fib+w.e.----!Xo - and is thus unique up to* * weak equivalence, by x2.1(4)). More classically, a (simplicial) resolution for an object X 2 C is a resolutio* *n of the constant simplicial object c(X)o (cf. x1.2) in sC. 2.19. Functorial resolutions. The construction of [St, x2] provides canonical r* *esolutions in sC, defined as follows: consider the comonad L : C ! C given by a a [ a a (2.20) LY = MOE CM ; ff2F^OE2HomC(M;Y ) ff2F^2HomC(CM;Y ) by which we mean the the coproduct, over all OE : M ! Y , of the colimits* * of the various diagrams consisting of an inclusion MOE! CM for each : CM ! Y s* *uch that |M= OE. The counit " : LY ! Y is "evaluation of indices", and the comult* *iplication # : LY ,! L2Y is the obvious "tautological" one. Note that LY 2 F for any Y 2* * C by our assumptions on F (x2.7). Given X 2 C, we define its canonical resolution Qo ! X by Qn := Ln+1X, w* *ith the degeneracies and face maps induced as usual by " and # (see [Gd , App., x3]). The construction can be modified so as to yield resolutions for arbitrary Yo * *2 sC, and not only c(X)o. Moreover, it has the advantage that ssFg : ssFQn ! ssFYn is cle* *arly surjective for all n, so g can be changed into a fibration (Def. 2.16(iii)) by simply chan* *ging each Qn up to homotopy, which yields the factorization needed for x2.1(3). An alternative (noncanonical) construction of a resolution is given in Proposi* *tion 2.41 below. 2.21. representing objects for sC. Just as the spheres "represent" the weak equ* *ivalences in the usual model structure on T*, for example, in the sense that a map f : X ! Y* * is a weak equivalence if and only if it induces an isomorphism f* : [Sn; X] ! [Sn; Y ] fo* *r each n 0, we may similarly define representing objects for the resolution model category (co* *mpare [DKS2 , x5.1]): 2.22. Definition.Given a model category C and a subcategory F as above, for eac* *h n 0, the n-dimensional simplicial F-sphere, denoted by SnF, is the subcategory n* *F of sC, whose objects are of the form nX := X ^ Sn for X 2 F, where Sn = [n]= _[n]* * is the usual simplicial n-sphere (see x2.3(i)). Note that each such nX is cofibrant (in fact, free) in the resolution model * *category sC. Moreover, by the definition of the simplicial structure on sC (x2.3), nX is* * also a cogroup object in sC. 8 DAVID BLANC Given Yo 2 sC, choose some fibrant replacement Xo (that is, factor Yo ! * as Y* *o cof+w.e.----! Xo fib-!*, using x2.1(3)) and define ^ssnYo(also written [SnF; Yo]) to be * *the F^-graded set ss0map (SnF; Xo). This definition is independent of the choice of Xo. We define a map f : Xo ! Yo in sC to be an F-equivalence if it induces isomo* *rphisms in ^ssn(-)for all n 0. 2.23. fibration sequences. Let F C be as in x2.7, and Xo ! Yo a fibration* * in the resolution model category sC (x2.16), with fiber Fo (x2.2). Then as usual we * *have the long exact sequence of the fibration: (2.24) . .!.^ssn+1Yo@*-!^ssnFo! ^ssnXo! ^ssnYo! . .!.^ss0Yo; (see [Q1 , I,3.8]), which in fact may be constructed in this case as for S* (se* *e [May , 7.6]). 2.25. Definition.Given Xo 2 sC, , we define the n-cycles object of Xo, written * *ZnXo, to be the fiber of ffin : Xn ! MnXo (see x2.4), so ZnXo = {x 2 Xn| dix = 0 fori * *= 0; : :;:n} (cf. [Q1 , I,x2]). Of course, this definition really makes sense only when ff* *in is a fibration in C. Similarly, the n-chains object of Xo, written CnXo, is defined to be* * the fiber of ffi1n: Xn ! M1nXo. Note that for any W 2 C and fibrant Xo 2 sC we have natural adjunction iso* *morphisms Hom sC(W ^ Sn; Xo) ~=Hom C(W; ZnXo) and HomsC(W ^ Dn; Xo) ~=Hom C(W; CnXo) (* *where Dn := n=0[n] 2 S is a simplicial model for the n-disc). If Xo is fibrant, the map d0 = dn0:= d0|CnXo: CnXo ! Zn-1Xo is the pullbac* *k of ffin : Xn ! MnXo along the inclusion : Zn-1Xo ! MnXo (where (z) = (z; 0; : :* *;:0)), so d0 is a fibration (in C), fitting into a fibration sequence jXn d0 (2.26) . .Z.n-1Xo ! ZnXo -! CnXo -! Zn-1Xo (see [DKS2 , Prop. 5.7]). Moreover, there is an exact sequence of F-algebras (d0)# q (2.27) ssFCn+1Xo ---! ssFZnXo -!^ssnXo! 0; (see [DKS2 , Prop. 5.8]), which provides a (relatively) explicit way to recover* * s^snXofrom Xo. Finally, the composition of the boundary map @* : Zn-1Xo ! ZnXo of the fibra* *tion sequence (2.26)with d0 is trivial, so by (2.27)it induces a map of F-algebras* * from ^ssn-1Xo~=^ssn-1Xo(x2.10) to ssFZnXo which, composed with the map q in (2.27)* *, de- fines a "shift map" s : ^ssn-1Xo! ^ssnXo(see [DKS2 , Prop. 6.2]). 2.28. the simplicial F-algebra. Applying the functor ssF dimensionwise to any * *simplicial object Xo 2 sC yields a simplicial F-algebra Go = ssFXo, which is in particular* * an ^F-graded simplicial group; its homotopy groups form a sequence of ^F-graded groups which* * we denote by (ssnssFXo)1n=0, and each ssnssFXo is a F-algebra. Note that as for any (graded) simplicial group, the homotopy groups of Go may * *be computed using the Moore chains C*Go, defined CnGo := \ni=0Ker{di: Gn ! Gn-1} (cf. x* *2.25 and [May , 17.3]), and we have the following version of [Bl8, Prop. 2.11] 2.29. Lemma. For any fibrant Xo 2 sC, the inclusion : CnXo ,! Xn induces a* *n isomor- phism ? : ss*CnXo ~=Cn(ss*Xo) for each n 0. ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 9 Proof.(a) First, note that any trivial cofibration j : A ,! B in S induces * *a fibration j* : MBXo ! MAXo in C. To see this, by assumption 2.13 it suffices to consider C = D, (since by [Bl* *5, Def. 4.13], f is a fibration in C if and only if Uf is a fibration in D), and in fact the o* *nly nontrivial case is when D = G (where the fibrations are maps which surject onto the identity * *component - see [Q1 , II, 3.8]). Note that in internal simplicial dimension k we have * * (MAXo)intk~= Hom sGp(F A; (Xo)intk) (see x1.2) for A 2 S, where F denotes the (dimensionwis* *e) free group functor. Since F A is fibrant in sGp, F j : F A ,! F B has a left inverse * * r : F B ! F A, so j* : (MBXo)intk! (MAXo)intkhas a right inverse r*, so in particular is on* *to. Since this is true in each simplicial dimension k, j* : MBXo ! MAXo is a fibration in G. * *(Note that di: Xn ! Xn-1 is always a fibration.) (b) In addition, kn= j* : M0nXo ! MknXo is a fibration for all 0 k n, as o* *ne can see by considering (2.5)(since ffin-1 surjects onto the identity component by assum* *ption). (c) Given j 2 Cn(ssffXo), represented by h : M ! Yn, with djh ~ 0 fo* *r 1 j n, note that for 1 k n, MknXo is the pullback of (dk;:::;dk)k ffikn-1 Mk+1nXo -----! Mn-1Xo --- Xn-1; in which (dk; : :;:dk) is a fibration by (a) if k 1, so this is in fact a * *homotopy pullback square (see [Mat , x1]). By descending induction on 1 k n, (starting with ff* *inn= dn), we may assume ffik+1nO h : M ! Mk+1nXo is nullhomotopic in C, as is dkO h, * *so the induced pullback map, which is just ffiknO h : M ! MknXo, is also nullhomotopic b* *y the universal property. We conclude that ffi1nO h ~ 0, and since ffi1n: Xn ! M1nXo is a fibra* *tion by (b), we can replace h by a homotopic map h0: M ! Xn such that ffinh0= 0. Thus * *h0 lifts to ZnYo = Fib(ffin), so ? is surjective. (d) Even though the retraction r : F [n] ! F 0nin (a) is not canonical, it ma* *y be chosen independently of the internal simplicial dimension k to yield a section r* for* * ffi1n= j* : Xn!! * * ffi0n1 M1nXo. The long exact sequence in [M; -] for the fibration sequence CnYo i* *-!Yn -! MnYo (cf. [Q1 , I,x3]) then implies that i# is monic, so ? is, too. The argument * *lifts from D = G to C because the objects M are in the image of the adjoint of U : C ! D, by* *_assumption 2.13. |_* *_| This Lemma, together with (2.27), yields a commuting diagram: (d0)# ssFCn+1Xo____________-ssFZnXo______-_-^ssnXop | | ppp | | pp |~= ^ | ppph ? | ?| pp | | ppp |? dssFXo0 |? |? Cn+1(ssFXo)__________-Zn(ssFXo)____-_-ssnssFXo Figure 1 which defines the dotted morphism of F-algebras h : ^ssnXo! ssn(ssFXo) (this wa* *s called the "Hurewicz map" in [DKS2 , 7.1]). Note that for n = 0 the map ^?is an isomorp* *hism, so h is, too. 2.30. An exact couple. If Xo 2 sC is Reedy fibrant, the long exact sequences * *(2.24)for the fibrations Cn+1Xo ! ZnXo fit into an (N; ^F)-bigraded exact couple (D1*;* *ff; E1*;ff) with D1k;ff~=ssffZkXo and E1k;ff~=ssffCkXo for k 0 and M 2 ^F. As in [DKS2* * , x8] the 10 DAVID BLANC derived couple has D2k;ff~=(^sskXo)ffand E2k;ff~=ssk(ssffXo) (using Lemma 2.29)* *, which fit into a "spiral exact sequence" (2.31) . .!.ssn+1ssFXo @-!^ssn-1Xos-!^ssnXoh-!ssnssFXo ! . .^.ss0Xoh-!ss0ssFX* *o ! 0 as in [DKS2 , 8.1], so by Reedy fibrant replacement (x2.22), one has such an ex* *act sequence for any Yo 2 sC. Of course, ^ss-1Xo:= 0; and at the right hand end we have h : ^ss* *0Xo~=ss0ssFXo, as noted above. We immediately deduce the following 2.32. Proposition.A map f : Xo ! Yo in sC is a weak equivalence in the resoluti* *on model category - i.e., induces an isorphism in ssnssF for all n 0 (x2.16(i)) * *- if and only if it is an F-equivalence - i.e., induces an isomorphism in ^ssnfor all n 0 (s* *ee x2.22). 2.33. Resolutions. By Definition 2.18, a resolution of an object X 2 C is a sim* *plicial object Qo over C which is cofibrant and has a weak equivalence f : Qo ! c(X)o. Note * *that such an f is detemined by an augmentation " : Q0 ! X in C (with d0 O " = d1 O "* *); by Proposition 2.32, f is a weak equivalence if and only if the augmented F^-grade* *d simplicial group "* : ssFQo ! ssFX is acyclic (i.e., has vanishing homotopy groups in al* *l dimensions 0). The long exact sequence (2.31)then implies that (2.34) ^ssnQo~=nssFX for alln 0: 2.35. Definition.A CW complex over a pointed category C is a simplicial object * * Ro 2 sC, together with a sequence of objects Rn (n = 0; 1; : :):- called a CW basis for* * Ro - such that Rn = Rnq LnRo (x2.4), and di|Rn= 0 for 1 i n. The morphism dn0: Rn ! Zn-1Ro * *is called the n-th attaching map for Ro (compare [Bl1, x5]). A CW resolution of a simplicial F-algebra Ao is a CW complex Go 2 sF -Alg, * * with CW basis (Gn )1n=0such that each Gn is a free F-algebra, together with a weak * *equivalence OE : Go ! Ao. 2.36. Definition.In the situation of x2.7, a simplicial object Ro 2 sC is ca* *lled a CW resolution of Xo 2 sC if Ro is a CW complex with each Rn in F, up to homoto* *py (so in particular Ro is indeed cofibrant), equipped with a weak equivalence f : Ro ! X* *o. 2.37. Remark.It is easy to see that one can inductively construct a CW resoluti* *on for every simplicial F-algebra Ao, since in order for OE : Go ! Ao to be a weak equiv* *alence it is necessary and sufficient that ZnOE take ZnGo onto a set of representatives of s* *snAo in ZnAo, and the attaching map dn0 map Gn onto a set of representatives for Ker(ssnOE) * *in Zn-1Go. Thus we can let Gn be the free F-algebra (x2.8) generated by union of the unde* *rlying sets of ZnAo and Ker(Zn-1f), say. The "topological" version of this requires a little more care. In particular, * *[Bl8, Remark 3.16] implies that not every free simplicial F-algebra Ao is realizable in the sense* * that there is a Ro 2 sC with ssFRo ~=Ao. In order to see what can be said on this context, a* *ssume given a fibrant and cofibrant simplicial object Po with an augmentation " : P0 ! X.* * For each ff 2 ^F, consider the long exact sequence (dm0)# @m-1 (jm)# (2.38) : :s:sff+1Cm Po ---! ssff+1Zm-1Po ---!ssffZm Po ---! ssffCm Po: : : for the fibration dm0, where Z0Po := P0. By definition, Po ! X is a resolution* * if and only if ssissFPo = 0 for each i 0, where the homotopy groups are understod in the augm* *ented sense ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 11 - that is, ss0ssFPo := Ker((d00)# : C0ssFPo ! Z-1ssFPo)= Im((d10)# : C1ssFPo ! * *Z0ssFPo). The key technical fact we shall need in this context is contained in the following 2.39. Lemma. An fibrant and cofibrant Po 2 sC with an augmentation Po ! X * *is a resolution of X if and only if for each m > 0: (jm)# (a) There is a short exact sequence 0 ! Im(@m-1) ,! ssFZm Po ---! Zm ssFPo ! 0* *, and (b) @m |Im(@m-1)is one-to-one, and surjects onto Im(@m ), and Im@0 ~=ssFX. Note that since @m shifts degrees by one, (a) and (b) together imply that I* *m(@m ) ~= m+1ssFX for each m. Proof.For any Po, the inclusion jm : Zm Po ! Cm Po induces a map of F-algebr* *as (jm )# : ssFZm Po ! ssFCm Po ~=Cm ssFPo (see Lemma 2.29), which factors through Zm ssFP* *o. Denote the boundary map for the chain complex C*ssFPo (which computes ss*ssFPo) by* * Dm := (jm-1)# O (dm0)#. If Po ! X is a resolution, we must have Im((jm )# O (dm+10)#) = Im(Dm+1) = * *Ker(Dm ) for each m 0, so in particular (jm )# maps onto Zm-1ssFPo. Moreover, since s* *sFC1Po ! ssFP0!!ssFX ! 0 is exact, Im@0 ~=ssFX and so if we assume by induction that * *(b) holds for m - 1, we see that Ker(jm )# = Im@m-1 is isomorphic to m ssFX, which * *proves (a). Moreover, if 0 6= fl 2 Ker@m = Im(dm+10)#, and fl 2 Im @m-1 = Ker(jm )#, th* *en we have fi 2 ssFCm+1Po with (dm+10)#(fi) = fl 6= 0 but Dm+1(fi) = 0 - contradicting (a)* * for m + 1. Finally, if (jm )#(fl) 6= 0, there is a fi 2 ssFCm+1Po with Dm (fi) = (jm )#(fl* *), by the acyclicity of ssFPo, so fl - (dm+10)#(fi) 2 Ker(jm )# = Im @m-1, and @m (fl - (dm+10)* *#(fi)) = @m (fl), which proves (b) for m. The identification of Im @0 is immediate from (2.38). Conversely, if (a) and (b) are satisfied for all m, for any element in i 2 Zm* * ssFPo, we have i = (jm )#(fl) for some fl 2 ssFZm Po. Thus there is a 2 ssFZm-1Po with @* *m (@m-1()) = @m (fl), by (b), so fl . @m-1()-1 is in Ker@m = Im(dm+10)#; thus (jm )#(fl* * ._@m-1()-1)_= i bounds, and ssFPo is acyclic. * * |__| It should be pointed out that the fundamental short exact sequence (jm)# (2.40) 0 ! m ssFX ~=Im(@m-1) ,! ssFZm Po ---!! Zm ssFPo ! 0 for a resolution Po is actually split, as a sequence of graded groups, because * *(jm )#|Im(dm+10)#= (jm )#|Ker@m is one-to-one, by (b), and surjects onto Zm ssFPo by the acyclic* *ity. However, Im(dm+10)# = Ker@m need not be a sub-F-algebra of ssFZm Po, since @m is not a* * morphism of F-algebras. With the aid of Lemma 2.39 we can now show: 2.41. Proposition.Under the assumptions of x2.7 and 2.13, any X 2 C has a CW re* *solution Ro 2 sC. Proof.Let Qo 2 sC be the functorial resolution of x2.19; we may assume that the* * augmentation "Q : Q0 ! X is a fibration. We start off`by`choosing a set T*0 ssFQ0 of F-algebra generators (x2.8), such* * that if we let R00:= ff2F^fi2T0ffMfi, then "Q# maps the free F-algebra ssFR00 ss* *FQ0 onto ssFX. We may assume T*0 is minimal, in the sense that no sub-graded set gener* *ates a free F-algebra surjecting onto ssFX - so that "Q#(fi) 6= 0 for all fi 2 T*0. The inclusion OE : ssFR00,! ssFQ0 defines a map f00: R00! Q0 with (f00)# = OE,* * and we let 0 Q 0 R0 0 i "R * * Q "R := " O f0; factoring " by 2.1(3) as R0 -!R0 -! X and usng the LLP for* * i and " yields f0 : R0 ! Q0 commuting with ". 12 DAVID BLANC Now assume by induction that we have constructed a fibrant and cofibrant Ro * *through simplicial dimension n - 1 0, together with a map trn-1f : trn-1Ro ! trn-1Q* *o which induces an embedding of F-algebras (trn-1f)#. We assume that Ro satisfies (a* *) and (b) of Lemma 2.39 for 0 < m < n (and of course Qo satisfies them for all m > 0)* *. If we map the short exact sequence (a) for Ro to the corresponding sequence for Qo by * *f*, we see that Zn-1(f#) = Zn-1OE : Zn-1ssFRo ! Zn-1ssFQo is one-to-one, so (Zn-1f)# : s* *sFZn-1Ro ! ssFZn-1Qo is, too. Any non-zero element in Zn-1ssffRo is represented by fl 2 ssFZn-1Ro, by (2.40)* *for Rn-1. Let g : M ! Zn-1Qo represent f#fl 2 ssffZn-1Qo, with M(g) the corr* *esponding coproduct summand of Qn = LQn-1 in (2.20), with i(g): M(g)! Qn the incl* *usion. Then diO i(g)= i(di-1g)for 1 i n (in the same notation) and d0O i(g)= g, * * by x2.19. Thus the F-algebra generator 2 ssffQn is in CnssFQo, and (dn0)# = * *f#fl. Thus if we choose a set T*n of F-algebra generators for Zn-1ssFRo and set a a (2.42) Rn := M(fi); ff2F^fi2Tnff we have maps fn : Rn ! CnQo and d0 : Rn ! Zn-1Ro such that (jn-1)# O (dQ0)# O (* *fn)# = (jn-1)# O (Zn-1f)# O (d0)#. Now (2.40)implies that (jn-1)# is one-to-one on * * Imd0, so (dQ0)# O (fn)# = (Zn-1f)# O (d0)#. Because (dQ0)# is a fibration and ssFRn * * is free, this implies that one can choose fn so that dQ0O fn= Zn-1f O d0. Since Lnf : LnR* *o ! LnQo exists by the induction hypothesis, one can define fn : Rn ' LnRo q Rn ! Qn * *extending trn-1f to trnf : trnRo ! trnQo, with ffiRn: Rn ! MnRo a fibration. Since * *ssissFPo_= 0 then holds for i n - 1, (2.42)and (2.40)hold for m = n. * * |__| 2.43. Remark.We have actually proved a little more: given any minimal simplicia* *l CW res- olution of F-algebra's Ao ! ssFX (x2.35) of a realizable F-algebra, one can * *find a CW resolution Ro ! X realizing it: that is, ssFRo ~=Ao. (Minimality here is under* *stood to mean that we allow no unnecessary F-algebra generators in each An , beyond those nee* *ded to map onto Zn-1Ao.) By a more careful analysis, as in [Bl8, Thm. 3.19], one could in fact show tha* *t any CW resolution of ssFX is realizable. However, this will follow from Corollary 4.1* *1 below. 3.Postnikov systems and the fundamental group action We now describe Postnikov systems for simplicial objects in the resolution mod* *el category, and the fundamental group action on them. 3.1. Definition.If C is a category satisfying the assumptions of x2.7, a Postni* *kov system for an object Yo 2 sC is a sequence of objects PnXo 2 sC, together with maps 'n : * *Xo ! PnXo and pn : Pn+1Xo ! PnXo (for n 0), such that ^sskpnand ^ssk'nare isomorphi* *sms for all k n, and s^skPnXo= 0 for k n + 1 3.2. Remark.In general, such Postnikov towers may be constructed for fibrant X* *o using a variant of the standard construction for simplicial sets (cf. [May , x8]) due t* *o Dwyer and Kan in [DK2 , x1.2], and for arbitrary Xo by using a fibrant approximation. Note that if Qo 2 sC is a resolution of some X 2 C (see x2.33), then by * *(2.34) ^ssiPnQo~=issFX for n i 0, and s^siPnQo= 0 for i > n; so (2.31)implies that 8 >: 0 otherwise. ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 13 3.4. Postnikov towers for resolutions. It is actually easier to construct a cof* *ibrant version of the Postnikov tower for a resolution than it is to construct the resolution i* *tself: Given a CW resolution Qo of an object X 2 C, (constructed as in Proposition 2.41), w* *ith CW basis (Qk )1k=0, we construct a CW cofibrant approximation Yo !`Q(n)oas`follows. Let J* := ssFX, and choose some G 2 hoF (i.e., G ' ff2F^ TffM) h* *aving a surjection of F-algebras OE : ssFG!!n+1J*. Set Yn+2:= Qn+2 q G, with (d0* *|G)# = OE, mapping onto n+1J* ~= Im(@n) ,! ssFZn+1Qo = ssFZn+1Yo (see (2.40)). This * *defines ffin+2 * * n+2 Yn0+2:= Yn+1q Ln+2Yo --! Mn+2Yo, which we then change into a fibration. Sinc* *e (d0 )# : ssFCn+2Yo!!ssFZn+1Yo is surjective, we may assume by induction on k n + 2 th* *at ~= (3.5) (jk)* : ssFZkYo -!ZkssFYo and @k-1= 0; and thus we may choose Yk+12 hoF with d0 : ssFYk+1!!ssFZkssFYo, and see that* * (3.5)holds for k + 1 by (2.40). Note that Yo ' Q(n)ois constructed by "attaching cells" to Qo, as in the tra* *ditional method for "killing homotopy groups" (cf. [Gr, x17]), so we have a natural embedding* * ae : Qo ,! Yo, rather than a fibration. In fact, it is helpful to think of PnXo as a homotop* *y-invariant version of the (n + 1)-skeleton of Xo: starting with trn+1Xo, one completes it t* *o a full simplicial object by a functorial construction which (unlike the skeleton) depends only * *on the homotopy type of Xo. 3.6. -algebras and the fundamental group. Under our assumptions, the category* * C = F -Algis a CUGA, or category of universal graded algebras (see [BS , x2.1] a* *nd [Mc1 , V,x6]), so that sC, the category of simplicial F-algebras, has a model category struc* *ture defined by Quillen (see [Q1 , II, x4]). Equivalently, one could take the resolution mod* *el category on sC, starting with the trivial model category structure on F -Alg, and letting FF* * -Algbe the sub- category of all free F-algebras - as in x2.17. One thus has a concept of "sp* *heres" in sF -Alg - namely, ssFnM, for ff 2 ^F(cf. x2.22) - and (ssnAo)ff~=[nM; Ao]sF -* *Algfor any simplicial F-algebra Ao. Thus if we take homotopy classes of maps between (* *coproducts of) these spheres as the primary homotopy operations (see [W , XI, x1]), we c* *an endow the homotopy groups ss*Ao = (ssiAo)1i=0of Ao with an additional structure: that o* *f a (F -Alg)- -algebra, in the (somewhat unfortunate, in this case) terminology of [BS , x3* *.2]. By definition, this structure is a homotopy invariant of Ao. In our situation, however, because we are dealing with Postnikov sections, b* *y (3.3)we only need the very simplest part of that structure - namely, the action of the f* *undamental group ss0Ao on each of the higher homotopy groups ssnAo. Observe that because C has an underlying group structure, by assumption 2.13* *, the indexing of the homotopy groups of an object in sC should be shifted by one compared * *with the usual indexing in T*, so that ss0Ao is indeed the fundamental group, and in fac* *t the action we refer to is a straightforward generalization of the usual action of the funda* *mental group of a simplicial group (or topological space) on the higher homotopy groups. 3.7. J*-modules and J*-algebras. We shall be interested in an algebraic desc* *ription of this action: that is, we would like a category of universal algebras which model t* *his action, in the same sense that -algebras model the action of all the primary homotopy operat* *ions on the homotopy groups of a space. Just as in the case of ordinary -algebras, the ac* *tion in question is determined by the homotopy classes of maps of simplicial F-algebras. Thus we are led to consider two distinct "varieties of algebras", in the ter* *minology of [Mc1 , V, x6]): one modeled on the homotopy classes of maps, and one on the actual m* *aps. 14 DAVID BLANC 3.8. Definition.Given a F-algebra J*, let J*-Mod denote the category of uni* *versal algebras whose operations are in one-to-one correspondence with homotopy classe* *s of maps ssFnM ! ssF(nM q 0M), and whose universal relations correspond * *to the relations holding among these homotopy class in ho(sC). These model ssnAo, wit* *h the action of ss0Ao, for Ao 2 sF -Alg. An object K* 2 J*-Mod is itself a F-algebra, equipped with an action of an o* *peration : Jff00x Kff0! Kfffor each 2 [ssFnM; ssF(nM q 0M)]. Such a K** * will be called a J*-module, even though in general the category of such objects, which * *we shall denote by J*-Mod, need not be abelian (and it could depend on n). However, in the c* *ases that interest us, J*-Mod will be abelian, and will not depend on n > 0. 3.9. Definition.Given a F-algebra J*, let J*-Alg denote the category of univ* *ersal al- gebras whose operations are in one-to-one correspondence with actual maps ssFn* *M ! ssF(nM q 0M) as above, and whose universal relations correspond to t* *he relations holding among these maps in sC. The objects in J*-Alg, which are again F-algeb* *ras with additional structure, will be called J*-algebras. The category J*-Alg is generally very complicated; it is not abelian, and we c* *annot expect to know much about it, even for C = G, say. In particular, one may well have a dif* *ferent category for each n > 0 (although we surpress the dependence on n to avoid excessive not* *ation). Note, however, that maps ` : ssFnM ! Ao, for any simplicial F-algebra Ao, cor* *respond to elements in ZnAo, so that the A0-algebra structure on An restricts to a* *n action of of Z0Ao = A0 on ZnAo. 3.10. Remark.Let Qo be a resolution (in sC) of some object X 2 C, with J* := ss* *FX, and Yo ' PnQo its n-th Postnikov approximation. Then we have an action of ss0ssFYo* * ~=J* on ssn+2ssFYo ~=n+1J* which is a homotopy invariant of Yo, and thus in turn of * *Qo, so of X. It is not clear on the face of it whether the J*-module nJ* depends only on J*,* * though we shall see (in x4.5 below) this holds for n = 1, and hope to show in [BG ] that * *in fact this holds for all n. In any case it is describable purely in terms of the primary F-alge* *bra-structure of J*. In general, for any simplicial object Xo 2 sC, there is an action of s^s0Xo~=s* *s0ssFXo on the higher F-algebras ^ssnXo, defined similarly via homotopy classes of maps [SnF; * *S0FqSnF]sC (see x2.22); but there is no reason why this should define the same category of "^ss* *0Xo-modules" as that defined above. Thus we do not know (2.31)to be a long exact sequence of ^s* *s0Xo-modules. However, in our case, when Xo = Qo is a resolution, the isomorphism of (abelian* *) F-algebras ssn+2ssFYo ~=n+1J* is defined inductively by means of the connecting homomorphi* *sm of (2.31), and this yields the J*-module structure on nJ*. 3.11. Assumption.Under mild assumptions on the category C one may show that for* * any Ao 2 sF -Alg and n 1, the F-algebra ssnAo is abelian (see [BS , Lemma 5.2.1]). However, we shall need to assume more than this: namely, that J*-Mod as defin* *ed above is in fact an abelian category. We also assume that when Ao is a simplicial F-* *algebra, the action of ss0Ao on each ssnAo is induced by an action of A0 on An, and if * * Ao = ssFQo, then this in turn is induced by an action of Q0 on Qn. Moreover, ZnAo and * *CnAo are sub-A0-algebras of An, and d0 is a homomorphism of A0-algebras. 3.12. Proposition.These assumptions are satisfied for the categories listed in * *x2.14. Proof.As we shall see, all the categories in question are essentially special c* *ases of the first: (I) When C = G, the fundamental group action has an explicit description as fol* *lows: We define the generalised Samelson product of two elements x 2 Xp;ky 2 Xq* *;`(where, as in x1.2, p is the "external" dimension, k the "internal" dimension in a * *a bisimplicial ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 15 group Xo;o2 sG) to be the element <> 2 Xp+q;k+` 0 "(oe* *) 1 Y Y (3.13) <> := @ (sextaeq:s:e:xtae1sint:`:s:intx1; sextoeq:s:e:* *xtoe1sint'`:s:i:nt'1y)"(')A: (oe;ae)2Sp;q('; )2Sk;` Here Sp;qis the set of all (p; q)-shuffles - that is, partitions of * *{0; 1; : :;:p + q - 1} into disjoint sets oe1 < oe2 < . .<.oep, ae1 < ae2 < . .<.aeq - and "(oe) i* *s the sign of the permutation corresponding to (oe; ae) (see [Mc2 , VIII, x8]); Sp;qis ordere* *d by the reverse lexicographical ordering in oe. (a; b) denotes the commutator a . b . a-* *1 . b-1 (where . is the group operation). When p = q = 0, <> is just the usual Same* *lson product in X0;o2 G (cf. [C , x11.11]). We are mainly interested here in the case p = 0, so <> := <^x;>y* * for ^x:= sq-1. .s.0x 2 Xq;k. It is sometimes convenient to think of this as an "act* *ion" of x on y, setting tx(y) := <> . y (cf. [W , X, (7.4)]). The simplicial identities imply that if dintix = dintiy = 0 for all i, * * the same holds for , and if x = dint0z for some z 2 Cintk+1Xp;o, then = dint0, so that << ;>> induces a well-defined operation << ;>> : ssintkXp;oxssint`Xq;o! ssintk+`Xp* *+q;o, which is defined for any simplicial -algebra Ao*, with ff 2 Ap* and fi 2 Aq*, by: Y (3.14) <fi> := :s:oe1fi"(oe)2 Ap+q*: (oe;ae)2Sp;q Again when p = 0 we write off(fi) := <fi> . fi, so that off: Aq* **! Aq* is a group homomorphism in each degree (if ff 2 ZpAo*, fi 2 ZqAo*, then <fi> 2 Zp* *+qAo*). Now let Xo;o:= 0Sk q nS`, (where Sk is the k-sphere for G - x2.17) * *and let 0;k and n;`be -algebra-generators for sskX0;oand ss`S` ss*Xn;o, respect* *ively, so ss*Xn;ois generated by {^0;k; n;`}. Since djn;`= 0 (0 j n), we have a sh* *ort exact sequence of -algebras (3.15) 0 ! Znss*Xo;o! ss*(Skq S`) ! ss*Sk ! 0: When k; ` > 0, by [H , Theorem A] any element x 2 ss*Xn;o~=ss*(Sk q S`* *) can be written as a sum of elements of the form i# !(^0;k; n;`) (where !(x; y) = <* *: :<:x;>y;>:i:s: some iterated Samelson product), so x can be obtained by means of the "inte* *rnal" -Alg operations from expressions of the form off(n;`) (for ff 2 ss*X0;o). By passing to universal covers we have a similar description when ` > k * *= 0, since then any x 2 ssjXn;o (j 1) can be written as a sum of elements of the fo* *rm i# !(off1(n;`); : :;:offr(n;`) (for ffi2 ss*X0;o), and any other ff 2 ss*X0* *;oacts on this by permuting the generators offi(n;`, so again tauff(-) is a group homomor* *phism. When k = ` = 0, we are reduced to the case C = Gp (see (II) below). When k > 0 and ` = 0, let us write 'ff(fi) := <ff> for ff 2 s* *s*X0;o and fi 2 ss0Xn;o, so that we are thinking of the usual (internal) action of th* *e fundamental group ss0Xn;oas a function of fi. This is not a homomorphism, since we hav* *e 'ff(fi .fl) = 'ff(fi) + 'ff(fl) + <^ff>>> by [W , III, (1.7) & X, (7.4)]. But <fi> is a cycle (i.e., in Znss*Xo;o), by (3.15), so <<<fi>;* *>fl> ~ 0 in ssnss*Xo;o for any fl 2 ss0Xn;oby [BS , 5.2.1], which means that 'ffinduces a homom* *orphism on ssnss*Xo;o. In summary, an J*-algebra (x3.9), for any J* 2 -Alg, is just a -algebra * *K* together with an action of each ff 2 J*, which may be expressed in terms of the (deg* *ree-shifting) 16 DAVID BLANC homomorphisms off, or the functions 'ff, respectively, satisfying whatever* * relations hold among these (and the internal -algebra operations) in ss*Xn;o. A J*-module, on the other hand, is an abelian -algebra K*, together w* *ith homo- morphisms off: K* ! K* or 'ff: K* ! K* for each ff 2 J*, satisfying th* *e identities occuring in ssnss*Xo;o. These identities could be described more or less explicitly in the categ* *ory -Alg, in terms of suitable Hopf invariants (cf. [Ba1, II, x3]). Compare [Ba2, x3]). (II) When C = Gp, sC models the homotopy theory of (connected) topological s* *paces, and J*-Mod, defined (as noted above) through the usual action of the fun* *damental group, is equivalent to the category of (left) modules over the group ring* * Z[ss0Ao] (for Ao 2 sC G). (III)When C = Lie, the situation is similar to C = Gp, with Samelson products r* *eplaced by Lie brackets. (IV) When C = dL sLie, one has a generalized Lie bracket defined for bisimpl* *icial Lie algebras as in (3.13), with commutators replaced by Lie brackets (see [Bl7* *, x2.6]). (V) When C = R-Mod , sC is equivalent to the category of chain complexes over* * R, so there is no action of ss0 = H0 on the higher groups. __ |__| 3.16. Remark.It is possible to write down general conditions on category of un* *iversal alge- bras (or CUGA) C, defined in terms of operations and relations, which suffice * *to ensure that assumptions 3.11 hold: all one really needs is a suitable Hilton-Milnor theore* *m in sC (see, e.g., [Go ]). However, it seems simpler to state the conditions needed as abov* *e, and verify them directly in any particular case of interest. 4. Cohomology of F-algebras In this section we complete the description of the algebraic invariants used * *to distinguish homotopy types. To do so, we recall Quillen's definition of cohomology in a mo* *del category, in the context of F -Alg: 4.1. Definition.Let C be a model category with an abelianization functor Ab :* * C ! Cab, where Cab denotes of course the full category of abelian objects in C; we sh* *all usually write Xab for Ab(X) (see x2.10). In [Q1 , II, x5] (or [Q4 , x2]), Quillen defines* * the homology of an object X 2 C to be the total left derived functor LAb of Ab, applied to * *X (cf. [Q1 , I, x4]). Likewise, given an object M 2 Cab=X, the cohomology of X with coeffici* *ents in M is R HomCab=X(X; M) := Hom Cab=X(LAb X; M). 4.2. Quillen cohomology of F-algebras. When J* 2 C = F -Alg, we have the mo* *del category structure defined in x3.6 above, so we can choose a resolution Ao ! J* ** in sF -Alg as in x2.33, and define the i-th homology group of J* to be the i-th homotopy * *group ssi(Ab Ao) of the ^F-graded simplicial abelian group (Ao)ab - i.e., of the associated c* *hain complex (cf. [D , x1]). One must verify, of course, that this definition is independent of* * the choice of the resolution Ao ! J*. Similarly, if K* is an abelian J*-algebra, then the i-th cohomology group * *of J* with coefficients in K*, written Hi(J*; K*), is that of the cochain complex cor* *responding to the cosimplicial ^F-graded abelian group Hom J*-Alg(Ao; K*). 4.3. Remark.Here Hom J*-Alg(A; B) is the group of F-algebra homomorphisms whic* *h respect the J*-action; because we are mapping into an abelian object K*, Hom J*-Alg* *(Ao; K*) ~= Hom J*-Alg((Ao)0ab; K*) (where A0abdenotes the abelianization of A 2 J*-Algas * *an J*-algebra). ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 17 However, in the simplicial abelian J*-algebra (Ao)0abwe have a direct product * *decomposition (Ak)0ab= (A^0k)ab (LkAo)0abfor k 0, where (A^k)0ab:= Ck(Ao)0abis the the su* *b-abelian J*- algebra of (Ak)0abgenerated by (Ak)0ab(cf. [May , Cor. 22.2]) - and in fact (d^* *0)0ab: (A^n)0ab! (An-1)0abfactors through a map @^n: (A^0n)ab! (A^n-1)0ab(see [May , p. 95(i)]). Thus the n-cochains split as: Hom J*-Alg((An)0ab; K*) ~=Hom J*-Alg((A^n)0ab; K*) Hom J*-Alg(Ln(Ao)0a* *b; K*); so by [BK , X, x7.1] any cocycle representing a cohomology class in Hn(J*; K*)* * may be represented uniquely either by a map of abelian A0-algebras ^f: (A^n)0ab! K*,* * or by a map of A0-algebras f : An ! K*. Since CnAo contains the sub-A0-algebra of An generated by An (by assumptio* *n 3.11), f determines its restriction f|CnAo: CnAo ! K*, which determines ^f, which * *determines f in turn. We have thus shown that H*(J*; K*) may be calculated as the cohom* *ology of the (abelian) cochain complex HomA0-Alg(C*Ao; K*) (even though C*Ao is not i* *n general a homotopy invariant of Ao, in non-abelian categories). 4.4. obstructions to existence of resolutions. Given an object X 2 C, and a (* *suitable) simplicial resolution Ao ! J* of the F-algebra J* := ssFX, we have seen in Se* *ction 2 that one can construct a resolution Qo of X (in the resolution model category sC) re* *alizing Ao, in the sense that ssFQo ~=Ao. It is thus natural to ask whether any simplicia* *l F-algebra - or at least, any resolution Ao of an abstract F-algebra J* - is realizable in s* *C. One approach to this question in the topological setting (i.e., for C = G), in* * terms of higher homotopy operations, was given in [Bl3]. However, a glance at the proof of Prop* *osition 2.41 shows that one can instead consider obstructions to extending trnQo to the nex* *t simplicial dimension. For a homotopy-invariant description, we state this in terms of succ* *essive Postnikov approximations to Qo, since it is clear that, once we have constructed trnQo* *, it is always possible to obtain Yo ' Q(n-1)ofrom it by successive choices of free objects * *Yk+12 hoF (k = n; : :):mapping to ZkYo by a F-algebra surjection. 4.5. constructing the obstruction. Assume given a CW resolution Ao 2 sF -Alg * *of J*, with CW basis (An )1n=0, and choose corresponding free objects Qn 2 F C with s* *sFQn ~=An . We begin the induction with tr1Qo, and thus Q(0)o, constructed as in the proof * *of Proposition 2.41. Note that to obtain tr1Qo we do not in fact need to know X 2 C with s* *sFX ~=J* - or even to know that such an object exists! This implies that the J*-module str* *ucture on J* is uniquely determined. In the inductive stage we assume given trnQo (equivalently: Q(n-1)o), satis* *fying 2.39(a) and (b) for 0 < m n. Our strategy is to try to attach (n + 1)-dimensional "* *cells" to trnQo in such a way as to guarantee acyclicity of the resulting trn+1Qo in one * *more simplicial dimension - using Lemma 2.39 above. The key to the construction of trn+1Qo from* * trnQo thus lies in the extension of A0-algebras (2.40)(for Qo, rather than Po), * * in which the two ends are given to us. Observe that this extension determines the A0-algebra* * structure on nJ*, if more than one is possible. We want this extension to be "trivial" (that is, split as a semi-direct produc* *t of A0-algebras), in order to be able to lift the given map of A0-algebras dA0: An+1 ! ZnAo to* * a map dQ0: Qn+1 ! ZnQo, so the question is reduced from one about simplicial objects* * over C to one of algebraic objects, namely: A0-algebras. There is a close analagy to the* * classical theory of group extensions, where the triviality of an extension E : 0 ! A ! B ! G * *is measured by the characteristic class (E) 2 H2(G; A) (compare [Mc2 , IV, x6]). Similar* *ly, in our case the triviality of the extension is measured by the vanishing of a suitable coho* *mology class in Hn+2(J*; nJ*), defined as follows: 18 DAVID BLANC Because (jn)# : ssFZnQo!!ZnssFQo ~=ZnAo is surjective, and An+1 is a free F-* *algebra, we can choose a lifting in the following diagram: dAn+1 An+1 _________-0ZnAo___-0 ppp ppp |~ ppp ||= pp Q |? i |? (jn)# 0____-nJ*____-ssFZnQo_____-ZnssFQo____-0 Figure 2 and we can find a map ` : Qn+1! ZnQo realizing (again, because An+1 = ssFQn+1 * * is free). Combined with the "tautological map" Ln+1Qo ! Mn+1Qo (see x2.4), which depends * *only on trnQo, by setting Qn+1:= Qn+1qLn+1Qo we obtain an extension d0 : Qn+1! Qn of ` * *(which is a map of Q0-algebras), and thus an (n+1)-truncated simplicial object trn+1Qo* * over C, with Qn+1:= Qn+1q Ln+1Qo, and ssF trn+1Qo ~=trn+1Ao. In particular, dQn+10: Cn+1Qo * *! ZnQo induces a map ^ from ssFCn+1Qo = Cn+1Ao to ssFZnQo extending (and determined b* *y) the lifting : An+1! ssFZnQo of dAn+10. This is a map of A0 = ssFQ0-algebras, by A* *ssumption 3.11. Since (jQn)# O (^|Zn+1Ao) = 0, the map ^|Zn+1Aofactors through : Zn+1Ao ! Ker* *(jQn)# = nJ*, and composing with dAn+20: Cn+2Ao ! Zn+1Ao defines : Cn+2Ao ! nJ* - agai* *n, a map of A0-algebras: dAn+20 j dAn+10 Cn+2Ao________-Zn+1Ao_____-Cn+1Ao______-ZnAo___-0 | | | @ | | | @ | | | @ | | | | |^ |~= @ | | | @ | | | @ | | | | | | @@R|? i |? (jQn)# |? 0______-nJ* ____-ssFZnQo______-ZnssFQo__-0 The cochain = O dAn+20is clearly a cocycle in the cochain complex Hom J*-Mod* *(Ao; J*), so it represents a cohomology class On 2 Hn+2(J*; nJ*), called the characterist* *ic class of the extension. 4.6. Lemma. The cohomology class On is independent of the choice of lifting . Proof.Assume that we want to replace in x4.5 by a different lifting 0: An+1 !* * ssFZnQo, and choose maps `; `0: Qn+1! ZnQo realizing , 0 respectively; their extensions * *to maps Qn+1! Qn (which we may denote by d0, d00) agree on Ln+1Qo. We correspondingly h* *aving 0: Zn+1Ao ! nJ* and 0:= 0O dAn+20. ` Because Qn+1 := Qn+1 q Ln+1Qo is a coproduct of the form iM, by x2* *.13 the underlying group structure on any X 2 C induces a group structure on (4.7) Hom C(Qn+1; X) (and similarly for Hom F -Alg(An+1; ssFX)). Therefore, we can set h := (d0)-1. (d00) : Qn+1! Qn, and h induces a map j * *: Cn+1Ao ! ssFZnQo such that j|An+1= -1. 0. Moreover, because d0 and d00agree outside * *of Qn+1, (jQn)# O j = 0. Thus j factors through i : Cn+1Ao ! nJ*, which is a map of* * A0-algebras ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 19 because nJ* is an abelian A0-algebra (actually, a J*-module), and i is induc* *ed by group operations from the A0-algebra maps d0 and d00. Moreover, i|Zn+1Ao= - 0 in the abelian group structure on Hom J*-Mod(-; nJ*) * *(which * * __ corresponds to the group structure of (4.7)). Thus 0- = ^jO dAn+20is a cobound* *ary. |__| 4.8. Theorem. On = 0 if and only if one can extend Q(n-1)oto an n-th Postnik* *ov approxi- mation Q(n)oof a resolution of X. Proof.First assume that there exists Yo ' Q(n+1)owith trnYo ~=trnQo: by Lemm* *a 2.39 we know (jQn)#|Im(dn+10)#is one-to-one (and onto ZnssFQo), for dn+10: Cn+1Yo ! ZnY* *o = ZnQo, and thus Im (dn+10)# \ Im@Qn-1= {0}. But then we can choose : An+1! ssFZnQo t* *o factor through Im(dn+10)#, (and this will induce a map of A0-algebras because of x3* *.10), so that = 0 and thus = 0. Conversely, if On = 0, we can represent it by a coboundary = # O dAn+20for* * some A0- algebra map # : Cn+1Ao ! nJ*, and thus get i O #|An+1: An+1! ssFZnQo. If we* * set 0:= . (i O #|An+1)-1, we have Im 0\ nJ* = {0}. We can therefore choose dQn+10: Qn* *+1! ZnQo realizing 0, and then (dQn+10)# avoids Im (@Qn-1) ~=nJ*, so that trn+1Qo so co* *nstructed yields Q(n+1)o, as required. In particular, this determines a choice of J*-mod* *ule structure_on n+2J* (if more than one is possible), via (2.40)for n + 1. * * |__| 4.9. notation. If we wish to emphasize the dependence on the choice of , we sh* *all write Q(n+1)o[] for the extension of Q(n)oso constructed. 4.10. Proposition.The class On depends only on the homotopy type of Q(n-1)oin* * sC. Proof.Assume Q(n-1)ohas been constructed, realizing a simplicial resolution of* * F-algebras Ao ! J* through simplicial dimension n, and let Bo ! J* be any other F-alge* *bra resolution: we then have a weak equivalence ' : Bo ! Ao in sF -Alg. Assume by* * induction on 0 m < n that we have constructed an m-truncated simplicial object trmRo ove* *r C, and a map f : trmRo ! trmQ(n-1)orealizing trm'. Moreover, assume that we have a ma* *p of the (split) short exact sequences (2.40)(in dimension m) for Ro and Qo: i (jmR)# 0____-m J*___-ssFZm Ro________-Zm ssFRo__-0 |= |(Z f) |Z (f ) = Z ' | | n # | n # n | | Q | |? i |? (jm )# |? 0____-m J*___-ssFZm Qo________-Zm ssFQo__-0 Now, in order to extend f to dimension n + 1, we must choose the map (dRm+1* *0)# : ssFRm+1 ! ssFZm Ro (lifting dBm+10: Bm+1 ! Zm Bo) in such a way that (Zm f)# O(* *dRm+10)# = (dRm+10)# O Zm '. Since Bm+1 = ssFRm+1 is free, it suffices to show that the * *obvious map from (jQm)# Zm' ssFZm Ro to the pullback of ssFZm Qo ---! Zm ssFQo = Zm Ao --- Zm Bo is a sur* *jection: given (a; b) 2 ssFZm QoxZm Bo with (jQm)#(a) = '(b), for any z 2 ssFZm Ro with * *(jRm)#(z) = b we have an ! 2 m+1J* ssFZm Ro such that (Zm f)#(z . !) = (Zm f)#(z0) . ! = b* * in the diagram above (where . is the group operation), so z . ! maps to (a; b). * *Thus we can choose dRm+10: Rm+1 ! Zm Ro in such a way that we can define trm+1Ro, togeth* *er with a map trm+1f : trm+1Ro ! trm+1Qo realizing trm+1'. Because ' was a weak equivalence of resolutions, it is actually a homotopy equ* *ivalence, with homotopy inverse : Ao ! Bo, say, and the above argument also yields a homot* *opy inverse 20 DAVID BLANC for f(m) (or trm+1f). Moreover, the characteristic classes we defined are cl* *early functorial with respect to maps in sC; since the characteristic class Om+1 2 Hm+3(J*; m+1J* **), defined for the resolution Ao ! J* by means of the lift dQm+10, must vanish, by Theo* *rem 4.8, the same holds for Ro, so by Theorem 4.8 again we can extend R(m)oto R(m+1)o, * *and_continue the induction as long as m < n. * * |__| We deduce the following generalization of Proposition 2.41: 4.11. Corollary.Given X 2 C, any CW F-algebra resolution Ao ! ssFX is reali* *zable as a resolution Qo ! X in sC. One could further extend Proposition 4.10 to obtain a statement about the natu* *rality of the characteristic classes with respect to morphisms of F-algebras : J* ! L*. H* *owever, such a statement would be somewhat convoluted, in our setting, and it seems better t* *o defer it to a more general discussion of the realization of simplicial F-algebras, in [BG ]. 4.12. realization of -algebras. If G : S* ! G denotes Kan's simplicial loop f* *unctor (cf. [May , Def. 26.3]), with adjoint W : G ! S* the Eilenberg-Mac Lane classifying* * space functor (cf. [May , x21]), and S : T* ! S* is the singular set functor, with adjoint * * k - k : S* ! T* the geometric realization functor (see [May , x1,14]), then functors S G (4.13) T* k-k S* G W induce isomorphisms of the corresponding homotopy categories (see [Q1 , I, x5])* *, so any homo- topy-theoretic question about topological spaces may be translated to one in G.* * In particular, in order to find a topological space X having a specified homotopy -algebra J** * ~=ss*X, it suffices to find the corresponding simplicial group X 2 G (with the F-algebra* * J* suitably re-indexed). If J* is realizable by such an X, any free simplicial resolution Q* *o ! X evidently provides a -algebra resolution ss*Qo of J* = ss*X. But the converse is also tr* *ue: if Qo 2 sG realizes some (abstract) -algebra resolution Ao 2 s -Alg of J*, then the co* *llapse of the Quillen spectral sequence of [Q2 ], with (4.14) E2s;t= sss(sstQo) ) sss+tdiagQo converging to the diagonal diagQo 2 G (defined (diagQo)k = (Qk)intk) implie* *s that ss*diagQo ~=J*. Thus J* is realizable by a simplicial group (or topological sp* *ace) if and only if some -algebra resolution Ao ! J* is realizable. The characteristic classes (On)1n=0(whose existence was promised in [DKS2 , x1* *.3] under the name of the "k-invariants for J*), thus provide a more succinct (if less explic* *it) version of the theory described in [Bl3, x5-6] (as simplified in [Bl6, x6]), for determining t* *he realizablity of a -algebra in terms of higher homotopy operations - which we summarize in 4.15. Theorem. Given an (abstract) -algebra J*, the following conditions are * *equivalent: (1) J* is realizable as ss*X for some topological space X 2 T*. (2) Any CW -algebra resolution Ao ! J* is realizable by a simplicial space Qo. (3) The (inductively defined) characteristic classes On 2 Hn+2(J*; nJ*) (n = * *0; 1; : :): all vanish. Of course, the characteristic class On+1 is determined by the choice of some e* *xtension Q(n)o of Q(n-1)o, so as usual our obstruction theory requires back-tracking if at s* *ome stage we find On 6= 0. We shall now show how we can use other cohomology classes to determin* *e the choices of extensions at each stage: ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 21 4.16. distinguishing between different resolutions. A more interesting question* *, perhaps, is how one can distinguish between non-equivalent realizations Qo; Ro 2 sC of* * a fixed F- algebra resolution Ao ! J* of a realizable F-algebra J* ~=ssFX. Of course, i* *f Qo and Ro are both resolutions (in the resolution model category sC) of weakly equiv* *alent objects X ' Y in the model category C, then by definition Qo is weakly equivalent * *(actually: homotopy equivalent) to Ro. Thus we are looking for a way to distinguish betw* *een objects in C, using the iterative construction of a resolution Qo ! X (or equivalentl* *y, the Postnikov system for Qo). There are a number of possible approaches to this question: one could try to c* *onstruct a homotopy equivalence Qo ! Ro by induction on the Postnikov tower for Ro, us* *ing an adaptation to sC of the classical obstruction theory for spaces (cf. [W , V, x* *5]). Alternatively, one could try directly to construct a map Qo ! Y realizing the augmentation * *ssFAo ! J* (see [Bl3, x7], and compare [B , x5]). A description more in this spirit will b* *e given in [BG ]. Here our strategy is similar to that of x4.4: rather than assuming that we are* * given X and Y to begin with, we try to construct all different realizations (up to homotop* *y equivalence in sC) of a given simplicial F-algebra Ao (which is assumed to be a resolution o* *f a realizable F-algebra J*). We start our construction as in x4.5, and in the induction step * *we have assume given trnQo - or equivalently Q(n-1)o, satisfying the assumptions of x4.5 (* *see the proof of Proposition 2.41). We ask in how many different ways we can attach (n + 1)-di* *mensional "cells" to extend the realization one further dimension. Again the key lies in the extension of F-algebras of (2.40). Of course, we may* * assume that the characteristic class On 2 Hn+2(J*; nJ*) vanishes, so that it is possible to* * find "splittings" for (2.40), given by various liftings in Figure 2 - all of which yield the s* *ame cohomology class On by Lemma 4.6. As in the classical case of groups, we find that the dif* *ference between two such "semi-direct products" is represented by suitable cohomology classes, * *in dimension lower by one than the characteristic classes (see [Mc2 , IV, x2]). 4.17. Definition.Assume given two liftings ; 0: An+1! ssFZnQo in Figure 2 above* *, which define extensions of trnQo - so that, as in the proof of Theorem 4.8, we may as* *sume without loss of generality that the corresponding maps ; 0: Zn+1Ao ! nJ* vanish. As in * *the proof of Lemma 4.6, we extend , 0 to face maps d0; d00: Qn+1! Qn, define j : Cn+1Ao ! ss* *FZnQo with (jQn)# O j = 0, and lift to a map of A0-algebras i : Cn+1Ao ! nJ*. A* *gain i|Zn+1Ao= - 0, which is zero, so i is a cocycle in Hom J*-Mod(Ao; J*), rep* *resenting a cohomology class ffi;0 2 Hn+1(J*; nJ*), which we call the difference obstru* *ction for the corresponding Postnikov sections Q(n)o[] and Q(n)o[0] (in the notation of x4.9). Just as in the proof of Proposition 4.10, one can show that the classes ffin+1* *;0n+1in question do not in fact depend on the choice of F-algebra resolution Ao ! J*, but only* * on the homotopy type of Q(n-1)oin sC. Their significance is indicated by the following 4.18. Theorem. If ffi;0 = 0 then the corresponding Postnikov sections Q(n)o[] a* *nd Q(n)o[0] are weakly equivalent. Proof.If i is a coboundary, there is a map # : CnAo ! nJ* such that i = # * *O dAn0. Composing with the inclusion i : nJ* ,! ssFZnQo yields a morphism of A0-algeb* *ras ' : An ! ssFZnQo. If, as in the proof of Proposition 2.41, we set Q0n:= Qn q LnQo* *, we may realize ' by a map z0 : Q0n! ZnQo. Since we assumed Q0n is actually a copro* *duct of objects in ^F, it is a cogroup object in C by x2.7(i), so using the resulting* * group structure on Hom C(Q0n; Qn) we may set s0:= k . z : Q0n! Qn, where k : Q0n,! Qn is the inclu* *sion. Since k is a trivial cofbration and Qn is fibrant in C, we have a retraction r : Qn* * ! Q0n(which is a weak equivalence). Let s := s0O r : Qn ! Qn. 22 DAVID BLANC Recall from x2.13 that we have a faithful forgetful functor U^: C ! D, where f* *or simplicity we may assume D = G or D = sR-Mod (the other cases are trivial). We therefo* *re have a further forgetful functor U0 : D ! S, and we denote U0O ^U simply by U : C !* * S. The group operation map, while not a morphism in C or D, is a map m : UQnx UQn ! UQ* *n in S. Thus the following diagram commutes in S: id > U(z0O r O j) m|ZnQo UZnQo ___________________-UZnQox UZnQo __________-UZnQo | | | Uj | |U(j x j) |Uj | | | |?Ur Uk > U(j O z0) |? m |? UQn ____-'UQ0n____________-UQn x UQn ____________-UQn @@__________________________________________ Us Since U is faithful, this implies that s O j : ZnQo ! Qn factors through a m* *ap t : ZnQo ! ZnQo in C. Moreover, because we assumed that each M 2 ^F is of the* * form M = F M0 for some M02 S (where F = ^FOF 0is adjoint to U : C ! S), * *any map b : M ! ZnQo corresponds under the adjunction isomorphism to ^b: M0! U* *ZnQo, and thus t#fi = fi . (i O (jQn)#fi) for any fi 2 ssFZSnQo (since the group o* *peration . in ssFZnQo is induced by m - cf. [Gr, Prop. 9.9]). Now if ` : Qn+1! ZnQo realizes , we have (t O `)# = (` . (z0O `))# = . (# O (* *jQn)# O ) = . (-1. 0) = 0: An+1! ssFZnQo. Thus we have a comutative diagram 0 (jQn)# ssFQn+1 = An+1_______-ssFZnQo______-ZnssFQo = ZnAo | | | id| |t# |id | | | |? |? (jQn)# |? ssFQn+1 = An+1_______-ssFZnQo______-ZnssFQo = ZnAo which yields a map of (n+1)-truncated objects ae : trn+1Qo[] ! trn+1Qo[0] (or e* *quivalently, Q(n)o[] ! Q(n)o[0]). Clearly ae induces an isomorphism in sskssF for k n + 1. Now for any choice of lifting we have ssn+2ssFQ(n)o[] ~=Im(@Qn), and since (# O (jQn)#)|Im(@Qn-1)= 0; we find (t#)|Im(@Qn-1)= id, so by 2.39(b) the diagram @Qn ssff+1ZnQo_______-ssffZn+1Qo t | |id #| | | Q | |? ________-@n |? ssff+1ZnQo ssffZn+1Qo commutes. Thus ae induces an isomorphism on Im (@Qn), so that (ae)* : Q(n)o[] !* *_Q(n)o[0]_is a weak equivalence. |* *__| 4.19. Remark.Given a (realizable) F-algebra J*, a CW resolution Ao 2 sF -Al* *g of J*, and a fixed (but arbitrary) choice object X 2 C with ssFX ~=J*, by Corollary 4.* *11 we have a corresponding resolution Qo ! X. If X02 C is another realization of J* with co* *rresponding ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 23 Q0o! X0, we may assume without loss of generality that Yo0:= (Q0o)(n)' Yo := Q(* *n)ofor some n 0, with ; 0: An+2! ssFZn+1Qo ~=ssFZn+1Q0othe respective liftings. 4.20. different realizations of a -algebra. Assume given an abstract -algebra J* **, which is known to be realizable (e.g., by the cohomological criterion of Theorem 4.15* *). We wish to distinguish between the various non-weakly equivalent realizations of J* b* *y topological spaces (or simplicial groups). The spectral sequence (4.14)implies that in orde* *r for two such X; X02 G (with ss*X ~=J* ~=ss*X0) to be weakly equivalent, it suffices that the* *ir corresponding resolutions Qo ! X and Q0o! X0 be weakly equivalent (and thus homotopy equiv* *alent) in the resolution model category. This is in fact the main reason for consideri* *ng this model category structure on sG in the first case (and justifies its original name o* *f "E2-model category" in [DKS1 ]). Note, however, that this is not a necessary condition; an alternative model st* *ructure on sS (or sG), defined in [Mo ], has as weak equivalences precisely those maps in * *sC inducing an equivalence on the realizations. The difference obstructions ffi;0 , which yield an inductive procedure for d* *istinguishing between various realizations of a given -algebra resolution Ao ! J*, thus aga* *in provide an alternative to the theory described in [Bl3, x7] (as simplified in [Bl4, x4.9])* * for distinguishing between different realizations of a given -algebra, in terms of higher homotopy* * operations. To state this explicitly, assume given an (abstract) -algebra J*, a CW resol* *ution Ao 2 s -Alg of J*, and two realizations Qo; Q0o2 sG of Ao, determined as in x* *4.16 by successive choices of lifts k+1: Ak+1 ! ssFZkQo and 0k+1: Ak+1 ! ssFZkQ0o. B* *y x4.12, we know that the realizations X := diagQo and X0 := diagQ0o are two realizat* *ions of J*. If ffi0;00= 0, there is a weak equivalence f0 : (Q0o)(0)' Q(0)o, which* * we can use to push forward 01: A2 ! ssFZ1Q0o to 001: A2 ! ssFZ1Qo so it is meaningful to c* *onsider ffi1;01:= ffi1;0012 H2(J*; J*). Proceeding in this way we obtain the following 4.21. Theorem. Assume given a -algebra J*, a CW resolution Ao 2 s -Alg of * *J*, and two topological spaces X; X0 2 T* realizing J*, corresponding to X; X0* * 2 G under (4.13). Let Qo; Q0o2 sG be CW resolutions of X; X0 respectively, determine* *d as in x4.16 by successive choices of lifts n+1 : An+1 ! ssFZnQo and 0n+1: An+1 ! ssFZnQ0* *o. If the difference obstructions ffin+1;0n+12 Hn+2(J*; n+1J*) vanish for all n 0, t* *hen X and X0 are weakly equivalent. Again, these classes satsify certain naturality conditions, which are more eas* *ily stated for simplicial F-algebras: see [BG ]. 4.22. Remark.Theorem 4.21 provides a collection of algebraic invariants - sta* *rting with the homotopy -algebra ss*X - for distinguishing between (weak) homotopy types of sp* *aces. As with the ordinary Postnikov systems and their k-invariants, these are not actua* *lly invariant, in the sense that distinct values (i.e., non-vanishing difference obstructions)* * do not guarantee distinct homotopy types. Thus we are still far from a full algebraization of ho* *motopy theory - even if we disregard the fact that -algebras, not too mention their cohomolog* *y groups, are rather mysterious objects, and no non-trivial naturally occurring examples are * *fully known to date. Note, however, that we have a considerable simplification of the theory in the* * case of the rational homotopy type of simply-connected spaces: in this case the F-algebras * *in question are just connected graded Lie algebras over Q, and the cohomology theory reduce* *s to the usual Cartan-Eilenberg cohomology of Lie algebras. The obstruction theory we define a* *ppears to be the Lie algebra version of the theory for graded algebras due to Halperin and S* *tasheff in [HS ]. See also [O , xIII] and [F]. 24 DAVID BLANC Another such simplification occurs when we consider only the stable homotopy t* *ype: in this case F-algebras are just graded modules over the stable homotopy ring ss := ss* *S*S0, and the cohomology groups in question are Ext*ss(J*; -nJ*). Here we have no acti* *on of the fundamental group to worry about. Furthermore, the spectral sequence of (4.14)implies that if Q(n)o~=(Q0o)(n), * * then also (diagQo)(n)~=(diagQ0o)(n), so one can also use the theory described above "with* *in a range". References [Ba1] H.J. Baues, Commutator Calculus and Groups of Homotopy Classes, London Ma* *th. Soc. Lec. Notes Ser. 50, Cambridge U. Press, Cambridge, UK, 1981. [Ba2] H.J. Baues, "On the cohomology of categories, universal Toda brackets and* * homotopy pairs", K-Theory 11 (1997) No. 3, pp. 259-285. [Ba3] H.J. Baues, "(New) Combinatorial Foundation of Homology and Homotopy", pr* *eprint 1997/98. [Bl1] D. Blanc, "Derived functors of graded algebras", J. Pure Appl. Alg. 64 (1* *990) No. 3, pp. 239-262. [Bl2] D. Blanc, "Abelian -algebras and their projective dimension", in M.C. Tan* *gora, ed., Algebraic Topol- ogy: Oaxtepec 1991 Contemp. Math. 146, AMS, Providence, RI 1993. [Bl3] D. Blanc, "Higher homotopy operations and the realizability of homotopy g* *roups", Proc. Lond. Math. Soc. (3) 70 (1995), pp. 214-240. [Bl4] D. Blanc, "Homotopy operations and the obstructions to being an H-space",* * Manus. Math. 88 (1995) No. 4, pp. 497-515. [Bl5] D. Blanc, "New model categories from old", J. Pure & Appl. Alg. 109 (1996* *) No. 1, pp. 37-60. [Bl6] D. Blanc, "Loop spaces and homotopy operations", Fund. Math. 154 (1997), * *pp. 75-95. [Bl7] D. Blanc, "Homotopy operations and rational homotopy type", preprint 1996. [Bl8] D. Blanc, "CW simplicial resolutions of spaces, with an application to lo* *op spaces", to appear in Topology & Appl.. [BG] D. Blanc & P.G. Goerss, "Cohomology invariants for simplicial spaces", pr* *eprint 1999. [BS] D. Blanc & C.R. Stover, "A generalized Grothendieck spectral sequence", i* *n N. Ray & G. Walker, eds., Adams Memorial Symposium on Algebraic Topology, Vol. 1, Lond. Math.* * Soc. Lec. Notes Ser. 175, Cambridge U. Press, Cambridge, 1992, pp. 145-161. [B] A.K. Bousfield, "Homotopy spectral sequences and obstructions", Isr. J. M* *ath. 66 (1989), No. 1-3, pp. 54-104. [BK] A.K. Bousfield & D.M. Kan, Homotopy Limits, Completions, and Localization* *s, Springer-Verlag Lec. Notes Math. 304, Berlin-New York, 1972. [C] E.B. Curtis, "Simplicial homotopy theory", Adv. in Math. 2 (1971), pp. 10* *7-209. [D] A. Dold, "Homology of symmetric products and other functors of complexes"* *, Ann. Math., Ser. 2 68 (1958), PP. 54-80. [Dr] W. Dreckman, "On the definition of -algebras", C. R. Acad. Sci., Paris, S* *er. I, Math. 321 (1995), No. 6, pp. 767-772. [DK1] W.G. Dwyer & D.M. Kan, "Function complexes for diagrams of simplicial set* *s", Proc. Kon. Ned. Akad. Wet. - Ind. Math. 45 (1983), No. 2, pp. 139-147. [DK2] W.G. Dwyer & D.M. Kan, "An obstruction theory for diagrams of simplicial * *sets", Proc. Kon. Ned. Akad. Wet. - Ind. Math. 46 (1984), No. 2, pp. 139-146. [DKS1]W.G. Dwyer, D.M. Kan, & C.R. Stover, "An E2model category structure for p* *ointed simplicial spaces", J. Pure & Appl. Alg. 90 (1993) No. 2, pp. 137-152. [DKS2]W.G. Dwyer, D.M. Kan, & C.R. Stover, "The bigraded homotopy groups ssi;jX* * of a pointed simplicial space", J. Pure Appl. Alg. 103 (1995), No. 2, pp. 167-188. [F] Y. Felix, Denombrement des types de k-homotopie: theorie de la deformatio* *n, Mem. Soc. Math. France 3, Paris, 1980. [Gd] R. Godement, Topologie algebrique et theorie des faisceaux, Act. Sci. & I* *nd. No. 1252, Publ. Inst. Math. Univ. Strasbourg XIII, Hermann, Paris 1964. [Go] P.G. Goerss, "A Hilton-Milnor theorem for categories of simplicial algebr* *as", Amer. J. Math. 111 (1989), pp. 927-971. [Gr] B. Gray, Homotopy Theory: An Introduction to Algebraic Topology, Academic* * Press, New York, 1975. [HS] S. Halperin & J.D. Stasheff, "Obstructions to homotopy equivalences", Adv* *. in Math. 32 (1979) No. 3, pp. 233-279. [H] P.J. Hilton, "On the homotopy groups of the union of spheres", J. Lond. M* *ath. Soc. 30 (1955), pp. 154-172. ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 25 [Hi] P.S. Hirschhorn, Localization of model categories, preprint 1996. [K1] D.M. Kan, "On monoids and their dual", Bol. Soc. Mat. Mex. (2) 3 (1958), * *pp. 52-61. [K2] D.M. Kan, "On homotopy theory and c.s.s. groups", Ann. Math., Ser. 2 68 (* *1958), No. (1), pp. 38-53. [Mc1] S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag Gr* *ad. Texts Math. 5, Berlin- New York 1971. [Mc2] S. Mac Lane, Homology, Springer-Verlag Grund. math. Wissens. 114, Berlin-* *New York 1963. [Mat] M. Mather, "Pull-backs in homotopy theory", Can. J. Math. 28 (1976) No. 2* *, pp. 225-263. [May] J.P. May, Simplicial Objects in Algebraic Topology, U. Chicago Press, Chi* *cago-London, 1967. [Mi] J.W. Milnor, "On the construction FK", in J.F. Adams, editor, Algebraic T* *opology - A Student's Guide, London Math. Soc. Lecture Notes Series 4, Cambridge U. Press, Camb* *ridge, 1972, pp. 119-136. [Mo] I. Moerdijk, "Bisimplicial sets and the group-completion theorem", in J.F* *. Jardine & V.P. Snaith, eds, Algebraic K-Theory: Connections with Geometry and Topology, NATO ASI, Ser* *. C 279, Kluwer Ac. Publ., Dordrecht 1989, pp. 225-240. [O] A. Oukili, Sur l'homologie d'une algebre differentielle de Lie, Ph.D. the* *sis, Univ. de Nice, 1978. [Q1] D.G. Quillen, Homotopical Algebra, Springer-Verlag Lec. Notes Math. 20, B* *erlin-New York, 1963. [Q2] D.G. Quillen, "Spectral sequences of a double semi-simplicial group", Top* *ology 5 (1966), pp. 155-156. [Q3] D.G. Quillen, "Rational homotopy theory", Ann. Math. 90 (1969) No. 2, pp.* * 205-295. [Q4] D.G. Quillen, "On the (co-)homology of commutative rings", Applications o* *f Categorical Algebra, Proc. Symp. Pure Math. 17, AMS, Providence, RI, 1970, pp. 65-87. [R] C.L. Reedy, "Homotopy theory of model categories", preprint, 1975(?). [St] C.R. Stover, "A Van Kampen spectral sequence for higher homotopy groups",* * Topology 29 (1990) No. 1, pp. 9-26. [W] G.W. Whitehead, Elements of Homotopy Theory, Springer-Verlag Grad. Texts * *Math. 61, Berlin-New York, 1971. Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel E-mail address: blanc@math.haifa.ac.il