MAPPING SPACES AND M-CW COMPLEXES DAVID BLANC Abstract.We describe a procedure for recovering X from the space of maps* * from M into X, when M is constructed by cofibers of self-maps. This can be use* *d to define an M-CW approximation functor. The case when M is a Moore space is discu* *ssed in greater detail. 1. introduction The concept of "homotopy groups with coefficients", in which spheres are rep* *laced by a Moore spaces as the representing objects, were first studied by Peterson in [* *P ], and in greater detail by Neisendorfer in his thesis ([N1 ]; see [N2 ]). Much of homoto* *py theory can be redone in this spirit, with an arbitrary but fixed space M and its suspensio* *ns replacing the spheres not only in the definition of homotopy groups, but also in that of * *a CW - complex, loop space, and so on. In particular, an M-CW complex is a space const* *ructed inductively by successively attaching M-cells. This concept has been studied (* *under various names) by Bousfield, Dror-Farjoun, and others (cf. [Bo1 , Bo2, BT1 , Ch* * , D ]. Some of the properties of ordinary CW complexes carry over mutatis mutandis to * *M- CW complexes - e.g., the Whitehead theorem - but others do not. Compare [BT* *2 ]. In this note we address the question of recovering the space X from the mapp* *ing space XM , for a special class of "self-map resolvable" spaces M (see x2.1 be* *low), a question analogous to the classical one of recovering X from nX (x3.6). Jus* *t as for loop spaces, one needs some additional structure on XM in order to do so.* * Our procedure for recovering X is given recursively by a sequence of homotopy colim* *its, in Theorem 2.13. We may also think of this procedure as another construction of an* * M- CW approximation functor. Our approach can be made more explicit in the case of* * the mod pr Moore space (see Theorem 3.8). 1.1. notation and conventions. T* will denote the category of pointed CW comple* *xes with base-point preserving maps, and by a space we shall always mean an object * *in T*, for which we shall use boldface letters: X, Sn. Then n-fold smash product wi* *ll be Def denoted X^n = X ^ : :^:X. The space of (pointed) maps between X and Y will be written YX , rather than map*(X; Y), and the map induced by f : A ! B will be written f# : XB ! XA. When f is the degree k map between spheres, we write simply k : nX ! nX. Let M 2 T* be some d-dimensional "model space", which we shall assume to be* * a suspension. We adopt the stable convention that Mr denotes r-dM (so that Mr does not necessarily exist for r < d). ____________ Date: July 7, 1996. 1991 Mathematics Subject Classification. Primary 55P99; secondary 55Q70, 55P6* *0, 55S37. Key words and phrases. mapping space, homotopy with coefficients, Moore space* *, cellular space, M-CW complex. I would like to thank the Institute of Mathematics at the Hebrew University o* *f Jerusalem for its hospitality during the period when this paper was written. 1 2 DAVID BLANC Definition 1.2. The k-th homotopy group with coefficients in M of any space X * *2 T* Def k M is defined to be ssk(X; M) = [M ; X] ~=ssk-dX , for k d. We say that a m* *ap f : X ! Y in T* is an M-equivalence if f* : sst(X; M) ! sst(Y; M) is an isomorp* *hism for all t d - that is, if fM : XM ! YM is a weak homotopy equivalence (s* *ince M was assumed to be a suspension). Definition 1.3. The class CM is defined to be the smallest class of spaces i* *n T* containing M and closed under homotopy equivalences and arbitrary pointed homot* *opy colimits (cf. [D , Ch. 2, D.1]). If X 2 CM , we say X is an M-CW complex. Because M is a suspension, any M-CW complex may be constructed inductively, starting with a discrete space, by taking cofibers of maps from suspensions of * *M (see [D , Ch. 2, E.3]). Definition 1.4. An M-CW complex X^ equipped with an M-equivalence f : ^X! X is called an M-CW approximation for X. Any such ^X will be denoted by CWM X; it is unique up to homotopy equivalence, by the analogue of the Whitehead Theorem * *([W , V, Thm 3.8] and [D , Ch. 2, Thm E.1]). There are a number of different constructions of CWM X - see [Bo1 ], [CaP* *P ], and [D , Ch. 2, B.1 & E.6]. 1.5. organization. In section 2 we describe a recursive procedure for recoverin* *g X from XM when M is "self-map resolvable" - that is, constructed by taking cofibers* * of self- maps. In section 3 we specialize to the case where M is a Moore space, and desc* *ribe M-CW complexes explicitly. In section 4 we describe the simplicial resolution o* *f [Stv] in our context, and discuss its relevance to the construction of mapping spaces* * (Remark 4.14, as well as to M-CW approximations (Proposition 4.15). Acknowledgements. I would like to thank the referee for many useful comments and suggestions. 2. Mapping spaces We first consider the following question: given a space M as in x1.1, and a* * homotopy equivalence Y ' XM , what information regarding Y is needed in order to recov* *er X from it? Note that, by Definition 1.2, we can only hope to recover X up to M- equivalence. The problem of recovering X from XM appears to be a hard one for arbitrary* * M, so we restrict attention to the following special case: Definition 2.1. We say that a space V 2 T* is self-map resolvable (cf. [BT2 ,* * x2.1]) if there is a sequence of spaces {V(m)}nm=-1, with V = V(n), such that for * *each m 0 there is a self-map vm : dmV(m - 1) ! V(m - 1) with cofiber V(m). We always start with a (possibly localized) sphere V(-1) = Sj (or Sj(p)), and * *assume v0 : V(-1) ! V(-1) is the degree k map (k 2), so V(0) is the (j +1)-dimensional mod k Moore space. For simplicity we assume that each vm is a suspension, and* * all spaces are simply connected finite CW complexes. Thus each V(m) satisfies Bousfield's n-supported J-torsion condtion (see [Bo* *2 , x7.1]) for J = {p : p|k} and some n 1. The most useful case is when each space V(m - 1) is a p-local space of type * *m, and the map vm : dmV(m - 1) ! V(m - 1) is a vm -self-map (see [R , x1.5] or [HS ,* * Thm. 5.12]). Such spaces and maps play a central role in the definition of vn-perio* *dicity. MAPPING SPACES AND M-CW COMPLEXES 3 2.2. notation. To simplify the notation, for the rest of this section we fix a * *self-map v : Mt+d! Md, and denote its homotopy cofiber by V. By abuse of notation, any suspension of v will also be denoted by v; thus r-fold composites of suspensio* *ns of v will be written simply vr (with dimensions understood from the context). The c* *ofiber of vr : Mrt+s! Ms will be denoted by Vrt+s+1r, and we shall write simply Vr* * for Vrt+d+1r(so V = V1= Vt+d+11). Since we want V to be a co-H-space, we shall ass* *ume that v : Mt+d! Md is a suspension. Thus the problem at hand is that of recovering X from XV, where V is self-* *map resolvable. We assume inductively that we have a suitable procedure for recove* *ring X (up to M-equivalence, so in particular up to V-equivalence) from XM , so in* * fact the inductive step, which we consider in this section, involves recovering XM * * (up to V-equivalence) from XV. Remark 2.3. Just as in the case of loop spaces, one needs some structure on XV,* * beyond its bare homotopy type (in addition to the loop structure, which comes from the* * fact that V is a suspension), in order to recover X. As in Stasheff's approach to d* *elooping, our approach is iterative; so we shall not describe in advance the full structu* *re on XV which we postulate; the additional data needed at each stage in the process is * *given in x2.5. Of course, such data depends on the homotopy type of X, not only on that * *of XV. Again in analogy with the case of loop spaces, one really expects only to re* *cover CWVX from XV (since by definition the augmentation f : CWVX ! X induces a homotopy equivalence fV : (CWVX)V ! XV). Thus we may as well assume that X itself is a V-CW complex to begin with. This may be restated as follows: The additional structure on Y = XV which we need for our procedure consist* *s of homotopy invariants of X (see x2.5 below). However, on the face of it these ne* *ed not be invariant under V-equivalences, and choosing a different space X0 which is V-eq* *uivalent to X - so that we may also view Y as (X0)V - may yield different invariant* *s, and thus different constructions for recovering X0. We do not claim that our proc* *edure works for any such choice of X0(and thus of the invariants), but only for the "* *canonical" invariants - those associated to viewing Y as (CWVX)V. 2.4. cofibration sequences. The iterates of v fit into commutative diagrams, in* * which both the rows and the columns are cofibration sequences (up to homotopy): = 2rt+d_________-*_______________-*2rt+d+1 M2rt+d ___________-M * M |vr |v2r | | r | | d |* |v |? vr |? ir |? jrt+d+1r |? Mrt+d ___________-Md _____________-Vrt+d+1r________-Mrt+d+1 |irt+d |id | |irt+d+1 | r 2rt+d+1 | 2r 2rt+d+1 |= | r |? r |? r |? firt+d+1r|? V2rt+d+1r_________-V2rt+d+12r_____-Vrt+d+1r________-V2rt+d+2r |j2rt+d+1 |j2rt+d+1 | |jrt+d+2 | r | 2r |* | r |? = |? * |? * |? M2rt+d+1 _________-M2rt+d+1________-*_____________-M2rt+d+2 Figure 1 Thus we have: (a)fisr= is+1rO jsr(so that fis+1rO fisr~ *). (b)jsr= js2rO srand isr= s+2rt+1rO is2r(so that fisr= s+2rt+12rO fis2rO sr). 4 DAVID BLANC Remark 2.5. Mapping the third row of Figure 1 into X yields a fibration sequenc* *e: #r V #r V 2rt+d+1 : :X:Vr-! X 2r-! X r (see [W , I, 7.8]), and thus a long exact sequence 2rt+d+1 fi*r Vr *r V *r V 2rt+d+1 : :!:ss2rt+d+2(X; V) ~=ss1XVr -! ss0X -! ss0X 2r-! ss0X r : Since Vr, V2r, and all the maps in Figure 1 were assumed to be suspension* *s, this is actually an exact sequence of groups, which may be continued to the right as a * *sequence of pointed sets (cf. [Sp, 7,x2,Thm. 10]). In particular, one has a short exact * *sequence of groups: (2.6) 0 ! Coker(fi*r) ! ss0XV2r! Ker(fi*r) ! 0 for fi*r: ss2rt+d+2(X; V) ! ssrt+d+1(X; V) = ss0XVr and fi*r: ss2rt+d+1(X; V* *) ! rt+d rt+d rt+d+1 ssrt+d(X; V) = ss0XVr , where Vr = Vr = Vr and fir= fir. Note rt+d * # that ss0XVr is just a pointed set, but Ker(fir) = Im(r ) is a group. Thus Coker(fi*r) and Ker(fi*r) determine ss0XV2r= [V2rt+d+12r; X] = [V2* *r; X] as a set, but since we shall also need its group structure, we must assume the exten* *sion in (2.6) is given, in addition to the homotopy class of the map fi#r: XV2rt+d+1! * *XVrt+d, as part of the additional structure needed for our recovery procedure (see x2.3) Remark 2.7. Moreover, again because Figure 1 desuspends, there is a topological* * group i H and a closed normal subgroup G ,! H such that the following diagram commutes up to homotopy: fi#r # # : :r:t+d+2XVr' XVrrt+d+2_______-XVr_______-rXV2r_____-rXVr |6' |6' |6 |6 | | | | | i | q | Bi | : :G:____________-H ________-H=G ______-BG Figure 2 where the rows are fibration sequences, the two left vertical maps are homotopy* * equiva- lences, and the two right vertical maps are inclusions of components (up to hom* *otopy), by [M3 , Prop 7.9]. Since all the components of XV2r are homotopy equivalent,* * we have a (2.8) XV2r' (H=G)(0)x [V2rt+d+12r; X] ' (H=G)(0) [V2r;X] (as a loop space), where (H=G)(0) is the 0-component of H=G, say, and the gr* *oup [V2rt+d+12r; X] is assumed to be known, since the extension (2.6)is given as p* *art of our initial data. Moreover, H=G is homotopy equivalent to the realization B(H; G; *) of th* *e geo- metric bar construction B*(H; G; *), which is the simplicial space with Bj(H; G* *; *) = H x Gn x {*} (and the obvious face and codegeneracy maps - see [M3 , x7]). B* *y [D , Ch. 2, D.16] we have thus exhibited XV2r as a homotopy colimit of a diagram d* *efined rt+d+2 # in terms of XVr, XVr , the map fir and the extension (2.6). The cofibration sequences of Figure 1 fit into an inverse system: MAPPING SPACES AND M-CW COMPLEXES 5 4 2 : :_:_____-vM4t+d________-vM2t+d________-vMt+d |v4 |v2 |v | | | : :_:_____-|?= = |? = |? Md ___________-Md ___________-Md |i |i |i | 4 | 2 | 1 : :_:_____-|?4 2 |? 1 |? V4t+d+14______-V2t+d+12______-Vt+d+11 |j |j |j | 4 | 2 | 1 : :_:_____-|?v4 v2 |? v |? M4t+d+1 _______-M2t+d+1______-Mt+d+1 Figure 3 and thus the fibration sequences obtained by mapping the above diagram into a f* *ixed space X form a directed system as follows: 4)# (v2)# v# : :_:______oe(vXM4t+d' 4tXM __________oe2tXM__________oetXM |6 4 # |6 2 # |6# |(v ) |(v ) |v | | | : :_:______oe=XM____________oe=XM___________oe=XM |6# |6# |6# |i4 |i2 |i1 # | # | # | : :_:______oe4XV4___________oe2XV2__________oe1XV |6# |6# |6# |j4 |j2 |j1 (v4)# | (v2)# | v# | : : :________oe4t+1XM ________oe2t+1XM________oet+1XM Figure 4 Remark 2.9. The diagram in Figure 4 can be changed up to homotopy so that all t* *he st+1M j#2s Vs i#2sM (v2s)#2stM maps #2sare cofibrations, and 2 X -! X2 -! X ---! X is a fibration sequence. Definition 2.10. Recall from [Bo2 , x1] that, given a fixed space W, a space X * *is called W-local (or W-periodic) if XW ' ?. A map f : A ! B is called a PW -equival* *ence f# A if XB -! X is a (weak) homotopy equivalence for every W-local space X. Fin* *ally, a map ' : X ! "X is a W-localization if "Xis W-local and ' is a PW -equivale* *nce. Such localizations exist for any W. A functorial version of W-localization * *is denoted by PW X; see [D , Ch. 1, xB] and [Bo2 , x2]. Proposition 2.11. If X is a V-CW complex, and we set Def V #1 V2 #2 V4 V1 X = hocolim{X -! X -! X : :}:; then the map i# : V1 X ! XM , induced by the maps ir : XVr ! XM , is a weak homotopy equivalence. 6 DAVID BLANC Def t M v# 2t M (v2)#4t M (v4)# Proof. Set M1 X = hocolim{ X -! X ---! X ---! : :}:, and take the colimits of each row in Figure 4 to obtain a fibration sequence j# i# M v# M1 X -! V1X -! X -! M1 X: (To see this is indeed a fibration sequence, consider the colimits as unions of* * simplicial groups; note that the homotopy colimits used to define V1 X and M1 X are ju* *st ordinary colimits, if we apply Remark 2.9). Now # (v2)# ssiM1 X = colim {ssi(X; M) v-!ssi+t(X; M) ---! ssi+2t(X; M) : :}: for i 0 (cf. [Gr , Prop. 15.9]), and the right hand side is by definition the i-th v-pe* *riodic homo- topy group of X, usually written v-1ssi(X; M). Taking Vn-1 = M, ! = v and thus Vn = V in [Bo2 , x11.3], by [Bo2 , Def* *. 9.1 & Thm. 9.12], we have for a suitable space Wn (see [Bo2 , 10.1]), * *so the periodization map X ! PVX, which is a PV-equivalence, is also a Pvn-equivalence* * (see [Bo2 , x10.2]), and thus induces an isomorphism v-1ss*(X; M) ~=v-1ss*(PVX ; M)* * = 0 by [Bo2 , Thm 11.5]. However, since X is a V-CW complex, PVX is contractible (cf. [D , Ch. 3, P* *rop.___ B.1]). We deduce that M1 X is (weakly) contractible, and so i1 : V1 X ' XM * *. |___| Remark 2.12. If OE : XV ! V1 X is the obvious map, then i1 O OE ' i#1: XV ! * *XM ; thus OE is up to homotopy the inclusion of a closed normal subgroup, with quo* *tient map : V1 X ! XM =XV (as in Remark 2.7 above). Now let ! : V1X=XV ! BXV denote the classifying map of (which, up to homotopy, is just j1). Then the * *following diagram commutes up to homotopy: OE _______- _______-! XV ________-V1X V1X=XV BXV |= i1|' |' |= |? i#1 |? v# |? B(j#1) |? XV ________-XM ________-tXM ________-BXV By 2.4(a) we have B(fi#1) ' tOE O !. This implies we can recover the column* *s of Figure 3 from the spaces {XV2s}1s=0and maps between them. We may summarize the results of this section in the following Theorem 2.13. Suppose v : tM ! M is a suspended self-map with cofiber V, and Y = XV is a mapping space; then XM is V-equivalent to the homotopy colimit * *of XV ! XV2! : : :as in Proposition 2.11, and each XV2s+1ris a disjoint union as* * in (2.8), where each component is given by the homotopy colimit B(XV2sr; rtXV2sr;* * *)(0) as in x2.7. At the s-th stage, the data needed to determine the diagrams over w* *hich we take these homotopy colimits consists of the map fi#2srand the extension (2.6). Corollary 2.14. If V = V(n) is self-map resolvable, and Y ' XV, CWVX may be recovered from Y by a countable sequence of homotopy colimits. Proof. Since we assumed that X was a V-CW complex (x2.3), Proposition 2.11 impl* *ies that V1X ' XM . Since any V-CW complex is in particular an M-CW complex, the * * ___ same holds throughout the inductive application of Theorem 2.13. * * |___| Remark 2.15. The procedure we defined above is one for recovering X (up to V-eq* *ui- valence) from XV, rather than recognizing when Y ' XV. Of course this yiel* *ds MAPPING SPACES AND M-CW COMPLEXES 7 an implicit method for recognizing spaces of maps from a self-map resolvable sp* *ace V: apply the procedure to all possible candidates for the maps fi#r : rt+d+2Y* *(' XVrrt+d+2) ! Y(' XVr), etc., and verify that the resulting space X satisfies X* *V ' Y. In fact, one can say something about which maps b : rt+d+2Y ! Y could be of t* *he form fi#r (assuming Y ' XV), by checking b# : ssi+rt+d+2(Y; V) ! ssi(Y; V) = ssi-(rt+d+1)YV = ssi-(rt+d+1)(XV ^V* *): However, this is still far from an explicit recognition principle for spaces of* * maps from V, comparable to those of May [M2 ], Cobb [Co ], Smith [Sm ], and others. 3. Moore CW complexes We now specialize to the simplest example of a self-map resolvable space, th* *e d- dimensional mod pr Moore space M = Md(pr) = Sd-1 [pred. Throughout this d(pr) r section we shall write XM(r) for the mapping space XM , and sst(X; p) f* *or sst(X; M) = [Mt(pr); X]. In this case we can say explicitly when a space X 2 T* is of the homotopy t* *ype of an M-CW complex: Proposition 3.1. A space X 2 T* is an Md(pr)-CW complex if and only if X is (d - 2)-connected, ss*X is p-torsion, and pr. ssd-1X = 0. Proof. (I) If we assume that X is an M-CW complex, then since M is a suspensi* *on, we have X = hocolimnXn, whereWeach Xn+1 is obtained as the cofiber of a map fn : Mkn! Xn, starting with X0 ' fl2 Mkfl(see [D , Ch. 2, E.3]). Since H"i(Xn; Z) =* * 0 for each i d - 2 and "Hi(Xn; Z) is a p-group for all i, the same holds fo* *r ssiXn by [Sp, IX, x6, Thm 20], and thus for ssiX by [BoK , XII, 5.7]. Moreover, by the* * Blakers- Massey Theorem (cf. [W , VII, Thm 7.12]) ssd-1Xn+1 is a quotient of ssd-1Xn, so* * it has exponent pr by induction, and thus pr. ssd-1X = 0 by [Gr , Prop. 15.9]. pr d-1 d * * r (II) Now let X 2 T* be arbitrary: the cofibration sequence Sd-1-! S ! M (* *p ) then yields a long exact sequence pr r pr (3.2) : :!:sstX -! sstX ! sst(X; p) ! sst-1X -! sst-1X ! : : : and thus a short exact sequence (3.3) 0 ! sstX Z=pr ! sst(X; pr) ! T or(sst-1X; Z=pr) ! 0 fort > d: (See [N2 , x1] for the case t = d = 2). This short exact sequence implies t* *hat, if f : X ! X is the (d - 2)-connected cover of X, then f is an M-equivalenc* *e, and thus CWM (f) : CWM X ! CWM X is a homotopy equivalence. Similarly, t* *he p-localization X ! X(p) is an M-equivalence. Thus we may assume without loss * *of generality that X is p-local and (d - 2)-connected. In [Bo2 , Thm 5.2] Bousfield shows that, for W = Md+1(pr) we have 8 < ssiX if i < d (3.4) ssiPW X = ssdX=(p-torsion) if i = d : ss iX Z[1=p] if i > d; Thus if X is (d - 1)-connected and ss*X is p-torsion (i.e., all elements ar* *e of orders which are powers of p) then PW X ' *. In this case [D , Ch. 3, Prop. B.3] imp* *lies that CWM X ' X. 8 DAVID BLANC Now let ss be an (abelian) p-group; then ~="Hd-1(Md+s(pr); ss) = T or(Z=pr; ss) ssd+s(K(ss; d - 1); M) for s = 0, and ssd+s(K(ss; d - 1); M) = 0 for s > 0. Thus the inclusion i* *nduces an M-equivalence K(T or(Z=pr; ss); d - 1) ! K(ss; d - 1). By (I) we know that ss*^X is p-torsion for any M-CW complex ^X. On the othe* *r hand, CWM K(ss; d - 1) is a GEM (generalized Eilenberg-Mac Lane space) by [D , Ch. 5,* * Thm E.1], since M is a suspension and YM ' *. This implies that CWM K(ss; d - * *1) ' K(T or(Z=pr; ss); d - 1), by (3.3). Now if X is any (d - 2)-connected space with ss*X p-torsion, such that p* *r. ss = 0 for ss = ssd-1X, and X is the (d - 1)-connected cover of X, then we hav* *e a map of fibration sequences i CWM f F ibp(CWM f)_____-CWM X __________-CWM K(ss; d - 1) ppp ppp |h |k ' |` ppp || || || ' |? g |? f |? CWM (X) _______-X__________-X____________-K(ss; d - 1) in which ` is a homotopy equivalence, by the above. If F = F ib (h) is the ho* *motopy fiber of h, then F ib (k) ' F, and since k (and `) are M-equivalences, so is * *h, so F is M-local - that is, ss*(F; pr) = 0 for i d. Moreover, by [D , Ch. 5, xE.7], F is a 2-stage Postnikov system; but since X* * and CWM X are (d - 2)-connected, F is (d - 3)-connected, and thus by (3.3)we ha* *ve ssiF = 0 for i 6= d - 1; d - 2, and ssd-1F has no p-torsion. But since ss*X is * *p-torsion by assumption, and ss*CWM X is p-torsion by (I), the same is true of F, so that* * in fact F = K(ss0; d - 2) for some p-torsion group ss0, which fits into the short exact* * sequence 0 ! ssd-1CWM X ! ssd-1X ! ss0! 0. By comparing the long exact sequences (3.2) for CWM X and X, we see that* * ___ ss0~=ssd-1X=T or(ssd-1X; Z=pr). This completes the proof. * * |___| An alternative proof of the Proposition may be obtained by using [Ch , Thm. * *20.9]. Note that the Proposition does not provide us with any obvious construction of a Md(pr)-CW approximation functor (but compare Bousfield's p-cocompletion functor in [Bo2 , x14.1]). Corollary 3.5. If (p; q) = 1, and we let M = Md(prqs), M0 = Md(pr), and 0 M00 M00= Md(qs), then XM ' XM x X and CWM X ' CWM0X _ CWM00X for any X 2 T*. Proof. By [N2 , Prop. 1.5] we have M ' M0_ M00, so a map is M-equivalence if * *and only if it is both an M0-equivalence and an M00-equivalence. The Corollary then* *_follows_ from the Proposition. |__* *_| 3.6. n-fold loop spaces. To start the inductive sequence of procedures describe* *d in section 2 for recovering X from XV, we need to consider the initial case, whe* *n V = V(-1) = Sn: Recall from [M2 , x4] or [BV ] the little n-cubes operad Cn, which operate* *s on any n-fold loop space Y ' nX (cf. [M2 , x5]); conversely, any (connected) space o* *n which Cn operates is weakly equivalent to an n-fold loop space: in fact, May defines * *an "n-fold MAPPING SPACES AND M-CW COMPLEXES 9 classifying space" functor B(Sn; Cn; -) which recovers X from Y ' nX (if Y * *is connected, of course - cf. [M2 , Thm 13.1], and see also [Be ]). d-1 r d r r Lemma 3.7. For M = S and v = p (and thus V = M (p ), the mod p Moore space), the classifying space B(Sd-1; Cd-1; V1 X=XVr) (see x2.12) is * *a V-CW complex. Proof. Since ssiXVr~= ssd+i(X; pr), we see by (3.3)that (a) ssiXVr is p-torsion for all i 0, and (b) pr. ss0XVr= 0 (In fact, ss*XVr has exponent pr for p > 2, by [N2 , Prop. 7.1]). By considering the homtopy colimits used to define them (see x2.7, Propositi* *on 2.11, x2.12), we conclude that each of the spaces XV2r; XV4r; : :,: as well as V1 * *(which is the colimit of XVr! XV2r! : :):, and V1 X=(XVr), satisfies properties (a) a* *nd (b) above. The Lemma then follows from Proposition 3.1, since B(Sd-1; Cd-1; -) t* *akes__ values in (d - 2)-connected spaces. * *|___| When M is a Moore space, we thus obtain from Theorem 2.13 a more explicit de- scription of the procedure for recovering X from XM : Theorem 3.8. Let p be a prime, V = Md(pr) the d-dimensional mod pr Moore space (d 3), and Y = XVr, and assume that [V; X] = 0; then CWVX ' B(Sd-1; Cd-1; V1 X), where V1 X is the sequential homotopy colimit of XVr ! XV2r ! : : :as in Proposition 2.11, and each XV2s+1r is the homotopy colimit B(XV2sr; XV2sr; *), in the notation of x2.7. If we do not assume that [V; X] = 0, we must include the extensions (2.6)i* *n the construction of XV2s+1rfrom XV2sr. Corollary 3.5 allows one to generalize the * *Theorem to the mapping space from any mod k Moore space (k 2). 4. M-CW approximations In a sense, Corollary 2.14 provides a way of constructing an M-CW approximat* *ion for any space X, when M is self-map resolvable. However, this procedure is some* *what unsatisfactory, because it requires the full mapping space XM as part of the * *initial data. We now show how one may construct CWM X out of simpler building blocks: 4.1. the mapping cotriple. One obvious candidate for such a functorial and rela* *tively efficient (i.e., countable) construction for the M-CW approximation of a space * *is the M- analogue of Stover's construction of "simplicial resolutions by spheres": for * *any space M 2 T*, one can define a functor J : T* ! T*, as in [Stv, x2], by _1 _ _1 _ (4.2) JX = ( iMf _ CiMF)= ~ i=0 f2HomT*(iM;X) i=0 F2HomT*(CiM;X) where for each A = iM, the subspace @(CAF) ~=A of CAF (which is the copy of the cone on A indexed by F : CA ! X) is identified under ~ with Af, the c* *opy of A indexed by f = F |@CA. Note that JX is homotopy equivalent to a wedg* *e of copies of M and its suspensions. J is clearly a comonad (cf. [M , VI, x1]) on T*, with the obvious counit " :* * JX ! X - namely, "evaluation" - and comultiplication : J(X) ! J2X - where |iMf* * is 10 DAVID BLANC an isomorphism onto the copy of iM in L2X indexed by the inclusion iMf ,! JX, for any f : iM ! X; and similarly for CiMF. Now given X 2 T*, one may define a functorial simplicial space Jo = Jo(X) by setting Jn = Jn+1X, with face and degeneracy maps induced by the counit and comultiplication respectively (cf. [Go , Appendix, x3]). The counit also indu* *ces an augmentation " : Jo ! X. Moreover, if M is a suspension, then one has: "# Proposition 4.3. For any t d, the augmented simplicial group sst(Jo; M) -! sst(X; M) is acyclic - that is, sss(sst(Jo; M)) = 0 for s 1, and ss0(s* *st(Jo; M)) ~= sst(X; M). * * ___ Proof. Same as that of [Stv, Prop. 2.6]. * * |___| The realization of a simplicial space Yo (cf. [M2 , x11.1]) is its homotop* *y colimit, denoted by kYok, and constructed analogously to the geometric realization of * *a sim- plicial set. In particular, for Jo = Jo(X) as above we see that kJok is an* * M-CW complex. Recall from [BoF , Thm B.5] & [BrL , App.] that for any suitable sim* *plicial space Yo there is a first quadrant spectral sequence with (4.4) E2s;t= sss(sstYo) ) sss+tkYok: ("Suitable" includes the cases where Yo is a simplicial loop space, or where * *each Yn is connected). Applying the mapping space functor (-)M to Yo yields a simpl* *icial space YMo, and we have: Corollary 4.5. If M = M0 is a suspension, then for any space X 2 T* we have k(Jo(X)M k ' XM . 4.6. the map ". What one would really like to conclude from Proposition 4.3 is * *that the augmentation " : kJok ! X is an M-equivalence, and thus kJok ' CWM X. Unfortunately, this is not true in general: To see why, let M = Md(p) be the d-dimensional mod p Moore space, (so that e* *ach Jn is (d - 2)-connected). Let Ko be the simplical Eilenberg-Mac Lane space * *with Kn = K(ssd-1Jn; d - 1), and go : Jo ! Ko the obvious map of simplicial spaces* * made into a fibration, with (dimensionwise) fiber "Jo= Jo, the (d - 1)-connec* *ted cover of Jo. kgok By [A ] one then has a fibration sequence k"Jok ! kJok --! kKok, and thus* * a fibration sequence g M (4.7) k"JokM ! kJokM -! kKok : Recall from x2.4(a) that fisr= is+1rO jsr, so from the long exact sequenc* *e (3.2) Def d # we see that n = ssd-1Kn = ssd-1(Jn) ~= ssd(Jn; M)=Im(fir) . Since the spe* *ctral sequenceQ(4.4)for Ko collapses at E2, if we set Gn = ssn(ssd-1Ko) we see * *that kKok ' 1n=d-1K(Gn+d-1; n) (since Ko is a simplicial GEM, kKok is a GEM, too). Moreover, for any G we have sstK(G; n)M ~= "Hn-t(M; G), and K(G; n)M is aga* *in a GEM (cf. [T , Thm. 3]), so 1Y (4.8) kKokM ' (K(Gn+d-1; n) x K(Gn+d; n)) n=0 MAPPING SPACES AND M-CW COMPLEXES 11 since pr . Gn = 0 for all i (as in the proof of 3.7), and thus H"d-1(M; Gn) * *~= Hom(Z=pr; Gn) ~=Gn, and "Hd(M; Gn) ~=Ext(Z=pr; Gn) ~=Gn by [Sp, V, x5, Thm. 3]. One similarly sees that KMn' K(n; 0), so that again the spectral sequence * *for the simplical space KMo collapses, and 1Y (4.9) kKMok ' K(ssno; n - d + 1): i=d-1 On the other hand, applying realization to the fibration sequence of simplic* *ial spaces "JMo! JMo ! KMo yields a fibration sequence (by [A ] again), which maps into th* *at of (4.7)by naturality: k"JMok______-kJMok____-kKMok f|'l |ffi |j |? |? |? k"JokM _____-kJokM____-kKokM Figure 5 The map fl : k"JokM ! k"JokM is a weak equivalence, by [BT1 , Lemma 6.1]. * *Now Proposition 4.3 implies that the spectral sequence for JMo collapses, and thus (4.10) sstkJMok ~=ss0sstJMo~=sst+d(X; M): 4.11. M--algebras. We now show how to interpret the groups appearing in the rig* *ht hand side of (4.8)and (4.9)as derived functors, which allows us to show they of* *ten do not vanish, and thus to conclude that " : kJok ! X is not in general an M-equiv* *alence. Definition 4.12. Recall that a -algebra is an algebraic object modeled on the h* *omo- topy groups of a space, together with the action of the primary homotopy operat* *ions on them. If we replace the spheres representing ordinary homotopy groups by a m* *odel space M = Md (as in x1.1), we get M-homotopy operations corresponding to each homotopy class ff 2 ssr(Mn1_ : :_:Mnk ; M) (subject to the universal relations* * among such operations, corresponding to compositions of maps). We then define an M--algebra to be a graded set {Xi}1i=d, together with an a* *ction of the M-homotopy operationsWon them. As usual, the free M--algebras are those isomorphic to ss?( ff2AMrff; M) for some (possibly infinite) wedge of model * *spaces. (In the case of Moore spaces, one can be more explicit - cf. [Bl2, x5.6]). The category of M--algebras will be denoted by M--Alg; since it is a categor* *y of universal (graded) algebras, one has a concept of free simplicial resolutions, * *and thus of left derived functors LnT : M--Alg ! AbGp for any functor T : M--Alg ! AbGp (see [Q , I,x4] or [BS , x2.2.4] for more details). In particular, given an M--algebra X* = {Xi}1i=d, one has a functor T : * *M-- Alg ! AbGp defined T (X*) = Xd=Im{(fidr)# : Xd+1 ! Xd}. Now Proposition 4.3 shows that ss*(Jo; M) ! ss*(X; M) is a free simplicial resolution, so that * *ssd-1(Jn) ~= Def ssd(Jn=Im(fidr)#; M) = T (ss*(Jn; M)), and thus Gn = ssn(ssd-1Jo) = (LnT )ss* **(X; M). Of course, we may use any resolution of ss*(X; M) to calculate these derived * *functors. 12 DAVID BLANC Lemma 4.13. Let M = Md(pr) with d 4, p > 2, and X = K(Z; n) with n < min{2p - 4; 2d - 4}; then " : kJok ! X is not an M-equivalence. Proof. By (3.3), ss r i(X; M) = Z=p for i = n, and 0 otherwise. Since the on* *ly M- homotopy operations in dimensions n are the Bocksteins (fiir)# for i = d; * *: :n:- 1 (cf. [Y ]), we have ssk(Mk ; M) = Z=pr, ssk-1(Mk ; M) = Z=pr<(fikr)#>k* *, and sst(Mk ; M) = 0 for k < t < 2(p - 2). In the stable range it suffices to find an ordinary chain-complex resolution* * A* ! ss*(X; M) (cf. [Bl1, Lemma 6.10]), so we may choose Ai~=ss*(Mn-i ; M) in dimens* *ions n for i n - d, with din-i = (fin-i+1r)# n-i+1. Thus Gi = (LiT )ss*(X; M* *) ~= ssi(T A*) = Z=pr for i = n - d (and Gi= 0 for 0 i < n - d). Thus the map* * j in Figure 5 cannot be an equivalence, so the fiber of j, and thus of ffi, is not_t* *rivial._The Lemma follows. |___| Remark 4.14. We have shown that the simplicial space Jo does not in general pro* *vide an M-CW approximation functor. Nevertheless, as long as M is a suspension, it does* * give us a (relatively) explicit construction of XM as the homotopy colimit of a di* *agram of copies of spaces iM^j)M (i; j = 0; 1; : :):. In particular, if M is self-map* * resolvable, combined with Corollary 2.14 this gives a construction of CWM X from the spac* *es (M^j)M by a countable sequence of homotopy colimits. Q ThisVis because the Hilton-Milnor Theorem (cf. [W , XI, Thm. 6.7]) shows tha* *t JMn' nfini M W nff fi( i=1 M) (since Jn ' ff M). As this is in fact a weak product, and all finite products may be expressed as pointed homotopy colimits by [D , Ch. 2* *, Thm D.16], the statement follows by induction. We observe that for most Moore spaces this result can be slightly improved: Proposition 4.15. Let M = Md(k) be the d-dimensional mod k Moore space, where k is odd or 4|k. Then CWM X may be constructed by homotopy colimits as in The* *orem 2.13 from the simplicial space Jo(X)M of x4.1, where each JMn is homotopy equi* *valent to a product of mapping spaces of the form (iM)M . Proof. By [N2 , Cor. 6.6] we have Mr(k) ^ Ms(k) ' Mr+s-1(k) _ Mr+s(k) if r; * *s 3 and k is odd or 4|k. 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