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. Thus the Proposition follows from the above Remark, agai*
*n_by_
induction on the dimensions. |_*
*__|
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Dept. of Mathematics and Computer Science, Haifa University, Haifa 31905, Is*
*rael
E-mail address: blanc@mathcs2.haifa.ac.il