STABILIZATION AS A CW APPROXIMATION
A. D. Elmendorf
Purdue University Calumet
January 21, 1999
Abstract. This paper describes a peculiar property of the category of Smo*
*dules con
structed by the author, Kriz, Mandell, and May: the full subcategory of su*
*spension spec
tra (all of which are Smodules) forms a precise copy of the category of t*
*opological spaces.
Consequently, the "classical" homotopy category of Smodules with morphism*
*s the actual
homotopy classes of maps contains a copy of unstable homotopy theory. Stab*
*ilization and
stable homotopy are induced by CW approximation as Smodules. One conseque*
*nce is that
CW complexes whose suspension spectra are CW Smodules must be contractibl*
*e.
This paper offers somewhat belated proofs of the following theorems, which h*
*ave by now
become "wellknown to the experts" [6]. Let MS be the category of Smodules con*
*structed
in [1]; MS is a symmetric monoidal closed category, and also a topological clos*
*ed model
category whose "derived" category DS (obtained by inverting the weak equivalenc*
*es) is
equivalent to Boardman's stable category as a symmetric monoidal category. We w*
*ill not
refer to DS as the homotopy category of MS, instead reserving 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):
In this paper all spaces and homotopies are based. We work in the category T of*
* compactly
generated weak Hausdorff spaces. Given any space X, there is a natural Smodule*
* structure
on the suspension spectrum 1 X, and this provides us with a continuous functor *
*1 :
T ! MS.
_____________
1991 Mathematics Subject Classification. 55P42.
Key words and phrases. spectra, homotopy, Smodules.
Typeset by AM S*
*TEX
1
2 A. D. ELMENDORF
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 stabi
lization of X, and nothing has been stabilized! We see immediately that hMS(1 X*
*; 1 Y )
is isomorphic to the unstable homotopy classes [X; Y ]. However, it is precise*
*ly this lack
of stabilization that prevents the functor 1 from contradicting the nonexisten*
*ce results
of Hastings [2] and Lewis [3]. Hastings's corollary 2a of [2] does not apply, s*
*ince our 1
doesn't even reach the SpanierWhitehead category. Lewis's axiom A5 of [3] is v*
*iolated,
since the right adjoint of 1 is given explicitly by MS(1 S0; __), and theorem *
*1 then
shows that "1 "1 X ~=X, rather than "1 "1 X ' QX, as required. (We will make
no further use of this right adjoint of 1 .)
The problem, 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.
Theorem 2. For based spaces X and Y where X is homotopic to a CW complex, the
composite 1 *
T (X; Y ) ! MS(1 X; 1 Y ) fl!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 Smodules 1 X are the*
*mselves
cofibrant. Say that an Smodule M is resolvent if it is homotopic (in the sense*
* of being
isomorphic in hMS) to a cofibrant Smodule. 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 based space based homotopy equivalent to a CW complex, and*
* 1 X
is a resolvent Smodule, 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 Smodules. 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 . By a spectrum E we mean a space EV for each finite dimension*
*al
subspace V of U, together with homeomorphisms EV ~=W E(V W ) whenever V ? W ,
STABILIZATION AS A CW APPROXIMATION 3
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 halfsmash product of [5], 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
halfsmash 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 Lspectra, written S[L].*
* It supports
a coherently commutative and associative smash product written ^L. An Smodule*
* is
an Lspectrum satisfying a unital condition that will be unimportant in this pa*
*per; it is
satisfied by the spectra 1 Y with their natural Lspectrum structure described*
* below.
The category MS of Smodules is simply the full subcategory of Smodules in S[L*
*].
Given a space Y , the spectrum 1 Y has the structure of an Lspectrum 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 theorem 2 and theorem 3 follow from theorem 1.
Proof of theorem 2. Since X has the homotopy type of a CW complex, we may CW
approximate 1 X as an Lspectrum by applying L and as an Smodule by applying
= S ^L L; this is a consequence of lemma I.5.4 of [5] and theorems I.4.6 and I*
*I.1.9
of [1]. We have the Lspectrum 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. Since is a homotopy equivalence by [1], II.1.9(iv), it follows that a*
*nd fl induce
equivalent maps on passage to ss0. We have a commutative diagram
T (X; Y )_______________j*________________//T (X; 1 1 Y )
1 ~=
fflffl * ~= fflffl
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 .
4 A. D. ELMENDORF
It follows that * in the above diagram is an isomorphism on passage to ss0. The*
*refore j*
is an isomorphism on passage to ss0. This in turn implies that X is contractibl*
*e; 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), since X is equivalent to a space with a nondegenerate basepoint. *
*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.
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. Call this composit*
*e .
The (1 ; 1 ) adjunction gives us a commutative square
1 ^* //
T (X; 1 1 Y ) _______//_T (X; 1 F [L(1); 1 Y ))
0
~= ~=
fflffl ^* fflffl
SU (1 X; 1 Y ) _____////_SU(1 X; F [L(1); 1 Y ))
where 0is defined so that the square with lower horizontal arrows commutes. By *
*naturality
and the Yoneda Lemma, 0 is induced by a map
j : 1 1 Y ! 1 F [L(1); 1 Y )
which will be identified explicitly below. We wish to show that in the total di*
*agram
1 ^* //
T (X; Y )___j*____//T (X; 1 1 Y )_______//_T (X; 1 F [L(1); 1 Y ))
SS 0
1  SSSS1SS ~ ~
 SSS = =
fflffl SS))S fflffl ^* fflffl
MS(1 X; 1 Y ) ____//_SU(1 X; 1 Y ) _____////_SU(1 X; F [L(1); 1 Y ))
STABILIZATION AS A CW APPROXIMATION 5
the top row is an equalizer and the left square commutes. Since the bottom row*
* is an
equalizer, this will force the left vertical arrow to be an isomorphism, establ*
*ishing theorem
1. The lower triangle in the left square commutes simply because 1 : T (X; Y *
*) !
SU (1 X; 1 Y ) factors through MS; this is what defines the left vertical arrow*
*. The top
triangle in the left square commutes since j is the unit of the (1 ; 1 ) adjunc*
*tion. All
that remains is to show that the top row is an equalizer. In the special case X*
* = S0, we
are assuming the left vertical arrow is an isomorphism, so the top row is force*
*d to be an
equalizer which we can display explicitly as
j _1_^_//1 1
Y _____//1 1 Y __j__// F [L(1); Y )):
Since homfunctors preserve equalizers, the general case now follows by applyin*
*g T (X; __)
to this equalizer.
We are left with the proof of theorem 1 in the special case X = S0, which we*
* now address.
We need to identify explicitly the map j : 1 1 Y ! 1 F [L(1); 1 Y ) described i*
*n the
proof above. We use the fact that the structure map ^ on 1 X is by definition *
*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)
and the commutative diagram
SU (1 X; 1 Y ) ______________//_SU(F [L(1); 1 X); F [L(1); 1 Y ))
p* p*
fflffl fflffl
SU (1 (L(1)+ ^ X); 1 Y ) _____//SU(F [L(1); 1 (L(1)+ ^ X)); F [L(1); 1 Y ))
~= ~=
fflffl fflffl
SU (L(1) n 1 X; 1 Y ) ________//SU(F [L(1); L(1) n 1 X); F [L(1); 1 Y ))
XXXX
XXXXXXX *
~=XXXXXXXXXXX,,XX jfflffl
SU (1 X; F [L(1); 1 Y ))
to rewrite 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 displays j*
* as the
composite *
1 1 Y p! F (L(1)+ ; 1 1 Y ) ~=1 F [L(1); 1 Y ):
6 A. D. ELMENDORF
We must show that j : Y ! 1 1 Y is an equalizer for j and 1 ^. We do this in th*
*ree
steps: first, show that j composed with either map gives the same result, secon*
*d, show
that j is a closed inclusion, which implies it is a topological equalizer if it*
* is a settheoretic
equalizer, and finally, show that it is a settheoretic equalizer.
For the first step, a chase around the diagram
~=
SU (1 Y; 1 Y ) ______________//_T (Y; 1 1 Y )
p* p*
fflffl ~= fflffl
SU(1 (L(1)+ ^ Y ); 1 Y )_____//T (Y; F (L(1)+ ; 1 1 Y ))

~= 
fflffl 
SU (L(1) n 1 Y; 1 Y ) ~=

~ 
= 
fflffl ~= fflffl
SU (1 Y; F [L(1); 1 Y ))______//_T (Y; 1 F [L(1); 1 Y ))
starting at id1 Y gives the desired equality: note that ^ is reached in the l*
*ower right
corner.
The second step consists of a sequence of lemmas taken mainly from Gaunce Le*
*wis's
unpublished 1978 dissertation [4].
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 settheoretic 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 nonbasepoint 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 (; ) adjunctio*
*n. Conse
quently u is a closed inclusion, being the inclusion of a retract. (This is a b*
*asic property
STABILIZATION AS A CW APPROXIMATION 7
of compactly generated weak Hausdorff spaces.) We now have a commutative square
Y __u___//_Y
oe oe
fflfflu fflffl
Y _____//Y
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 j1k(j(C)) = uk(C), which is closed in kkY from above. This is pre*
*cisely
the criterion for j(C) to be closed in 1 1 Y , so j is a closed map.
It now suffices to show that j is the settheoretic 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 : {f} ! L(1) induces a map *f: F [L(1); 1 Y ) ! F [{f}; 1 Y ) = f*1 Y . Defin*
*e f
as the composite
^ *f
1 Y _____//F [L(1); 1 Y_)___//f*1 Y
We can consider the composite maps
p* 1 1 1 1 1 *f 1 *1
1 1 Y _____//F [L(1)+ ; Y ) ~= F [L(1); Y_)___// f Y
and
1 ^ 1 *f
1 f : 1 1 Y _____//1 F [L(1); 1 Y )____//1 f*1 Y;
and it suffices to show that j : Y ! 1 1 Y is the joint equalizer of all such p*
*airs. The
first map is the settheoretic equality, valid for all spectra,
1 E = E({0}) = E(f({0})) = (f*E)({0}) = 1 f*E;
8 A. D. ELMENDORF
applied when E = 1 Y . We wish to describe the second map, 1 f, equally explici*
*tly.
We have already shown that
j p* 1 1 1 1
Y _____//1 1 Y _____//F (L(1)+ ; Y ) ~= F [L(1); Y )
coincides with
j 1 ^ 1 1
Y _____//1 1 Y _____// F [L(1); Y );
so composing with 1 f : 1 F [L(1); 1 Y ) ! 1 f*1 Y shows that
j 1 1 1 *1
Y _____// Y = f Y
coincides with
j 1 f 1 *1
Y _____//1 1 Y _____// f Y:
We may therefore take the adjoint of
j 1 1 1 *1
Y _____// Y = f Y
to arrive at
f : 1 Y ! f*1 Y
and apply 1 to get our second map,
1 f : 1 1 Y ! 1 f*1 Y:
Using this description, we can compute 1 f explicitly as follows. Let OE be an *
*arbitrary
element of 1 1 Y , and suppose OE1 : SV ! Y ^ SV represents OE. Then 1 f(OE)*
* is
represented by the composite
1 OE1 1^f
SfV f! SV ! Y ^ SV  ! Y ^ SfV :
The proof of theorem 1 is finished by proving the following lemma.
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 by OE1 : 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 .
Then OE is also represented by the map OE2 given by the composite
OE1^1SW V W V W
OE2 : SV W ~= SV ^ SW ! Y ^ S ^ S ~=Y ^ S :
STABILIZATION AS A CW APPROXIMATION 9
Since f(V W ) = V W , 1 f(OE) is represented by
1 OE1^1 1^f
OE3 : SV W f! SV W  ! Y ^ SV W  ! Y ^ SV W :
Since f switches V and W , we see that OE3 is given by the composite
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 this composite co*
*incides
with OE1 ^ 1SW , which represents OE.
Next, we show that if OE 6= *, then OE1 sends nonbasepoints to nonbasepoin*
*ts. Let
(v; w) 2 SV W \ {1} ~=V W;
and suppose OE1(v) = *. Then
OE2[v; w] = (OE1 ^ 1SW )[v; w] = [*; w] = *
for all w 2 W , but also OE2[v; w] = OE3[v; w] = [OEf(w); v], so since v 6= 1, *
*OEf(w) = * for all
w 2 W . Therefore OE1 = *, since OEf is just a conjugate of OE1 by f, and so OE*
* = * as well.
Since OE = * is in the image of j, we may now assume OE 6= *, and since OE1 *
*was arbitrary,
any representative of OE sends nonbasepoints to nonbasepoints. Consequently,
OE2 : SV W \ {1} ! (Y ^ SV W ) \ {*};
that is,
OE2 : V W ! (Y \ {*}) x (V W ):
The coordinate functions may be written
OE2(v; w) = (OE01v; OE001v; w)
by the first characterization of OE2 as OE1 ^ 1SW . But this coincides with OE*
*3, which has
coordinate functions
OE3(v; w) = (OE03w; v; OE003w);
where OE03w 2 Y \ {*}, OE003w 2 W , and neither depends on v. From OE2 = OE3 we*
* conclude
that
(1) OE01v is constant, and
(2) OE001v = v.
This precisely characterizes OE1 as representing an element of the image of j, *
*so OE 2 imj.
10 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.Harold M. Hastings, Stabilizing tensor products, Proc. Amer. Math. Soc. 49 (*
*1975), 17.
3.L. G. Lewis, Jr., Is there a convenient category of spectra?, J. Pure Appl. *
*Algebra 73 (1991), 233246.
4.L. G. Lewis, Jr., The stable category and generalized Thom spectra (Universi*
*ty of Chicago dissertation,
1978).
5.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, SpringerVerla*
*g, 1986.
6.N. P. Strickland, Products on MUmodules, Trans. Amer. Math. Soc (to appear).
Department of Mathematics, Purdue University Calumet, Hammond, IN 46323
Email address: aelmendo@math.purdue.edu