STABILIZATION AS A CW APPROXIMATION
A. D. Elmendorf
Purdue University Calumet
June 7, 1997
Abstract. This paper describes a peculiar property of the category of S-mo*
*dules con-
structed by the author, Kriz, Mandell, and May: the full subcategory of su*
*spension spec-
tra (which are all S-modules) forms a precise copy of the category of topo*
*logical spaces.
Consequently, the "classical" homotopy category of S-modules with morphism*
*s the actual
homotopy classes of maps contains a copy of unstable homotopy theory. Stab*
*ilization and
stable homotopy are then induced by CW approximation as S-modules. One con*
*sequence is
that CW complexes whose suspension spectra are CW S-modules must be contra*
*ctible.
This paper offers somewhat belated proofs of the following statements, which*
* have by
now become "well-known to the experts" [4]. Let MS be the category of S-modules*
* con-
structed in [1]; MS is a symmetric monoidal closed category, and also a topolog*
*ical closed
model category whose "derived" category DS (obtained by inverting the weak equi*
*valences)
is equivalent to Boardman's stable category, including the multiplicative struc*
*ture descend-
ing from MS. We will not refer to DS as the homotopy category of MS, instead re*
*serving
that term for the "classical" homotopy category hMS, which has the same objects*
* as MS
and morphisms defined by
hMS(M; N) := ss0MS(M; N):
Given any space X, (which we always assume to be compactly generated weak Hausd*
*orff),
there is a natural S-module structure on the suspension spectrum 1 X, and this *
*provides
us with a continuous functor 1 : T ! MS, where T is the category of based spac*
*es.
Theorem 1. For any based spaces X and Y, the map
1 : T (X; Y ) ! MS(1 X; 1 Y )
is a homeomorphism.
This result may seem unsettling at first, since 1 X is supposed to represent*
* the
stabilization of X, and nothing has been stabilized! In fact, we see immediate*
*ly that
_____________
1991 Mathematics Subject Classification. 55P42.
Key words and phrases. spectra, homotopy, S-modules.
Typeset by AM S-*
*TEX
1
2 A. D. ELMENDORF
hMS(1 X; 1 Y ) is isomorphic to the unstable homotopy classes [X; Y ]. The catc*
*h, of
course, is that the weak equivalences have not been inverted. All objects of MS*
* are fibrant
in the model category structure, so this amounts precisely to a lack of CW (i.e*
*., cofibrant)
approximation. Let fl : 1 X ! 1 X be any CW approximation of 1 X in MS. We
have the following reassuring result.
Proposition 2. For based spaces X and Y where X is homotopic to a CW complex, t*
*he
composite
1 fl*
T (X; Y ) --! MS(1 X; 1 Y ) -! MS(1 X; 1 Y )
induces the usual stabilization map [X; Y ] ! {X; Y } on passage to homotopy cl*
*asses.
This provides conclusive evidence that very few of the S-modules 1 X are the*
*mselves
cofibrant. Say that an S-module M is resolvent if it is homotopic (in the sense*
* of being
isomorphic in hMS) to a cofibrant S-module. The following result follows from o*
*ne proved
by (in alphabetical order) Mike Hopkins, Norio Iwase, John Klein, and Nick Kuhn*
* in
response to a question I posted on the Algebraic Topology Discussion List.
Theorem 3. If X is a space homotopy equivalent to a CW complex, and 1 X is a
resolvent S-module, then X is contractible.
Overall, the situation is one that is familiar in algebra, but perhaps not s*
*o much so
in stable homotopy. There is a good notion of homotopy in MS, and consequently*
* a
homotopy category hMS, but it is very far from the stable category. The invers*
*ion of
weak equivalences works a profound change in transforming hMS into DS _ it resu*
*lts in
stabilization itself.
* * *
We turn now to the proofs, and begin by reviewing the basic definitions of t*
*he theory
of S-modules. More details can be found in [1].
We fix our attention on a particular "universe" U, which the reader is welco*
*me to
consider as R1 . First, by a spectrum E we mean a space EV for each finite dime*
*nsional
subspace V of U, together with homeomorphisms EV ~=W E(V W ) whenever V ? W ,
subject to an evident associativity diagram. Given a based space Y , we write *
*1 Y for
the spectrum whose V 'th space is colimW?VW V W Y ; we write S for 1 S0. We wr*
*ite SU
for the category of spectra. Next, we write L(1) for the space of linear isome*
*tries from
U to itself (it is the first space in the linear isometries operad on U, which *
*explains the
notation.) Using the twisted half-smash product of [3], there is a monad L in S*
*U defined
by LE = L(1) n E. Using the twisted function spectrum right adjoint to the twi*
*sted
half-smash product, L has a right adjoint L* defined by L*E = F [L(1); E). As t*
*he right
adjoint of a monad, it is a comonad, and the algebras over L can be identified *
*with the
coalgebras over L*. Either one defines the category of L-spectra, written S[L].*
* It supports
a coherently commutative and associative smash product written ^L. An S-module*
* is
an L-spectrum satisfying a unital condition that will be unimportant in this pa*
*per; it is
satisfied by the spectra 1 Y with their natural L-spectrum structure described*
* below.
The category MS of S-modules is simply the full subcategory of S-modules in S[L*
*].
STABILIZATION AS A CW APPROXIMATION 3
Given a space Y , the spectrum 1 Y has the structure of an L-spectrum with *
*the
structure map
: L(1) n 1 Y ~=1 (L(1)+ ^ Y ) -p*!1 Y
where p : L(1)+ ! S0 is the collapse map. We will be mostly interested in its*
* dual
L*-structure given by the adjoint map
^ : 1 Y -! F [L(1); 1 Y ):
We show first that proposition 2 and theorem 3 follow from theorem 1.
Proof of proposition 2. Since X has the homotopy type of a CW complex, we may CW
approximate 1 X as an L-spectrum by applying L and as an S-module by applying
= S ^L L; this is a consequence of lemma I.5.4 of [3] and theorems I.4.6 and I*
*I.1.9
of [1]. We have the L-spectrum structure map : L1 X ! 1 X and the unit map
: 1 X = S ^L L1 X ! L1 X. The composite O is the CW approximation map
fl, so since is a homotopy equivalence by [1], II.1.9(iv), it follows that an*
*d fl induce
equivalent maps on passage to ss0. We have a commutative diagram
T (X;?Y ) -j*! T (X; 1?1 Y )
?y1 ?y~=
* ~=
MS(1 X; 1 Y ) -! S[L](L1 X; 1 Y ) -! SU (1 X; 1 Y )
for any based space Y , and since 1 is a homeomorphism by theorem 1, the desir*
*ed result
now follows on passage to ss0.
The proof of theorem 3 now follows directly.
Proof of theorem 3. By theorem 1, the left vertical arrow in the above diagram *
*is a home-
omorphism. Since 1 X is resolvent, fl is a homotopy equivalence, and therefore *
*so is ,
and it follows that * in the above diagram is an isomorphism on passage to ss0.*
* Therefore
j* is an isomorphism on passage to ss0. This in turn implies that X is contrac*
*tible; we
give a modification of an argument due to John Klein.
First, X must be connected, because otherwise
j* : [X; S0] ! [X; 1 1 S0]
can't be onto. Next, let X+ be X with a new disjoint basepoint. Then 1 1 (X+ *
*) '
1 1 (X _ S0), so we get
[X; X] ~=[X; X _ S0] ~=[X; 1 1 (X _ S0)] ~=[X; 1 1 (X+ )] ~=[X; X+ ] ~={*}:
Therefore X is contractible.
We begin the proof of theorem 1 with a reduction.
4 A. D. ELMENDORF
Lemma 4. Theorem 1 follows from the special case in which X = S0.
Proof. MS(1 X; 1 Y ) is by definition the equalizer of two maps
SU (1 X; 1 Y ) ! SU (1 X; F [L(1); 1 Y ));
the first induced by the structure map ^ : 1 Y ! F [L(1); 1 Y ), and the second*
* given
by the composite
SU (1 X; 1 Y ) ! SU (F [L(1); 1 X); F [L(1); 1 Y )) ! SU (1 X; F [L(1); 1 Y )*
*);
where the second map is induced by the structure map of 1 X. By adjunction, the*
*se are
homeomorphic to two maps
T (X; 1 1 Y ) ! T (X; 1 F [L(1); 1 Y ));
natural in both X and Y , and the special case X = S0 reduces theorem 1 to the *
*statement
that j : Y ! 1 1 Y is the equalizer of two maps
1 1 Y ! 1 F [L(1); 1 Y ):
The general case now follows by applying the functor T (X; __) to this equalize*
*r, since
hom-functors preserve equalizers.
We are left with the proof of theorem 1 in the special case X = S0, which we*
* now
adddress. We need to identify explicitly the two maps 1 1 Y ! 1 F [L(1); 1 Y )
described in the proof above, and since 1 is naturally isomorphic to SU (S; __*
*), the first
map must be 1 ^. Next, we use the fact that the structure map ^on 1 X is by def*
*inition
the composite
1 X -j!F [L(1); L(1) n 1 X) ~=F [L(1); 1 (L(1)+ ^ X)) -p*!F [L(1); 1 X)
to rewrite the composite defining the second map as
*
SU (1 X; 1 Y ) -p!SU (1 (L(1)+ ^ X; 1 Y )
~=SU (L(1) n 1 X; 1 Y ) ~=SU (1 X; F [L(1); 1 Y )):
Specializing again to X = S0 and using the 1 -1 adjunction, this in turn can *
*be
rewritten as *
1 1 Y -p! F (L(1)+ ; 1 1 Y ) ~=1 F [L(1); 1 Y ):
We must show that j : Y ! 1 1 Y is an equalizer for these two maps. We do this *
*in
three steps: first, show that j composed with either map gives the same result,*
* second,
show that j is a closed inclusion, which implies it is a topological equalizer *
*if it is a
set-theoretic equalizer, and finally, show that it is a set-theoretic equalizer.
STABILIZATION AS A CW APPROXIMATION 5
For the first step, a chase around the diagram
* ~=
SU (1 Y;?1 Y ) -p! SU (1 (L(1)+?^ Y ); 1 Y ) -! SU (L(1) n?1 Y; 1 Y )
?y~= ?y~= ?y~=
* ~=
T (Y; 1 1 Y ) -p! T (Y; F (L(1)+ ; 1 1 Y ))-! T (Y; 1 F [L(1)+ ; Y ))
starting at id1 Y shows that the dual structure map ^: 1 Y ! F [L(1); 1 Y ) is *
*adjoint
to the composite
* ~=
Y -j!1 1 Y -p! F (L(1)+ ; 1 1 Y ) -! 1 F [L(1); 1 Y ):
However, this composite coincides with
1 ^
Y -j!1 1 Y ---! 1 F [L(1); 1 Y )
since both are dual to ^.
The second step consists of a sequence of lemmas taken mainly from Gaunce Le*
*wis's
unpublished 1978 dissertation [2].
Lemma 5. A map of spaces is an equalizer if and only if it is a closed inclusio*
*n.
Proof. Clearly any equalizer is a closed inclusion, by construction. Conversely*
*, let f : A !
B be a closed inclusion. Then f is an equalizer of the two maps B ! B=f(A), one*
* the
canonical map, the other the trivial (basepoint) map.
It follows immediately that a closed inclusion that is a set-theoretic equal*
*izer of two
continuous maps is a topological equalizer of the two maps.
Lemma 6. The unit map u : Y ! Y is a closed inclusion.
Proof. I am indebted to Gaunce Lewis for the following proof, which corrects a *
*flaw in his
dissertation.
Let oe : S0 ! S1 send the non-basepoint to 1_22 S1. Then smashing with oe g*
*ives a
natural closed inclusion oe : Y ! Y . Next, observe that the composite
Y -u-!Y -e!Y
is the identity, where the evaluation map e is the counit of the - adjunction, *
*and
consequently u is a closed inclusion, being the inclusion of a retract. (This l*
*ast is a basic
property of compactly generated weak Hausdorff spaces.) We now have a commutat*
*ive
square
Y? u-! Y ?
oe?y ?yoe
Y -u-! Y
6 A. D. ELMENDORF
in which all arrows except u are known to be closed inclusions. The composite *
*u O oe
is therefore a closed inclusion, and thus an equalizer. Since oe : Y ! Y is mon*
*ic
(being an equalizer) and oe O u is an equalizer, it follows that u is an equali*
*zer, and thus a
closed inclusion.
From lemma 5 and the fact that is a right adjoint, we deduce that if f is a*
* closed
inclusion of based spaces, so is f. Lemma 6 now implies that all the maps in th*
*e system
Y -u!Y -u-!22Y -! . . .
are closed inclusions.
Lemma 7. The map into the colimit, j : Y ! 1 1 Y , is a closed inclusion.
Proof. Since j is clearly injective, it suffices to show that it is a closed ma*
*p. Let uk :
Y ! kkY and jk : kkY ! 1 1 Y be the canonical maps, and let C Y be a
closed subset. Then j-1k(j(C)) = uk(C), which is closed in kkY from above; but *
*this
is precisely the criterion for j(C) to be closed in 1 1 Y , so j is a closed ma*
*p.
It now suffices to show that j is the set-theoretic equalizer of the two map*
*s, and for
this purpose, we consider an arbitrary linear isometry f 2 L(1). Then the inc*
*lusion
{f} L(1) induces a map F [L(1); 1 Y ) ! F [{f}; 1 Y ) = f*1 Y , and we can con*
*sider
the composite maps
*
1 1 Y -p! F (L(1)+ ; 1 1 Y ) ~=1 F [L(1); 1 Y ) ! 1 f*1 Y
and 1 ^
1 1 Y ---! 1 F [L(1); Y ) ! 1 f*1 Y:
It suffices to show that j : Y ! 1 1 Y is the joint equalizer of all such pairs*
*. The first
map is merely the canonical identification, valid for all spectra,
1 E = E({0}) = E(f({0})) = (f*E)({0}) = 1 f*E;
applied when E = 1 Y . The second is constructed from the first by composing wi*
*th j,
Y -j!1 1 Y = 1 f*1 Y;
taking adjoints,
f : 1 Y ! f*1 Y;
and passing to 1 again:
1 f : 1 1 Y ! 1 f*1 Y:
Using this description, we can compute 1 f explicitly as follows. Let OE : SV !*
* Y ^ SV
be an element of V V Y representing an arbitrary element of 1 1 Y . Then 1 f(OE)
is represented by the composite
-1 OE 1^f
SfV -f-! SV -! Y ^ SV - -! Y ^ SfV ;
which is an element of fV fV Y . The proof of theorem 1 is finished by proving*
* the
following lemma.
STABILIZATION AS A CW APPROXIMATION 7
Lemma 8. Let OE 2 1 1 Y and suppose 1 f(OE) = OE for all f 2 L(1). Then OE 2 im*
*j.
Proof. Let OE be represented as OE : SV ! Y ^ SV , and choose W orthogonal to V*
* with the
same dimension. We can choose an f 2 L(1) such that f(V ) = W and f(W ) = V . T*
*hen
OE is also represented by the map
OE^1SW V W V W
SV W ~= SV ^ SW -----! Y ^ S ^ S ~=Y ^ S ;
to which we can apply 1 f. Since f(V W ) = V W , we get
-1 OE^1 1^f
SV W -f-! SV W - -! Y ^ SV W - -! Y ^ SV W :
Since f switches V and W , this version of 1 f(OE) is of the form
OEf^1SV W V V W
SV W ~= SW ^ SV - ----! Y ^ S ^ S ~= Y ^ S
for some map OEf : SW ! Y ^ SW . Our assumption implies that the displayed com*
*posite
coincides with OE ^ 1SW .
Next, we show that OE sends non-basepoints to non-basepoints. Let
(v; w) 2 SV W \ {1} ~=V W;
and suppose OE(v) = *. Then
1 f(OE)[v; w] = (OE ^ 1SW )[v; w] = [*; w] = *
for all w 2 W , but also 1 f(OE)[v; w] = [OEf(w); v], so since v 6= 1, OEf(w) =*
* * for all
w 2 W . Therefore OE = *, since OEf is just a conjugate of OE by f.
Since OE = * is in the image of j, we may now assume OE 6= *, and consequent*
*ly
OE : SV W \ {1} ! (Y ^ SV W ) \ {*};
that is,
OE : V W ! (Y \ {*}) x (V W ):
The coordinate functions may be written
OE(v; w) = (OE1v; OE2v; w)
by the first characterization of OE as OE ^ 1SW . But this coincides with 1 f(*
*OE ^ 1SW ),
which has coordinate functions
1 f(OE ^ 1SW )(v; w) = (O^E1w; v; ^OE2w);
where ^OE1w 2 Y \{*}, ^OE2w 2 W , and neither depends on v. From OE^1SW = 1 f(*
*OE^1SW )
we conclude that
(1) OE1v is constant, and
(2) OE2v = v.
This precisely characterizes the image of j, so OE 2 imj.
8 A. D. ELMENDORF
References
1.A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, Modules, and *
*Algebras in Stable
homotopy Theory, Mathematical Surveys and Monographs v. 47, American Mathema*
*tical Society,
Providence, RI, 1997.
2.L. G. Lewis, Jr., The stable category and generalized Thom spectra (Universi*
*ty of Chicago dissertation,
1978).
3.L. G. Lewis, Jr., J. P. May, and M. Steinberger (with contributions by J. E.*
* McClure), Equivariant
Stable Homotopy Theory, Lecture Notes in Mathematics v. 1213, Springer-Verla*
*g, 1986.
4.N. P. Strickland, Products on MU-modules, preprint available at http://hopf.*
*math.purdue.edu.
Department of Mathematics, Purdue University Calumet, Hammond, IN 46323
E-mail address: aelmendo@math.purdue.edu