LOCALIZATION OF ANDR'EQUILLENGOODWILLIE
TOWERS, AND THE PERIODIC HOMOLOGY OF
INFINITE LOOPSPACES
NICHOLAS J. KUHN
Abstract. Let K(n) be the nthMorava Ktheory at a prime p, and
let T(n) be the telescope of a vnself map of a finite complex of type n.
In this paper we study the K(n)*homology of 1 X, the 0thspace of
a spectrum X, and many related matters.
We give a sampling of our results.
Let PX be the free commutative Salgebra generated by X: it is
weakly equivalent to the wedge of all the extended powers of X. We
construct a natural map
sn(X) : LT(n)P(X) ! LT(n) 1 ( 1 X)+
of commutative algebras over the localized sphere spectrum LT(n)S. The
induced map of commutative, cocommutative K(n)*Hopf algebras
sn(X)* : K(n)*(PX) ! K(n)*( 1 X),
satistfies the following properties.
It is always monic.
It is an isomorphism if X is nconnected, ßn+1(X) is torsion, and
T(i)*(X) = 0 for 1 i n1. It is an isomorphism only if K(i)*(X) =
0 for 1 i n  1.
It is universal: the domain of sn(X)* preserves K(n)*isomorphisms,
and if F is any functor preserving K(n)*isomorphisms, then any nat
ural transformation F(X) ! K(n)*( 1 X) factors uniquely through
sn(X)*.
The construction of our natural transformation uses the telescopic
functors constructed and studied previously by Bousfield and the au
thor, and thus depends heavily on the Nilpotence Theorem of Devanitz,
Hopkins, and Smith. Our proof that sn(X)* is always monic uses Topo
logical Andr'eQuillen Homology and Goodwillie Calculus in nonconnec
tive settings.
1.Introduction and main results
In algebraic topology, homotopical aspects of topological spaces are stud
ied by means of generalized homology and cohomology theories. Such theo
ries are themselves determined by spectra, the objects of the stable category.
____________
Date: June 3, 2003.
2000 Mathematics Subject Classification. Primary 55P43, 55P47, 55N20; Second*
*ary
18G55.
This research was partially supported by a grant from the National Science F*
*oundation.
1
2 KUHN
One can then pass between the worlds of unstable and stable homotopy by
means of the adjoint pair of functors ( 1 , 1 ), where 1 Z denotes the
suspension spectrum of a based space Z, and 1 X denotes the 0th infinite
loopspace of the spectrum X.
Though 1 preserves homology (and cohomology), the homological be
havior of 1 is much more subtle, and one has the basic problem: given a
spectrum E, to what extent, and in what way, is E*( 1 X) determined by
E*(X)?
There is a related, more subtle, problem: to what extent, and in what way,
is LE 1 1 X determined by LE X? Here LE denotes Bousfield localization
with respect to E*.
In this paper, we develop new techniques allowing for a thorough study
of these questions when E* is a periodic homology theory. The key to our
methods is to combine two of the major strands of homotopy theory of the
past two decades: the flowering of powerful new techniques in homotopical
algebra, many following the conceptual model offered by T. Goodwillie's
calculus of functors [G1 , G2 , G3 ], and the deepening of our understanding
of homotopy as organized from the chromatic point of view, in the wake of
the Nilpotence Theorem of E. Devanitz, M. Hopkins, and J. Smith [DHS ].
The tools from modern homotopical algebra that we use are Topological
Andr'eQuillen homology of E1 ring spectra, as developed via the Good
willie calculus framework, together with a good theory of Bousfield local
ization of structured objects. These concepts require that we work within
a nice model category of spectra. Thus, for us, spectra will mean objects
in S, the category of Smodules as in [EKMM ], and so, e.g., commutative
Salgebras will serve as E1 ring spectra.
The input from chromatic homotopy theory comes from our use of the
telescopic functors, constructed by Bousfield and the author in [K3 ], [B1 ],
and [B4 ], which factor certain periodic localization functors through 1 .
The classic stable splitting of the space 1 1 Z [Kah ] provides a model
for our main results when presented as follows. Let Z+ denote the union
of a space Z with a disjoint basepoint. If X is an Smodule, 1 ( 1 X)+
is naturally a commutative Salgebra. Another example is PX, the free
commutative algebra generated by X. This is weakly equivalent to the
wedge, running over r 0, of DrX = E r+ ^ r X^r, the rth extended
power of X. Then, for all spaces Z, there is a natural map of commutative
Salgebras
s(Z) : P( 1 Z) ! 1 ( 1 1 Z)+
satisfying the following properties.
(1) s(Z)* : E*(P( 1 Z)) ! E*( 1 1 Z) is monic for all theories E*.
(2) s(Z) is an equivalence if Z is connected.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 3
Now let K(n) be the nth Morava Ktheory spectrum at a fixed prime p,
and T (n) its `telescopic' variant: the telescope of a vnself map of a finite
complex of type n. (See [R ] for background on this material.) The Telescope
Conjecture is the statement that , i.e. T (n)*acyclics are
K(n)*acyclics; in any case, the converse holds, so that LK(n)LT(n)= LK(n).
We show that, for all spectra X, there is a natural map of commutative
LT(n)Salgebras
sn(X) : LT(n)P(X) ! LT(n) 1 ( 1 X)+
satisfying the following properties.
(1) sn(Z)* : E*(P(X)) ! E*( 1 X) is monic for all X, if .
(2) sn(X) is an equivalence if X is suitably connected and T (i)*(X) = 0 for
1 i n  1. It is an equivalence only if K(i)*(X) = 0 for 1 i n  1.
(3) sn is universal in the sense that any natural transformation from a func
tor invariant under T (n)*equivalences to LT(n) 1 ( 1 X)+ will canonically
factor through sn.
Homological consequences are most precise when E* = K(n)*. The first
property then says that, for all X,
sn(Z)* : K(n)*(P(X)) ! K(n)*( 1 X)
is an inclusion of commutative, cocommutative, K(n)*Hopf algebras. The
second property and work of Hopkins, Ravenel, and Wilson [HRW ] combine
to say that, if X is an Smodule with T (i)*(X) = 0 for 1 i n  1, then
there is an isomorphism of K(n)*Hopf algebras
n+1O
K(n)*( 1 X) ' K(n)*(PX) K(n)*(K(ßj(X), j)).
j=0
Our main theorems also have consequences for E*n, where En is fundamen
tal pcomplete integral height n complex oriented commutative Salgebra
appearing in the work of Hopkins and his collaborators, since it is known
[H1 ] that K(n)*(X) = 0 if and only if E*n(X) = 0. Indeed our work here,
combined with work by many, beginning with [HKR ] and [Hu ], on E*n(DrX),
gets us most of the way towards calculations of E*n( 1 X) generalizing Bous
field's extensive functorial calculations [B4 ] of E*1( 1 X) = K*( 1 X; Zp).
Our theory of localized Andr'eQuillen towers also yields the following
theorem, a significant generalization of the main result of [K1 ]: if Z is a
connected space, and f : 1 Z ! X is an E*isomorphism, then ( 1 f)* :
E*( 1 1 Z) ! E*( 1 X) is monic.
We describe our results in detail in the next section.
Versions of our main theorem, Theorem 2.5, date from 2000, and have
been reported on in various seminar and conference talks since then in both
4 KUHN
the United States and Europe.
Acknowledgements
Many people deserve thanks for helping me with aspects of this work.
I thank Greg Arone and Mike Mandell for many tutorials, taught from
complementary prospectives, on Andr'eQuillenGoodwillie towers of vari
ous sorts. What I have learned from them has also been enhanced by the
ideas of Randy McCarthy and his students, and by perusing recent versions
of [G3 ] that Tom Goodwillie kindly supplied me. Some of the conversations
with Mike Mandell were during a visit to the University of Chicago during
the fall of 2000, and I thank the Chicago Mathematics Department for its
support. I thank Charles Rezk for helping me through a point of confusion,
and Steve Wilson for helping me reach a point of clarity, both related to
proofs of results in x2.5.
Finally I need to thank Pete Bousfield. He has been a constant guide to
my understanding of the strange behavior of periodic localization. This has
been true for twenty years, but email exchanges dating from mid 2000 have
particularly helped me keep straight the technical details of this project. I
particularly recommend his recent paper [B7 ] for discussions of problems
closely related to those studied here.
2. Main results
In this section, we describe our results. It ends with a discussion of the
organization of the remainder of the paper, where proofs and more detail
are given.
Throughout we will use the following convention: if A and B are objects in
a model category C, a weak map A f!B will mean either a pair A g~C h!B,
or a pair A h!C g~B. A weak map in C induces a well defined morphism
in the homotopy category ho(C), and we say that a diagram of weak maps
in C commutes if the induced diagram in ho(C) does.
2.1. Commutative Salgebras and the stable splitting of QZ, revis
ited. Let Z+ denote the union of a space Z with a disjoint basepoint. If
X is an Smodule, 1 ( 1 X)+ is naturally a commutative Salgebra aug
mented over the sphere spectrum S. We denote by Alg the category of such
objects. Another example is PX, the free commutative algebra generated
by X. There is a natural weak equivalence:
1`
PX ' DrX.
r=0
Given A 2 Alg , let I(A) be the homotopy fiber of the augmentation
A ! S. We view I(A) as the augmentation ideal, and we are interested
in two associated objects. The first is bA2 Alg, which arises as the inverse
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 5
limit of an Andr'eQuillen tower in Alg, and can be viewed as the I(A)adic
completion of A. The second is an Smodule taq(A), a form of Topological
Andr'eQuillen homology, and can be viewed as I(A)=I(A)2. There is a
convergence result: if A is 0connected then the canonical map A ! Abis
an equivalence.
Applied to the examples above, if A is either PX or 1 ( 1 X)+ , with
X 1connected in the latter case, then taq(A) ' X. The natural map
I(A) ! taq(A) identifies in the first case with projection onto the first
factor, and in the second with ffl(X) : 1 1 X ! X, the counit of the
adjunction. There is also a natural weak equivalence:
1Y
bPX ' DrX.
r=0
We observe:
Proposition 2.1. Let f : A ! B be a map in Alg . If taq(f) : taq(A) !
taq(B) is an equivalence, so is fb: Ab! Bb. Thus, in this case, there is a
factorization by weak algebra maps
___canonical//_
A >> ?bA,?
""
>>> ""
f >>OEOE>""""
B
bf
where the unlabelled weak map is B ! bB~ Ab.
Observations like this are the basis for various of our splitting theorems.
We first illustrate this idea by giving a new formulation and proof of a very
highly structured version of the classical splitting of 1 QZ, where, as usual,
QZ denotes 1 1 Z. (See [K4 , Appendix B] for a discussion of some of
the different proofs this theorem of D.S.Kahn.)
Let j(Z) : Z ! QZ be the unit of the adjunction. In a straightforward
way, this then induces a weak map in Alg
s(Z) : P( 1 Z) ! 1 (QZ)+ .
Theorem 2.2. For all spaces Z, the map s(Z) induces an isomorphism on
completions, and thus there is a natural factorization of weak algebra maps
P( 1 Z) _______canonical____//_bP( 1 Z).
MMM qqq88
MMMM qqq
s(Z)MMM&&M qqqqt(Z)
1 (QZ)+
If Z is 0connected then all of these maps are weak equivalences.
6 KUHN
In more downtoearth terms, our theorem gives us a factorization by
weak maps
W 1 1 canonical //Q1 1
r=0 DrZP _____________________ r=077 DrZ
PPP nnnn
PPPP nnnn
s(Z) PPP''P nnnnt(Z)
1 (QZ)+
in which both the infinite wedge and product are equivalent to E1 ring
spectra, such that all maps in the diagram are E1 .
Our general theory leads to a short proof of the theorem as follows. The
diagram
(2.1) 1 717 1NZ
1 ''(Z)pppp NNffl(N1NZ)N
pppp NN
ppp NN''N
1 Z _______________________ 1 Z
commutes for all Z. By construction, this shows that taq(s(Z)) can be iden
tified with the identity map on 1 Z and thus is an equivalence. By the
proposition, so also is [s(Z), and the theorem follows.
In Appendix A, we check that our stable splitting agrees with others in
the literature.
In the case when Z is not connected, our theorem improves upon various
weaker versions in the literature, and our proof shows that most of the
technical issues confronted in these papers need no longer be part of the
story. See Remarks 4.3.
2.2. Bousfield localization and Andr'eQuillen towers. Now we mix
Bousfield localization with the general theory. If E is any Smodule, LE S
will be a commutative Salgebra, and we define LE (Alg ) to be the category
of commutative LE Salgebras, which are also augmented over LE S, and are
Elocal. Up to weak equivalence, objects have the form LE A, with A 2 Alg,
but not all morphisms are homotopic to one of the form LE f, with f 2 Alg.
Analogous to the nonlocalized theory, given A 2 LE (Alg ), one gets a
completion bLEA 2 LE (Alg ), and an LE Smodule taqE (A). For A 2 Alg,
LE (taq(A)) ' taqE (LE A), but it is not in general true that the natural map
LE bA! bLEA is an equivalence. This is illustrated by the example
1Y
bLEPX ' LE DrX,
r=0
which is often different than _
1Y !
LE bPX ' LE DrX .
r=0
Analogous to Proposition 2.1, we observe:
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 7
Proposition 2.3. Let g : LE A ! LE B be a map in LE (Alg ). If taqE (g) :
LE taq(A) ! LE taq(B) is an equivalence, so is bg: bLEA ! bLEB. Thus, in
this case, one gets a factorization by weak algebra maps
LE A ____canonical__//_bLEA.
FF ;;w
FFF wwww
gFFF##F wwww
LE B
In the early 1980's, the author proved that if f : 1 Z ! 1 W is an E*
isomorphism, with Z and W connected, then 1 f is an E*monomorphism
[K1 ]. As a first application of our general theory of localized towers, we
deduce the following stronger version.
Theorem 2.4. Let Z be a connected space. If a map of spectra f : 1 Z !
X is an E*isomorphism, then
( 1 f)* : E*(QZ) ! E*( 1 X)
is a monomorphism.
We leave to the reader the proof that the hypotheses can be weakened
slightly: the domain of f need just be `spacelike', i.e. a wedge summand of
a suspension spectrum. See also Appendix A for a version of the theorem
for nonconnected Z.
Examples illustrating this theorem were given in [K1 ]. Besides these, see
also its use in Appendix B.
2.3. The main theorem. Fixing a prime p and n 1, we now apply the
proposition of the last subsection to the case when E = T (n). Our theorem
has a statement and proof analogous to Theorem 2.2.
In place of (2.1), we use the following much deeper theorem: there is a
natural factorization by weak Smodule maps
(2.2) LT(n) 1 1 X
''n(X)oo77oo OOOLT(n)ffl(X)OO
oooo OOO
ooo OO''O
LT(n)X _________________________LT(n)X.
Such a factorization was constructed in the mid 1980's by Bousfield [B3 ],
when n = 1, and by the author, for all n 1 [K3 ]. A recent paper by
Bousfield [B7 ] revisits these constructions. These papers heavily use the
classification of stable vnself maps of finite complexes, and thus are using
(at least for n > 1) the work of Devanitz, Hopkins, and Smith [DHS , HS ]
on Ravenel's Nilpotence Conjectures.
8 KUHN
The natural transformation jn(X) then induces a weak map
sn(X) : LT(n)P(X) ! LT(n) 1 ( 1 X)+
in LT(n)(Alg ). Proposition 2.3 combines with (2.2) to prove the main theo
rem of the paper:
Theorem 2.5. For all spectra X, the map sn(X) induces an isomorphism
on LT(n)completions, and thus there is a natural factorization of weak al
gebra maps
LT(n)P(X) __________canonical________//bLT(n)P(X).
QQQQ mmmm66
QQQQ mmmm
sn(X)QQQQ((Q mmmm tn(Z)
LT(n) 1 ( 1 X)+
In more downtoearth terms, our theorem gives us a factorization by
weak maps
W 1 canonical Q 1
LT(n)( r=0DrX) ___________________________// r=0LT(n)DrX
SSSS kkkk55k
SSSSS kkkkk
sn(X)SSSS))S kkkk tn(X)
LT(n) 1 ( 1 X)+
in which both the infinite wedge and product are equivalent to commuta
tive LT(n)Salgebras, such that all maps in the diagram are LT(n)Salgebra
maps.
For applications to computing K(n)* and E*n, it is useful to let
sKn(X) = LK(n)sn(X) : LK(n)P(X) ! LK(n) 1 ( 1 X)+ .
The functor on Smodules sending X to P(X) preserves E*isomorphisms
for any generalized homology theory E*. Our natural transformations sn(X)
and sKn(X) yield the best possible `invariant' approximations to the functors
LT(n) 1 ( 1 X)+ and LK(n) 1 ( 1 X)+ in the following sense:
Proposition 2.6. Let F : S ! S be any functor preserving
T (n)*isomorphisms. Then any natural transformation T of the form
T (X) : F (X) ! LT(n) 1 ( 1 X)+
factors uniquely through sn. Similarly, sKn is the terminal natural transfor
mation from a functor preserving K(n)*isomorphisms.
This proposition will be an easy consequence of results described in x2.5.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 9
2.4. First homological corollaries. It is easily verified that the canonical
map from a wedge to the product of a family of spectra induces a monomor
phism on any homology theory. Thus Theorem 2.5 has the following theorem
as an immediate corollary.
Theorem 2.7. If E* is any homology theory such that T (n)*acyclics are
also E*acyclics, then sn induces a natural monomorphism
sn(X)* : E*(PX) ! E*( 1 X).
The commutative Hspace structure on 1 X, together with the diagonal,
induces a K(n)*Hopf algebra structure on K(n)*( 1 X).1 Meanwhile, the
usual product maps,
DiX ^ DjX ! Di+jX,
and transfer maps,
Di+jX ! DiX ^ DjX,
associated to the inclusion of groups ix j i+j make K(n)*(PX) into
a K(n)*Hopf algebra. When specialized to E* = K(n)*, the last theorem
refines as follows.
Theorem 2.8. sn(X)* : K(n)*(PX) ! K(n)*( 1 X) is a natural inclusion
of commutative, cocommutative K(n)*Hopf algebras.
Since the cohomological Bousfield class of En is the same as the homologi
cal Bousfield class of K(n), Theorem 2.5 also has the following consequence.
Theorem 2.9. For all spectra X, and all n 1, there is a factorization of
commutative E*nalgebras
L 1 * canonical //Q1 *
r=0En(DrX)Q ____________________ r=0En(DrX),66
QQQ mmmm
QQQQ mmmm
tn(X)*QQQQ(( mmmmsn(X)*
E*n( 1 X)
where the algebra structure on the infinite sum and product is induced by the
transfer maps. In particular, tn(X)* is a natural inclusion of commutative
E*nalgebras.
Here the map labelled tn(X)* is defined by applying E*nto the composite
tn(X)Y1 Yr
1 ( 1 X)+  ! DqX ! DqX
q=0 q=0
and then letting r go to 1.
We explain the appearance of the transfer maps, e.g. in this last theorem.
This is a consequence of the naturality of sn and tn, as applied to the
diagonal : X ! X x X. Since 1 commutes with products, one sees that
the product on E*n( 1 X) is induced by applying the functor 1 ( 1 ( )+ )
____________
1In the twisted sense described in [B4, Appendix], if p = 2.
10 KUHN
to . Meanwhile, it is well known (see [LMMS , Thm.VII.1.10] or [K2 ,
Prop.A.3]) that applying Dk( ) to the weak map
X ! X x X ~ X _ X
yields the product, over i + j = k, of the transfer maps
DkX ! DiX ^ DjX.
2.5. When is sn(X) an equivalence? One might now wonder how often
sn(X) and sKn(X) are equivalences. We have various results which together
give a good sense of what is happening.
We define classes of Smodules Sn SKn as follows.
LetSn = {X 2 S  sn(X) is an equivalence}
= {X 2 S  sn(X)* : T (n)*(PX) ~!T (n)*( 1 X)}.
Let SKn= {X 2 S  sKn(X) is an equivalence}
= {X 2 S  sn(X)* : K(n)*(PX) ~!K(n)*( 1 X)}.
We recall that EilenbergMacLane spectra are acyclic in T (n)*. From
this the following first observations are easily deduced: if S0 is either Sn or
SKn, then X 2 S0 if and only if X<1> 2 S0, and furthermore, a necessary
condition is that ß0(X) = 0. Here X denotes the dconnected cover of
an Smodule X.
Our results about Sn and SKn are most pleasantly described by first in
troducing two more classes of Smodules.
Let ~Sn= {X 2 S X 2 Sn for larged}.
Let ~SKn= {X 2 S  X 2 SKn for larged}.
We recall a concept from [HRW ]: say X is strongly E*acyclic if the spaces
1 c(X<1>) are E*acyclic for all large c.2 For example, Eilenberg
MacLane spectra are strongly T (n)*acyclic.
Following notation in various papers, e.g. [B4 ], let Lfn1denote localiza
tion with respect to T (0) _ . ._.T (n  1). We recall that this is smashing:
Lfn1X ' Lfn1S ^ X.
Armed with this terminology, we have the following theorem and propo
sition.
Theorem 2.10.
(1) ~Sn= {X 2 S  Lfn1X is strongly T (n)*acyclic}, and Sn ~Sn.
(2) ~SKn= {X 2 S  Lfn1X is strongly K(n)*acyclic}, and SKn ~SKn.
____________
2Note that 1 c(X<1>) is the cthspace in the connective cover of the spect*
*rum X.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 11
Proposition 2.11. There are implications (1) ) (2) ) (3) ) (4) ) (5).
(1) T (i)*(X) = 0 for 1 i n  1.
(2) Lfn1X is strongly T (n)*acyclic.
(3) Lfn1X is strongly K(n)*acyclic.
(4) (with n 2) X is strongly K(n  1)*acyclic.
(5) K(i)*(X) = 0 for 1 i n  1.
Let c(n) denote the smallest integer c such that T~(n)*(K(Z=p, c)) = 0.
Then c(n) n + 1, with equality certainly holding if the Telescope Conjec
ture is true, and perhaps even if not.
Theorem 2.12.
(1) Suppose X 2 S~n. Then X 2 Sn if ß0(X) = 0, ßj(X) is uniquely
pdivisible for 0 j c(n), and also ßc(n)+1(X)=(torsion) is uniquely p
divisible.
(2) Suppose X 2 S~Kn. Then X 2 SKn if and only if ß0(X) = 0, ßj(X) is
uniquely pdivisible for 1 j n, and also ßn+1(X)=(torsion) is uniquely
pdivisible.
Example 2.13. The conditions in Proposition 2.11 trivially hold for all X
if n = 1. We conclude that that ~S1= ~SK1= S, and S1 = SK1 is determined
by the second statement of Theorem 2.12. In particular, there is a natural
equivalence of commutative augmented LK(1)Salgebras
LK(1)P(X) ' LK(1) 1 ( 1 X)+
for all 1connected X, with torsion ß2. This fits perfectly with the many
results on K*( 1 X) proved by Bousfield beginning with [B1 ].
Example 2.14. The Telescope Conjecture holds for n = 1: T (1)*acyclics
are K(1)*acyclics. Thus the conditions in Proposition 2.11 are all equiv
alent when n = 2. We conclude that S~2= S~K2= {X  K(1)*(X) = 0},
and SK2 is the set of Smodules described by the second statement of The
orem 2.12. In particular, there is a natural equivalence of commutative
augmented LK(2)Salgebras
LK(2)P(X) ' LK(2) 1 ( 1 X)+
for all K(1)*acyclic, 2connected X, with torsion ß3.
Example 2.15. These results tell us precisely which plocal finite spectra
are in SKn and S~Kn. All finites are in S~K1, and F is in SK1 if and only if
ß0(F ) = ß1(F ) = 0 and ß2(F ) is torsion. If n 2, F is in ~SKnif and only if
F has type at least n, and F is in SKn if and only if F has type at least n
and also ßj(F ) = 0 for all 0 j n.
12 KUHN
Remarks 2.16. If the Telescope Conjecture is true for a pair (p, n), then the
second statement of Theorem 2.12 improves the first by one dimension. This
is due to the fact that in proving this second statement, we use computa
tional methods based on special properties of K(n)*.
Even if the Telescope Conjecture fails, it is still conceivable that some
of the conditions in Proposition 2.11 are equivalent. We note that, as ob
served in [HRW , Thm.3.14], if X is a BP module, then condition (5) implies
condition (1).
Our results imply that, in general, the two maps
sKn( 1 Z), LK(n)s(Z) : LT(n)P( 1 Z) ! LT(n) 1 QZ+
are not homotopic, since, by Theorem 2.2, the second map is an equivalence
whenever Z is connected, while the first map needn't be. By perturbing the
`usual' stable splitting of QZ, we have thus lost homotopy equivalence but
gained naturality with respect to stable maps between suspension spectra.
See Appendix C for more about this.
2.6. More homological corollaries. Hand in hand with the results of the
last subsection, are some more homological corollaries.
In the spirit of [B5 , x11], if EilenbergMacLane spectra are strongly E*
acyclic, one can define Evir*(Z), the virtual E*homology of a space Z, by
the formula
Evir*(Z) = E*(Z) for larged.
Methods of [HRW ] imply the following illuminating lemma.
Lemma 2.17. An Smodule X is strongly E*acyclic if and only if
~Evir*( 1 X) = 0.
The fact that P(X) ! PX is a K(n)*equivalence implies that there
is a canonical lifting
K(n)vir*(616X)
mmmmmm 
mmmm 
mm sn(X)* fflffl
K(n)*(PX) _____//_K(n)*( 1 X).
The next theorem is closely related to Theorem 2.10.
Theorem 2.18. For all X, there is a natural short exact sequence of com
mutative, cocommutative K(n)*Hopf algebras
sn(X)* vir 1 vir 1 f
K(n)*(PX) ! K(n)* ( X) ! K(n)* ( Ln1X).
Note that the first term here is a functor of LK(n)X; in constrast, the last
term is `invisible' to LK(n)X, as K(n)*(Lfn1X) = 0.
Since K(n)*(PX) = 0 exactly when K(n)*(X) = 0, we have the next
corollary, which strengthens [HRW , Cor.3.13].
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 13
Corollary 2.19. X is strongly K(n)*acyclic if and only if X is K(n)*
acyclic and also Lfn1X is strongly K(n)*acyclic. In particular, if X is
K(n)*acyclic, and also T (i)*acyclic for 1 i n  1, then X is strongly
K(n)*acyclic.
Our next result is a homological variant of Proposition 2.6.
Proposition 2.20. Let F : S ! K(n)*modules be any functor preserving
K(n)*isomorphisms. Then any natural transformation T of the form
T (X) : F (X) ! K(n)*( 1 X)
factors uniquely through sn*.
The T (n)* variant of this proposition also holds. Similarly, there is also
a E*nvariant that says that s*nis the initial functor from E*n( 1 X) to a
functor preserving E*nisomorphisms.
The next theorem has Theorem 2.12(2) as a consequence.
Theorem 2.21. Let X 2 ~SKn, i.e. Lfn1X is strongly K(n)*acyclic. Then
each of the maps
K(n)*( 1 X) ! K(n)*( 1 X)
is an inclusion of a normal subK(n)*Hopf algebra. This induces a decreas
ing filtration of finite length on K(n)*( 1 X), and there is an isomorphism
of filtered K(n)*Hopf algebras,
n+1O
K(n)*( 1 X) ' K(n)*(PX) K(n)*(K(ßj(X), j)),
j=0
that is natural on the level of associated graded objects.
Example 2.22. When n = 1, the theorem says that for all X, there is an
isomorphism of K(1)*Hopf algebras
O2
K(1)*( 1 X) ' K(1)*(PX) K(1)*(K(ßj(X), j)).
j=0
Example 2.23. When n = 2, the theorem says that, if K(1)*(X) = 0, then
there is an isomorphism of K(2)*Hopf algebras
O3
K(2)*( 1 X) ' K(2)*(PX) K(2)*(K(ßj(X), j)).
j=0
14 KUHN
Example 2.24. Suppose Lfn1X is strongly K(n)*acyclic, and also X is
nconnected with ßn+1(X) torsion. The theorem implies that
(2.3) K(n)*( 1 X) ! K(n)*( 1 X)
is an isomorphism.
When n = 1, the first hypothesis is always satisfied, and we recover [B1 ,
Thm. 2.4]. This is the key technical theorem of Bousfield's paper. In
recent email to the author, Bousfield has observed that (2.3) allows for an
addendum to [B7 , Thm. 8.1], analogous to the use of [B1 , Thm. 2.4] in the
proof of [B1 , Thm. 3.2].
Some hypotheses are necessary here. In [B7 , x8.7], Bousfield notes that if
X is the suspension spectrum of the Moore space M(Z=p, 3), then
K(2)*( 1 X<3>) ! K(2)*( 1 X)
has nonzero kernel.
Example 2.25. Let k(n)c = 1 ck(n), where k(n) is the connective cover
of K(n). k(n) 2 S~Kn, as it is a BP module that is K(i)*acyclic for 1
i n  1. Thus the theorem applies to say that the cofibration sequence of
spectra
n2 v
2p k(n) ! k(n) ! HZ=p
induces a short exact sequence of commutative, cocommutative K(n)*Hopf
algebras
K(n)*(k(n)2pn2+c) ! K(n)*(k(n)c) ! K(n)*(K(Z=p, c))
for all c > 0.
This is [BKW , Thm.1.1].
These last two examples also illustrate our next result.
Theorem 2.26. Suppose f : X ! Y is map of 0connected Smodules
with cofiber C, such that P(f)* : K(n)*(PX) ! K(n)*(PY ) is monic. (For
example, f might be a K(n)*isomorphism.) If X 2 SKn then there is a
short exact sequence of commutative, cocommutative K(n)*Hopf algebras
K(n)*( 1 X) ! K(n)*( 1 Y ) ! K(n)*( 1 C).
Remarks 2.27. It seems appropriate to comment on other results in the
literature that concern homological calculations of the sort we look at here.
In unpublished work, but in the spirit of [Str], N. Strickland has observed
that E*n(PX) is a functor of E*n(X), when restricted to X such that E*n(X) is
appropriately `profree' as an E*nmodule, and also E*n(X) is concentrated
in even degrees. It strikes the author as likely that the second of these
hypotheses is unnecessary, since the work of either J. McClure [BMMS ,
Chapter IX] or Bousfield [B4 ] shows this to be the case when n = 1.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 15
Related to this, it appears that when K(n)*(X) and K(n)*(Y ) are con
centrated in even degrees, if f : X ! Y is a K(n)*monomorphism, then so
is P(f).
In various papers culminating in [Ka ], Kashiwabara computes E*( 1 X)
if E = BP, En, or K(n), under suitable side hypotheses on X, e.g. if X
is 1connected with cells only in even dimensions. His methods are very
different than ours; in particular, he heavily uses BP Adams resolutions
of his spectra. Most of his results appear to have only limited naturality,
and many of his results are specialized to the case when X is a suspension
spectrum.
There is most obvious overlap between our results and those of Bousfield,
particularly in [B7 ]. It seems that any proof of our main theorem, Theo
rem 2.5, will depend crucially on the existence of a Goodwillie tower for
1 1 X. But once this theorem has been established, many of our other
results allow for alternate proofs using his work. See Appendix B.
2.7. The main theorem for rational homology. For completeness, we
note that a version of our main theorem holds when n = 0, i.e., with LHQ
replacing LT(n).
Analogous to (2.2), we have the following lemma: for all 0connected
spectra X, there is a natural factorization by weak Smodule maps
(2.4) LHQ771 1 XO
''0(X)pppp OOOLHQffl(X)O
pppp OOO
ppp OOO''
LHQ X _________________________LHQ X.
For X just 1connected, such a factorization also exists, but can not be
made to be natural.
j0(X) then induces a weak map s0(X) : LHQ P(X) ! LHQ 1 ( 1 X)+ in
LHQ (Alg ), natural for 0connected X. Proposition 2.3 combines with the
lemma to prove:
Theorem 2.28. For all 1connected spectra X, the map s0(X) induces
an isomorphism on LHQ completions, and thus there is a factorization of
weak algebra maps
LHQ P(X) __________canonical_______//_bLHQP(X).
PPPP mmm66m
PPPP mmmm
s0(X)PPP((PP mmmm t0(Z)
LHQ 1 ( 1 X)+
This is natural for 0connected X, and, in this case, all three maps are
equivalences.
16 KUHN
As a corollary, one recovers the known theorem: for 0connected spectra
X, the Hopf algebra ßS*( 1 X) Q is naturally isomorphic to the graded
symmetric algebra primitively generated by ß*(X) Q.
2.8. Organization of the paper. In x3, we develop the general theory
of Andr'eQuillen towers associated to commutative augmented Salgebras,
leading to proofs of Proposition 2.1 and Proposition 2.3. Included is a
subsection summarizing the basic properties of the functor 1 ( 1 X)+ .
The splitting theorems concerning 1 ( 1 X)+ , Theorems 2.2, 2.5, and
2.28, are then easily proved in x4, which includes discussion of (2.2) and
(2.4), and Theorem 2.4.
In x5, we explore the extent to which sn(X) is an equivalence, proving
the results in x2.5 and x2.6. Some of the proofs are a bit long and delicate:
we hope we have made them comprehensible.
In Appendix A, we check that our stable splitting of QZ agrees with
others in the literature. In Appendix B, we compare our constructions and
theorems to those of [B7 ]. In Appendix C, we compare s to sn, and make
some remarks about JamesHopf invariants.
3.The Andr'eQuillen tower of commutative algebras
3.1. Categories of commutative Salgebras. We work always within
the topological model category of Smodules as in [EKMM ]. This is a
symmetric monoidal category with unit the sphere spectrum S, and we recall
that associative, commutative, unital Salgebras are modern day versions of
E1 algebras. Given such an algebra R, we let R  Alg denote the category
of associative, commutative, unital, augmented Ralgebras. When R = S
we simplify the notation to Alg.
Given A 2 RAlg , let I0(A) denote the fiber of the augmentation A ! R.
As discussed in [Ba ], the functor I0 takes values in the category R  Alg0of
associative, commutative, nonunital Ralgebras, and is the right adjoint of
a Quillen equivalence between the model categories R  Alg and R  Alg0.3
We let I(A) denote a cofibrant replacement in R  Alg0of I0(A0), where A0
is a fibrant replacement of A 2 Alg.
The categories R  Alg and R  Alg0 are tensored and cotensored over
based topological spaces. (See [EKMM ] or [K5 ].) In particular, given I 2
R  Alg0, one can form the iterated suspension Sn I and the iterated
looping nI.
We note that both n (as a right adjoint) and homotopy colimits over
directed systems (see, e.g. [EKMM , Lemma VII.3.10]) commute with the
forgetful functor from R  Alg0to Rmodules.
The processes of taking coproducts and suspending (tensoring with S1)
certainly don't commute with this forgetful functor. Indeed, the coproduct
____________
3The left adjoint sends a nonunital Ralgebra I to the augmented algebra I0_*
* R.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 17
in R  Alg is the smash product, and thus in R  Alg0one has
I q J = I ^ J _ I _ J.
Regarding suspension, one has the following lemma, which serves as the ba
sis for many convergence results.
Lemma 3.1. If a cofibrant I 2 Alg0 is nconnected, then the natural map
I ! (S1 I)
is 2n + 1connected.
We feature two families of examples.
Example 3.2. Given an Smodule X, we let PX denote the free commuta
tive Salgebra generated by X [EKMM , p.40]. If X is cofibrant then there
is a natural weak equivalence [EKMM , p.64]:
1`
PX ' DrX.
r=0
PX is naturally augmented, and we let i : X ! I(PX) denote the natural
weak map. Using the freeness of PX, given any A 2 Alg, a weak map of
Smodules f : X ! I(A) induces a weak map in Alg , f~: PX ! A such
that the diagram of weak maps
X II
IIfI
i III
fflfflII$$(f~)
I(PX) _____//I(A)
commutes.
Given a commutative Salgebra R, and an Rmodule Y , there is an
analogous free object PR Y 2 R  Alg, satisfying the evident `change of
rings' formula
PR (R ^ X) = R ^ PX.
Example 3.3. Given an Smodule X, the E1 structure on the infinite
loopspace 1 X implies that 1 ( 1 X)+ takes values in Alg . See [M2 ,
Ex.IV.1.10] and [EKMM , xII.4]. The composite
I( 1 ( 1 X)+ ) ! 1 ( 1 X)+ ! 1 ( 1 X)
is a weak equivalence for all X. The natural map X<1> ! X of Smodules
induces an equivalence in Alg:
1 ( 1 X<1>)+ ~! 1 ( 1 X)+ .
18 KUHN
3.2. Topological Andr'eQuillen homology. One version of the Topo
logical Andr'e Quillen Homology of A 2 R  Alg is as the set of homotopy
groups of the following construction.
Definition 3.4. Given A 2 R  Alg, let taqR (A) 2 R  Alg0be defined by
taqR (A) = hocolimn!1 n(Sn I(A)).
Remark 3.5. An alternative construction, more reminscent of the work of
Andr'e and Quillen is to let taqR (A) = ZQ(A), where Q : Alg ! Rmodules
is defined by
Q(A) = I(A)=I(A)2,
and Z : R  modules ! Alg0is defined by letting Z(X) be a fibrant replace
ment of X given trivial multiplication. This is the construction explored by
M.Basterra in [Ba ], and she and M.Mandell have an unpublished proof that
these two constructions are equivalent.4 For our purposes, particularly as in
Example 3.9 below, the construction we use is most convenient.
We denote taqS(A) by taq(A). The following `change of rings' formula
follows easily from the definition.
Lemma 3.6. For all A 2 Alg, there is a natural weak equivalence in RAlg 0
R ^ taq(A) ' taqR (R ^ A).
Another basic property that we will use is the following.
Lemma 3.7. taqR takes homotopy cofibration sequences in R  Alg to ho
motopy cofibration sequences in Rmodules.
Proof.As will be elaborated on in x3.6, the functor taqR , defined via sta
blization as above, will be 1excisive in the sense of Goodwillie: it will
take homotopy pushout squares to homotopy pullback squares. But in R
modules, homotopy pullbacks are homotopy pushouts.
We now calculate taq(A) for our two key examples.
Example 3.8. Corresponding to Example 3.2, we claim that there is a
natural equivalence
taq(PX) ' X,
where X has trivial multiplication, such the natural map I(PX) ! taq(PX)
corresponds to the projection
`1
DrX ! D1X = X.
r=1
To see this, we make two observations.
Firstly, for all based spaces K and Smodules X, there is a natural iso
morphism
K P(X) = P(K ^ X).
____________
4The proof of the main theorem of [BM ] indicates some of the ideas, as does*
* S. Schwede's
earlier paper [Sch].
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 19
Secondly, for all Smodules X there are natural equivalences
(
X ifr = 0
hocolimn!1 nDr( nX) '
* ifr > 0.
(This is clear by a connectivity argument if X is connective, and then note
that an arbitrary Smodule is equivalent to a hocolimit of connective S
modules.)
Combining these observations, we compute:
taq(PX) = hocolimn!1 n(Sn I(PX))
= hocolimn!1 nI(P nX)
1`
' hocolim nDr( nX)
r=1 n!1
' X.
Example 3.9. If X is a 1connected Smodule, corresponding to Exam
ple 3.3, we claim that there is a natural weak equivalence
taq( 1 ( 1 X)+ ) ' X,
such that the natural map I( 1 ( 1 X)+ ) ! taq( 1 ( 1 X)+ ) corresponds
to the counit
ffl : 1 1 X ! X.
To see this, we again make a couple of observations.
Firstly, if Z is an E1 space, let BZ be the associated classifying space
[M1 ]. As surveyed in [K5 ], there are natural weak equivalences
S1 I( 1 Z+ ) ' I( 1 BZ+ )
such that the natural map I( 1 Z+ ) ! (S1 I( 1 Z+ )) corresponds to
1 Z ! 1 BZ.
Secondly, given a 1connected Smodule X, let Xn = 1 nX. Then
there is a natural equivalence BXn ~!Xn+1, and a natural weak equivalence
hocolimn!1 n 1 Xn ~!X
such that the inclusion of 1 X0 into the hocolimit corresponds to ffl.
Combining these observations, we compute:
taq( 1 ( 1 X)+ )= hocolimn!1 n(Sn I( 1 (X0)+ ))
' hocolimn!1 nI( 1 (BnX0)+ )
' hocolimn!1 nI( 1 (Xn)+ )
' hocolimn!1 n 1 Xn
' X.
20 KUHN
3.3. The Andr'eQuillen tower of an augmented Ralgebra. It seems
that various people have noted the existence of an `Andr'eQuillen tower'
associated to A 2 R  Alg. Intuitively, this tower is supposed to be the
augmentation ideal tower
. .!.A=Ir ! . .!.A=I2 ! A=I
constructed in a homotopically meaningful way. The next theorem lists the
properties we care about.
In this theorem, DRrY denotes the rth extended power construction in
the category of Rmodules, i.e. one uses the smash product ^R . Note that
there is an isomorphism R ^ Dr(X) = DRr(R ^ X).
Theorem 3.10. Given A 2 R  Alg, there is a unique natural tower of
fibrations in R  Alg under A
(3.1) ...


fflffl
PR,2A;;
ww
wwww p2
e wwww fflffl
w2ww kPR,1A55
www kkkk
wwwkke1kkkkkw p1
kkkkkkkwwe0ww fflffl
A ________________//_PR,0A,
with the following properties.
(1) PR,0A ' R so that e0 identifies with the augmentation.
(2) For r 1, the fiber of pr : PR,rA ! PR,r1A is naturally weakly equiv
alent to DRr(taqR (A)). Furthermore, I(e1) identifies with the natural map
I(A) ! taqR (A).
(3) Denoting PS,r(A) by Pr(A), there is a change of rings formula: Given
A 2 Alg and R a commutative Salgebra, there is a natural weak equivalence
of towers under R ^ A:
R ^ Pr(A) ' PR,r(R ^ A).
(4) If I(A) is 0connected, then er is rconnected.
The uniqueness statement means up to natural weak equivalence. The
weak equivalence in the second property is as Rmodules.5
____________
5Though we won't need nor prove it, if one gives DRr(taqR(A)) trivial multip*
*lication,
this equivalence is even as objects in R  Alg0.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 21
V.Minasian constructs a tower with these properties in the preprint [Min ],
following along the lines of [Ba ], and using her version of taqR (A).6 How
ever, the appearance (finally) of a finished version of [G3 ] allows the author
to feel comfortable with an alternative construction, suggested to him by
G.Arone.
Definition 3.11. Let the tower {PR,r( )} denote the Goodwillie tower as
sociated to the identity functor on R  Alg.
In the subsection x3.6 we sketch a proof that this tower has the properties
stated in the theorem. Assuming the theorem, we now follow up with some
consequences and an example.
Definition 3.12. Let bA= holimr!1PR,rA.
Corollary 3.13. If I(A) is 0connected, then the natural map A ! Ab is
an equivalence.
Example 3.14. Corresponding to Example 3.2, we have
1` 1`
(3.2) Pr(PX) ' ( DqX)=( DqX)
q=0 q=r+1
so that there is a natural equivalence
1Y
bPX ' DrX.
r=0
One way to see this is to note that both sides of (3.2), viewed as towers of
functors from Smodules to Alg, have the correct form to be a Goodwillie
tower of the functor sending X to PX, and thus agree, up to natural weak
equivalence.
The following lemma is well known and much used.
Lemma 3.15. The functor DRrpreserves weak equivalences of Rmodules.
Proof.Both ^R and ( )h r (homotopy rorbits) preserve weak equiva
lences.
Proposition 3.16. Let f : A ! B be a map in R  Alg. If taqR (f) :
taqR (A) ! taqR (B) is a weak equivalence, so is PR,r(f) : PR,r(A) !
PR,r(B) for all r, and thus also bf: bA! bB. Thus, in this case, one gets a
____________
6In email with the author, M.Mandell has also sketched this result.
22 KUHN
factorization by weak Ralgebra maps
___canonical//_
A >> ?bA,?
""
>>> ""
f >>OEOE>""""
B
bf
where the unlabelled weak map is B ! bB~ Ab.
Proof.Using the lemma, one proves this by induction up the Andr'eQuillen
tower.
This proposition specializes to Proposition 2.1 when R = S.
3.4. A summary of the properties of P(X) and 1 ( 1 X)+ . In this
subsection, we use our work thus far to summarize basic properties of P(X)
and 1 ( 1 X)+ , viewed as functors from Smodules to Alg.
Proposition 3.17. The functor P satisfies the following properties.
(1) P takes homotopy colimits in SmodulesWto homotopy colimits in Alg.
(2) I(P(X)) ! taq(P(X)) identifies with 1r=1DrX ! D1X = X. Thus
the composite X i!I(P(X)) ! taq(P(X))Wis an equivalence.Q
(3) P(X) ! bP(X) identifies with 1r=0DrX ! 1r=0DrX.
Proof.The first property follows formally from the fact that P is left adjoint
to the forgetful functor from Alg to Smodules, since the model category
on Alg is defined so that algebra maps are fibrations or weak equivalences
exactly when they are fibrations or weak equivalences when considered as
maps of Smodules. The other two properties were established above in
Example 3.8 and Example 3.14.
Proposition 3.18. The functor 1 ( 1 )+ satisfies the following proper
ties.
(1) 1 ( 1 )+ takes filtered homotopy colimits in Smodules to filtered
homotopy colimits in Alg.
(2) 1 ( 1 )+ takes coproducts in Smodules to coproducts in Alg.
(3) If X ! Y ! Z is a cofibration sequence Smodules, with X and Y
1connected and Z 0connected, then
1 ( 1 X)+ ) ! 1 ( 1 Y )+ ) ! 1 ( 1 Z)+ )
is a cofibration sequence in Alg.
(4) I( 1 ( 1 X)+ ) ! taq( 1 ( 1 X)+ ) identifies with ffl : 1 1 X !
X<1>.
Proof.The last property was established above in Example 3.9.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 23
To see that the first property holds, we first note that filtered homotopy
colimits in Alg are detected by viewing them as being in Smodules. (Com
pare with [EKMM , xII.7].) But, as a functor to Smodules, 1 ( 1 )+
certainly commutes with filtered homotopy colimits.
Thanks to the first property, it suffices to prove the second property for
finite coproducts. In Alg, we have equivalences
1 ( 1 (X _ Y ))+ ~! 1 ( 1 (X x Y ))+ = 1 ( 1 X)+ ^ 1 ( 1 Y )+ ,
which is the coproduct in Alg of 1 ( 1 X)+ and 1 ( 1 Y )+ .
The proof of the third property is more delicate. A cofiber sequence of
Smodules
X f!Y g!Z
will induce a commutative diagram in Alg:
1 ( 1 X)+ ____//_ 1 ( 1 Y )+________//_A
  
  h
  fflffl
1 ( 1 X)+ ____//_ 1 ( 1 Y )+____// 1 ( 1 Z)+
where A is the cofiber in Alg of 1 ( 1 f)+ . We wish to show the map h is
an equivalence.
If X, Y , and Z, are all 1connected, then applying taq to this diagram
yields the diagram
f
X _____//Y_____//taq(A)
  
  taq(h)
 f  g fflffl
X _____//Y_______//Z
where we have used property (4) above. The bottom horizontal sequence
is given as a cofibration sequence; by Lemma 3.7, so is the top horizontal
sequence. We conclude that taq(h) is an equivalence. By Proposition 2.1,
we conclude that bhis an equivalence. Under our connectivity hypothesis
that Z is also 0connected (so that ß0(f) is onto), one can deduce that
both A and 1 ( 1 Z)+ have 0connected augmentation ideals, and thus
are equivalent to their completions, by Theorem 3.10(4). We conclude that
h is an equivalence.
Remark 3.19. If one regards 1 1 X+ just as a functor taking values in S
modules, rather than in Alg, the fact that its Goodwillie tower has rth fiber
equivalent to Dr(X) has been known for awhile by Goodwillie and others
working with the calculus of functors. This tower appears explicitly in the
literature in [AK ].
24 KUHN
3.5. The localized tower. If E is an Smodule, let LE denote Bousfield
localization with respect to E*. It has long been usefully observed that
various constructions in infinite loopspace theory behave well with respect
to Bousfield localization. For example, [K1 ] heavily used the follow analogue
of Lemma 3.15.
Lemma 3.20. [K1 , Cor.2.3] The functor DRrpreserves E*isomorphisms.
Proof.Both ^R and ( )h r preserve E*isomorphisms.
This same fact is behind the beautiful and much more recent theorem
that if R is a commutative Salgebra, so is LE R [EKMM , Chap.VIII].
Lemma 3.21. The functor taqR preserves E*isomorphisms.
Proof.[K5 , Cor.7.5] says that taqR (A) is the colimit of an increasing fil
tration F1taqR (A) ! F2taqR (A) ! . . .by cofibrations, and identifies the
cofibers: there is an equivalence
FdtaqR (A)=Fd1taqR (A) ' ( Kd ^R I(A)^Rd)h d,
where Kd is a certain partition complex appearing in [AM ]. The functor on
the right of this equivalence certainly preserves E*isomorphisms, and thus
so does taqR (A).
The lemmas combine with induction up the Andr'eQuillen tower to prove
Corollary 3.22. The functors PR,r preserve E*isomorphisms.
Definition 3.23. Let LE (Alg ) be the full subcategory of LE S  Alg con
sisting of Elocal objects.
In the spirit of these last results, we have the following proposition.
Proposition 3.24. LE : Alg ! LE (Alg ) commutes with homotopy pushout
squares and filtered homotopy colimits in the following sense:
(1) the natural map
LE (hocolim{B A ! C}) ! LE (hocolim{LE B LE A ! LE C})
is an equivalence, and
(2) the natural map LE (hocolimiAi) ! LE (hocolimiLE Ai) is an equiva
lence.
Proof.As discussed on [EKMM , p.162], a model for the pushout of a diagram
of Ralgebras of the form B A ! C, where both maps are cofibrations,
is given by an appropriate bar construction fiR (B, A, C). This construction
preserves E*isomorphisms in all variables; in particular
LE (fiS (B, A, C)) ! LE (fiLES (LE B, LE A, LE C))
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 25
is an equivalence, establishing (1). The proof of (2) is similar.
Remark 3.25. It is easy to see that B 2 LE S  Alg is Elocal if and only if
it is weakly equivalent LE A, for some A 2 Alg. More precisely, B is Elocal
if and only if the natural weak map LE (I(B) _ S) ~!B is an equivalence.
Thus LE (Alg ) is equivalent to the category L0E(Alg ) with objects A 2 Alg,
and with morphisms from A to B equal to the LE S  Alg maps from LE A
to LE B.
Definitions 3.26. We define functors with domain L0E(Alg ) as follows.
(1) Let taqE (A) = LE (taqLES (LE A)).
(2) Define the natural tower of fibrations in LE (Alg ) under LE A,
(3.3) ...


fflffl
P2EA<<
xx
xxxx pE2
eE2 xxx fflffl
xxxx l5P1EA5
xxxeE1llllllx
xxx lllll pE1
lllllllxeE0xx fflffl
A ________________//_P0EA,
to be the tower obtained by applying LE to the Andr'eQuillen tower
(3.1) {PLES,r(LE A)}.
(3) Let bLEA = limr!1PrE(A).
We have the following analogue of Proposition 3.16. This proposition is
a slight elaboration of Proposition 2.3 of the introduction.
Proposition 3.27. Let f : LE A ! LE B be a map in LE (Alg ). If taqE (f) :
taqE (A) ! taqE (B) is a weak equivalence, so is PrE(f) : PrE(A) ! PrE(B)
for all r, and thus also fb: bLEA ! bLEB. Thus, in this case, one gets a
factorization by weak LE Salgebra maps
LE AF____canonical__//_bLEA,
FF w;;w
FFF www
f FF##F www
LE B
bf
where the unlabelled weak map is LE B ! bLEB ~ bLEA.
26 KUHN
Proof.This is proved by induction on r, using Lemma 3.20 and Theo
rem 3.10(2).
This is given added punch when combined with the next proposition.
Proposition 3.28. Given A 2 Alg there are natural weak equivalences
(1) taqE (A) ' LE (taq(A)).
(2) {PrE(A)} ' {LE Pr(A)}, as towers.
Proof.First note that if X is an Smodule, then each of the maps
X ! LE S ^ X ! LE X
is an E*isomorphism.
To prove (1), we have
taqE (A)= LE (taqLES (LE A))
' LE (taqLES (LE S ^ A)) by Lemma 3.21
' LE (LE S ^ taq(A)) by Lemma 3.6
' LE (taq(A)).
To prove (2), we have
PrE(A)= LE (PLES,r(LE A))
' LE (PLES,r(LE S ^ A)) by Corollary 3.22
' LE (LE S ^ Pr(A)) by Theorem 3.10(3)
' LE (Pr(A)).
In constrast to the equivalences in this last proposition, we note that it
is not necessarily true that the natural weak map
LE (Ab) ! bLE(A)
is an equivalence (particularly when E is not connective), and thus the
convergence of the localized tower is very problematic. For example, if
A = PX, this map has the form
_ 1 !
Y 1Y
LE DrX ! LE DrX,
r=0 r=0
which would usually not be an equivalence.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 27
3.6. Proof of the properties of the Andr'eQuillen tower. In the se
ries of papers [G1 , G2 , G3 ], Tom Goodwillie has developed his theory of
polynomial resolutions of homotopy functors. Although [G3 ] only explicitly
studies such resolutions of functors
F : A ! B
with A and B either spaces or spectra, essentially everything in his paper
makes sense in a much broader setting. In particular, his concepts and con
structions certainly make sense if A and B are (based) topological model
categories (a model category tensored over based topological spaces, sat
isfying properties as in [EKMM , VII.4]), and B furthermore is a category
in which a directed hocolimit of homotopy cartesion cubical diagrams is
again a homotopy cartesion cubical diagram. One such category is R  Alg.
Another is R  Mod , the category of Rmodules.
Recall that the tower {PR,r( )} is defined to be the Goodwillie tower
associated to the identity functor on R  Alg. In this subsection we in
dicate why this tower has the properties given in Theorem 3.10. We do
this by summarizing the main points of Goodwillie's work as they apply to
Theorem 3.10. Throughout we are citing the version of [G3 ] of June, 2002.
As in [G2 ], a functor is said to be rexcisive if it takes strongly homotopy
cocartesion (r+1)cubical diagrams to homotopy cartesian cubical diagrams.
In [G3 ], given a functor F , the tower {PrF ( )} is defined so that F ! PrF
is the universal arrow to an rexcisive functor, up to weak equivalence.
Goodwillie proves the existence of such a tower by an explicit construc
tion which amounts to modifying F so as to visibly force certain strongly
homotopy cocartesion (r + 1)cubical diagrams to transform to homotopy
cartesian diagrams. Readers looking to apply his paper in the setting where
the domain and range of F are topological model categories should write
`U+ X' whenever Goodwillie writes `X x U', with U a finite set, and also
recall that the domain category has an initial/terminal object. [G3 , The
orem 1.8] says that the tower constructed as he describes has the desired
universal properties.
For example, there is a strongly cocartesion diagram
S0 _____//_D+
 
 
fflffl fflffl
D _____//S1,
representing the circle as the union of two 1disks D+ and D . Then
P1F (X) is defined to be the homotopy colimit of
F (X) ! T1F (X) ! T1T1F (X) ! . . .
28 KUHN
where T1F (X) is the homotopy pullback of
F (D+ X)


fflffl
F (D X) _____//F (S1 X).
Since D+ and D are contractible, F (D+ X) and F (D X) are equivalent
to the initial/terminal object in the domain category, so that
T1F (X) ' F (S1 X)
and
P1F (X) ' hocolimn!1 nF (Sn X).
Already, just using this part of the theory, various parts of Theorem 3.10
are evident. Statement (1), saying that PR,0A ' R, is clear. Statement (3),
the change of rings formula, is also clear, noting that R ^ ( ) takes strongly
homotopy cocartesion cubes of Salgebras to strongly homotopy cocartesion
cubes of Ralgebras, and homotopy cartesion cubes of Salgebras to homo
topy cartesion cubes of Ralgebras (see below). Finally, part of statement
(2), that the fiber of p1 : PR,1A ! PR,0A is naturally equivalent to taqR (A),
follows from the above description of P1F .
In [G2 , G3 ], Goodwillie develops general theory and examples allowing
for connectivity estimates to be made for the maps F (X) ! PrF (X) in
terms of the connectivity of X. In particular, statement (4), stating that
er : A ! PR,rA is rconnected if I(A) is 0connected, can be deduced from
Lemma 3.1.
We are left needing to show the rest of statement (2): that DR,r(A), the
fiber of pr, is weakly equivalent to the rth extended power of taq(A), the
fiber of p1.
To show this, we begin by noting that homotopy pullback diagrams in
R  Alg are just diagrams in R  Alg that are homotopy pullbacks in R
modules. Thus the tower {PR,r( )}, with algebra structures forgotten, is
the Goodwillie tower of the inclusion functor
I : R  Alg ,! R  Mod .
The category R  modules is a stable model category, in the sense of
[H2 ]; in particular, homotopy cocartesian cubical diagrams are equivalent
to homotopy cartesion cubical diagrams. Thus one can apply Goodwillie's
analysis in [G3 ] of how DrF (A), the fiber of PrF (A) ! Pr1F (A), can be
computed by means of cross effects.
The bits of the general theory we need are the following. Let
F : A ! B
be a functor between topological model categories as above, with B stable.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 29
Let r = {1, 2, . .,.r}. In [G3 , x3], ØrF , the rth cross effect of F , is de*
*fined
to the the functor of r variables given as the total homotopy fiber
a
(3.4) ØrF (A1, . .,.Ar) = TotFib F ( Ai).
T r i2rT
Then [G3 , Theorem 6.1] says that D(r)F , the rth multilinearization of F ,
can be computed by the formula
(3.5)D(r)F (A1, . .,.Ar) ' hocolimn n1+...+nrØrF (Sn1 A1, . .,.Snr Ar).
i!1
Finally, [G3 , Theorem 3.5] says that there is a natural weak equivalence
(3.6) DrF (A) ' (D(r)F (A, . .,.A))h r.
We apply this theory to the case when F = I.
Since the coproduct in R  Alg is just the smash product ^R , we have
^
ØrI(A1, . .,.Ar) = TotFib( Ai).
T r i2rT
For example, Ø2(A, B) is the total homotopy fiber of the square
A ^R B _____//A
 
 
fflffl fflffl
B ________//R
Recall that the total homotopy fiber is isomorphic to the iterated homo
topy fiber. This makes the next lemma easy to check.
Lemma 3.29. The natural map
I(A1) ^R . .^.RI(Ar) ! ØrI(A1, . .,.Ar)
is an equivalence.
Corollary 3.30. There are natural weak equivalences of Rmodules
D(r)I(A1, . .,.Ar) ' taq(A1) ^R . .^.Rtaq(Ar),
and
DrI(A) = (taq(A)^Rr)h r.
We have finished our proof of Theorem 3.10, as this last equivalence is
just a restatement of the remaining unproven part of statement (2).
4.Proof of Theorem 2.2, Theorem 2.4, Theorem 2.5, and
Theorem 2.28
In this section we use the theory developed in the last section to prove
the splitting theorems of the introduction.
30 KUHN
4.1. Proof of Theorem 2.2.
Definition 4.1. If Z is a space, let s(Z) : P( 1 Z) ! 1 (QZ)+ be the
natural weak map in Alg induced by the weak natural map of Smodules
1 ''(Z) 1 ~ 1
1 Z ! QZ  I( (QZ)+ ).
We restate Theorem 2.2.
Theorem 4.2. For all spaces Z, the map s(Z) induces an isomorphism on
completions, and thus there is a natural factorization of weak algebra maps
P( 1 Z) _______canonical____//_bP( 1 Z).
MMM qqq88
MMMM qqq
s(Z)MMM&&M qqqqt(Z)
1 (QZ)+
If Z is 0connected then all of these maps are weak equivalences.
Proof.By Proposition 3.16, we just need to show that
taq(s(Z)) : taq(P( 1 Z)) ! taq( 1 (QZ)+ )
is an equivalence. To see this, consider the diagram:
1ZCXXXXXX___________________________________________51Z5l
 CCC XXXXXXXX''X ffllllll 
 CC XXXXXXX llll 
 CCC XXXXXXX llll 
 CiC X,,1 
 CC QZOO 
 CCC  
o CCCC o o
 !! I(s(Z))  
 I(P( 1 Z)) _____//I( 1 (QZ)+ ) 
 nn RRR 
 nnnn RRRR 
 nnnn RRRRR 
fflfflvvnn taq(s(Z)) R(( fflffl
taq(P( 1 Z)) _______________________________________//_taq( 1 (QZ)+ )
Proposition 3.17(2) says that the left edge is an equivalence, and the left
triangle commutes. Similarly, Proposition 3.18(4) shows that the right edge
is an equivalence, and the right quadrilateral commutes. Naturality shows
that the bottom quadrilateral commutes. The map s(Z) was defined so
that the middle quadrilateral commutes. Finally, the top triangle is just the
categorical factorization (2.1). Thus the diagram commutes, and inspection
of the outside square shows that taq(s(Z)) can be identified with the identity
map on 1 Z.
Remarks 4.3. When Z is connected, the realization that a weak equivalence
like s(Z) can be taken to be E1 dates back to the late 1970's, with the
first proof based on a point set analysis of JamesHopf maps [CMT ]. Proofs
working on the spectrum level were given in [LMMS , Thm.VII.5.5] and [K1 ,
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 31
Prop.4.3]. Both of these references construct s(Z) using ideas of [C ]: see
[K4 , Appendix B] for an updated account.
If Z is not 0connected, then the fact t(Z) can be constructed to be E1
seems to be new, though slightly weaker results were proved in [CMT ]. We
also manage to show the existence of t(Z) without appealing to properties
of group completions: again this is new.
Our proof does not use the combinatorial approximation CZ ! QZ,
though some of the ideas behind that approximation are obviously lurking
in the proofs of needed properties of 1 (QZ)+ . We relate our constructions
to those using CZ in Appendix A.
4.2. Proof of Theorem 2.4. Suppose that Z is a connected space, and
f : 1 Z ! X a map inducing an isomorphism on E*homology. The maps
in Alg,
s(Z) 1 1 1 f 1 1
P( 1 Z) ! (QZ)+ ! ( X)+ ,
induce a commutative diagram
LE 1 ( 1 f)+
LE P( 1 Z) _____~_____//LE 1 (QZ)+___________//LE 1 ( 1 X)+
  
  
fflffl fflffl fflffl
bLEP( 1 Z) _____~_____//bLE 1 (QZ)+_____~_____//bLE 1 ( 1 X)+ .
In this diagram, the top left horizontal maps is an equivalence by Theo
rem 2.2, while the bottom left horizontal map is similarly an equivalence
using Proposition 2.3. The lower right horizontal map is an equivalence by
Proposition 2.3, since our hypothesis that f is an E*isomorphism implies
that taqE (f) is an equivalence.
As the left vertical map is an E*monomorphism, we conclude that so is
the upper right horizontal map. Otherwise said,
( 1 f)* : E*(QZ) ! E*( 1 X)
is monic.
4.3. Telescopic functors, and the proof of Theorem 2.5. We fix a
prime p and work plocally. For n 1, let K(n) be the nth Morava K
theory, and T (n) be the telescope of a vnself map of a finite complex of
type n. It is known that the Bousfield class of T (n) is independent of choice
of vnself map, and that , with equality holding if and
only if the Telescope Conjecture holds. (See [B4 ] for background and more
references.)
Theorem 4.4. There exists a functor n : Spaces ! Smodules and a
natural weak equivalence n 1 X ' LT(n)X.
With the result stated at the level of homotopy categories, and with
K(n) replacing T (n), this is the main theorem of [K3 ]. However the sorts
of constructions given there, and in [B1 ] (for n = 1), yield the theorem as
32 KUHN
stated. In particular, in the recent paper [B4 ], Bousfield proves the theorem
as stated using the model category of spectra of [BF ]. But this category is
known [SS ] to be Quillen equivalent to the Smodules of [EKMM ].
As a corollary, we obtain a proof of (2.2), which we restate here.
Corollary 4.5. (Compare with [K3 ].) There is a natural factorization by
weak Smodule maps
LT(n) 1 1 X
''n(X)oo77oo OOOLT(n)ffl(X)OO
oooo OOO
ooo OO''O
LT(n)X _________________________LT(n)X.
Proof.Apply n to the commutative diagram
(4.1) 1 1771 XO
''( 1 X)ooooo OOO1Offl(X)O
ooo OO
ooo OO''O
1 X ________________________ 1 X.
Definition 4.6. If X is an Smodule, let
sn(X) : LT(n)P(X) ! LT(n) 1 ( 1 X)+
be the natural weak map in LT(n)(Alg ) induced by the weak natural map
of Smodules
''n(X) 1 1 ~ 1 1
LT(n)X ! LT(n) X  LT(n)I( ( X)+ ).
We restate Theorem 2.5.
Theorem 4.7. For all spectra X, the map sn(X) induces an isomorphism
on LT(n)completions, and thus there is a natural factorization of weak al
gebra maps
LT(n)P(X) __________canonical________//bLT(n)P(X).
QQQQ mmmm66
QQQQ mmmm
sn(X)QQQQ((Q mmmm tn(Z)
LT(n) 1 ( 1 X)+
Proof.Denote LT(n) by L. By Proposition 3.27, we just need to show that
taq(sn(X)) : Ltaq(P(X)) ! Ltaq( 1 ( 1 X)+ )
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 33
is an equivalence. To see this, consider the diagram:
LX@@XXXXXXXX________________________________________LX55jjj
 @@@ XXXXXXjnXXXX Lffljjjjjj 
 @@ XXXXXXX jjjjj 
 @@i XX++ jj 
 @@ L 1 1 X 
 @@@ OO 
o @@@@ o o
 ØØ I(sn(X))  
 LI(P(X)) _____//LI( 1 ( 1 X)+ ) 
 pp SSS 
 ppp SSSSS 
 pppp SSSS 
fflfflwwpp taq(sn(X)) S))S fflffl
Ltaq(P(X)) _______________________________________//Ltaq( 1 ( 1 )+ )
The top triangle commutes by Corollary 4.5, and the map sn(X) was
defined so that the middle quadrilateral commutes. The rest of the diagram
commutes for the same reasons as in the proof of Theorem 2.2.
4.4. Rational spectra and the proof of Theorem 2.28. We restate
(2.4) as a lemma.
Lemma 4.8. For all 0connected spectra X, there is a natural factorization
by weak Smodule maps
LHQ771 1 XO
''0(X)pppp OOOLHQffl(X)O
pppp OOO
ppp OOO''
LHQ X _________________________LHQ X.
For X just 1connected, such a factorization also exists, but can not be
made to be natural.
Assuming this for the moment, we continue as we did in the last subsec
tion.
Definition 4.9. If X is a 1connected Smodule, let
s0(X) : LHQ P(X) ! LHQ 1 ( 1 X)+
be the weak map in LHQ (Alg ) induced by the weak natural map of S
modules
''0(X) 1 1 ~ 1 1
LHQ X ! LHQ X  LHQ I( ( X)+ ).
We restate Theorem 2.28.
Theorem 4.10. For all 1connected spectra X, the map s0(X) induces
an isomorphism on LHQ completions, and thus there is a factorization of
weak algebra maps
LHQ P(X) __________canonical_______//_bLHQP(X).
PPPP mmm66m
PPPP mmmm
s0(X)PPP((PP mmmm t0(Z)
LHQ 1 ( 1 X)+
34 KUHN
This is natural for 0connected X, and, in this case, all three maps are
equivalences.
The theorem follows from the lemma in the usual way.
Proof of Lemma 4.8. The idea behind this lemma is that passing to ho
motopy groups is a full and faithful process on the homotopy category of
HQlocal Smodules.
A version of the lemma then follows easily as follows. Let
j(X)* : ß*(X) ! ß*( 1 1 X)
be the map induced by the canonical inclusion
j(X) : 1 X ! Q 1 X.
Note that j(X)* is a map of abelian groups if * > 0, but is only a map of
sets if * = 0. However, for all * 0, we have that ffl(X)* O j*(X) is the
identity.
Thus if X is 0connected, there is a canonical natural homotopy class of
maps
j0(X) : LHQ X ! LHQ 1 1 X
realizing j(X)* Q.
If X is just 1connected, one can still choose a Qlinear section to
ffl(X)* Q, and then realize this, defining j0(X). However, this cannot
possibly be taken to be natural by the following argument, which the author
learned from Pete Bousfield. If j0(X) were natural, then the maps j0(HV ),
with V a Qvector space, would define a natural section to the augmentation
Q[V ] ! V
defined on the category of Qvector spaces. (Here Q[V ] denotes the Q
vector space with basis V .) But it is well known, and easily verified, that
there exist no nontrivial natural transformations V ! Q[V ].
A careful reader may be wondering if, in the 0connected case, one can
construct a natural weak section at the model category level, and not just
a natural section in the homotopy category. This is also possible: one can
apply [McC , Theorem 4], which implies that if P 1(X) is the codegree 1
approximation to LHQ ( 1 1 X) (in the sense of dual calculus), then the
composite
LHQffl(X)
P 1(X) ! LHQ 1 1 X ! LHQ X
is an equivalence for 0connected X.
5. When sn(X) is an equivalence, and related matters
Recall that Sn = {Smodules X  sn(X) is an equivalence} and that
SKn = {Smodules X  sKn(X) is an equivalence}. Recall also that c(n)
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 35
denotes the smallest integer c such that ~T(n)*(K(Z=p, c)) = 0. The anal
ogous integer associated to K(n)* is n + 1: by the calculations in [RW ],
~K(n)*(K(Z=p, c)) = 0 if and only if c n + 1.7
The starting point for the detailed results about Sn and SKn given in x2.5
is the following partial result.
Theorem 5.1. Let X be an Smodule such that Lfn1X ' *.
(1) X 2 Sn if X is c(n)connected.
(2) X 2 SKn if X is n + 1connected.
The proof of this theorem is slightly long and delicate. We organize it
into the following steps:
Step 1 We show that if F is a finite Smodule of type n, then dF 2 Sn
for d 0.
Step 2 Assuming Step 1, we show that if F is a 0connected finite S
module of type n, then c(n)F 2 Sn, and n+1F 2 SKn.
Step 3 We show that the classes Sn and SKn are closed under various con
structions.
Step 4 We show that, starting from the finite Smodules shown in Step
2 to be in Sn and SKn, one can build all X's as in Theorem 5.1 using the
constructions of Step 3.
These will be proved in the next four subsections. Then, armed with
Theorem 5.1, we will systematically work our way through the proofs of the
various results stated in subsections x2.5 and x2.6.
5.1. Proof of Theorem 5.1: step 1.
Lemma 5.2. Let Z be a space whose suspension spectrum is finite of type
at least n. Then, for d 0, LT(n)s( dZ) ' sn( 1 dZ).
Postponing the proof momentarily, we note the following corollary.
Corollary 5.3. Let F be a finite of type at least n. Then for d 0, sn( dF )
is an equivalence.
Proof.Replacing F by a high suspension of F if needed, we can assume that
F ' 1 Z, for some space Z. By the lemma, for large enough d, sn( dF ) will
be homotopic to LT(n)s( dZ), which is an equivalence by Theorem 2.2.
____________
7When p = 2, the reference is [JW , Appendix].
36 KUHN
Proof of Lemma 5.2. Let Z be a space. By construction, the two maps of
LT(n)Salgebras
sn( 1 Z), LT(n)s(Z) : LT(n)P( 1 Z) ! LT(n) 1 (QZ)+
will be homotopic if and only if the two maps of LT(n)Smodules
jn( 1 Z), LT(n) 1 j(Z) : LT(n) 1 Z ! LT(n) 1 QZ
are homotopic.
Now suppose that 1 Z has type at least n, and is at least 0connected.
Then 1 Z is T (i)*acyclic for 0 i n  1, and thus the same is true for
1 QZ, since by Theorem 2.2, 1 (QZ)+ ' P( 1 Z).
If a spectrum X is T (i)*acyclic for 0 i n  1, then LT(n)X '
LfnX ' LfnS ^ X. Applying this to our situation, we see that jn( 1 dZ)
and LT(n) 1 j( dZ) correspond to homotopy classes of maps of Smodules
xn(d), x(d) 2 [ 1 Z, LfnS ^ d 1 Q dZ].
We now show that, if d is very large, then x(d) = xn(d). This follows
from three observations.
Firstly, the naturality of j and jn ensures that x(d) maps to x(d + 1),
and xn(d) maps to xn(d + 1), under the homomorphism
[ 1 Z, LfnS ^ d 1 Q dZ] ! [ 1 Z, LfnS ^ (d+1) 1 Q d+1Z]
induced by the evaluation map Q dZ ! Q d+1Z.
Secondly, the colimit
colim[ 1 Z, LfnS ^ d 1 Q dZ]
d!1
can be identified with [ 1 Z, LfnS ^ 1 Z].
Thirdly, the key properties of j and jn, (2.1) and (2.2), imply that un
der this identification, colim x(d) and colimxn(d) each correspond to the
d!1 d!1
canonical element: the unit S ! LfnS smashed with the identity on 1 Z.
We conclude that colim(x(d)  xn(d)) is zero in the colimit, and thus
d!1
x(d)  xn(d) is zero at a finite stage of this colimit. Otherwise said, for d
large, we have x(d) = xn(d).
5.2. Proof of Theorem 5.1: step 2.
Proposition 5.4. Let F be a 1connected finite Smodule of type n + 1.
(1) ~T(n)*( 1 c(n)F ) = 0.
(2) K~(n)*( 1 n+1F ) = 0.
As a corollary, one deduces the assertion of step 2.
Corollary 5.5. Let F be a 0connected finite Smodule of type n.
(1) c(n)F 2 Sn.
(2) n+1F 2 SKn.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 37
Proof.Let F be 0connected and finite of type n. Let v : dF ! F be a
vnmap, an isomorphism in both T (n)* and K(n)*. Let W be the fiber.
By statement (1) of the proposition, 1 c(n)W is T (n)*acyclic. By an
AtiyahHirzebruchSerre spectral sequence argument, it follows that cv
induces an isomorphism
v* : T (n)*( 1 d+cF ) ~!T (n)*( 1 cF )
if c c(n).
Now consider the diagram
sn( dN+c(n)F)
LP( dN+c(n)F ) ___________//L 1 ( 1 ( dN+c(n)F ))+
 
 
fflffl sn( c(n)F) fflffl
LP( c(n)F )_______________//L 1 ( 1 c(n)F )+
where L = LT(n), and the vertical maps are induced by vN . The above
discussion implies that the right vertical map is an equivalence. The left
vertical map is an equivalence because v is a T (n)*isomorphism. Finally,
for large enough N the top horizontal map is an equivalence as a consequence
of Step 1. Thus the bottom map is an equivalence, proving statement (1) of
the corollary. Statement (2) is proved similarly.
To prove Proposition 5.4, we use ideas from [HRW ].
If HZ=p is strongly E*acyclic, let cp(E) be the smallest c such that
~E*(K(Z=p, c)) = 0. The two statements of Proposition 5.4 are then just
special cases of the following proposition.
Proposition 5.6. Suppose a 1connected spectrum X has ptorsion ho
motopy groups. If X is strongly E*acyclic, then ~E*( 1 cp(E)X) = 0.
Proof.We argue as in [HRW , x3].
Firstly, the argument proving [HRW , Prop.3.4] shows that if 1 cX is
E*acyclic, then so is 1 c(X ^ Y ) for any 1connected spectrum Y .
Now let statement (c) be the statement that E~*( 1 cX) = 0. By as
sumption, statement (c) is true if c is very large. We complete the proof of
the proposition by showing that, if c cp(E), then statement (c+1) implies
statement (c).
Consider the fibration sequence of Smodules
~H! S ! HZ.
This induces a fibration sequence of spaces
1 c+1(X ^ 1H~) ! 1 cX ! 1 c(X ^ HZ).
By our inductive assumption, and the observation above, the first of these
spaces is E*acyclic. Thus the second map is an E*isomorphism. But
1 c(X ^ HZ) will be a weak product of EilenbergMacLane spaces of
38 KUHN
type K(A, c) where A is ptorsion and c cp(n). Thus this space is also
E*acyclic, and we conclude the same for 1 cX.
5.3. Proof of Theorem 5.1: step 3. The following proposition says that
Sn and SKn are closed under various constructions.
Proposition 5.7. Let S0 be either Sn or SKn.
(1) X 2 S0 if and only if X<1> 2 S0.
(2) If X 2 S0, then ß0(X) = 0.
(3) Let X ! Y ! Z be a cofibration sequence of Smodules with X and Y
1connected and Z 0connected. Then X, Y 2 S0 implies that Z 2 S0. In
particular, if X 2 S0 is 1connected, then X 2 S0.
(4) Let X be the filtered homotopy colimit of Smodules Xi. If Xi 2 S0 for
all i, then X 2 S0.
Proof.We will prove the statements when S0 = Sn; the proofs when S0 = SKn
are similar. Let L denote LT(n).
The first two properties follow from the fact that EilenbergMacLane
spectra are T (n)*acyclic.
In more detail, X<1> ! X is a T (n)*isomorphism, and thus so is
P(X<1>) ! P(X). Also 1 X<1> = 1 X. Thus in the square
sn*(X<1>)
T (n)*(P(X<1>)) __________//_T (n)*( 1 (X<1>)
 
 
fflffl sn*(X) fflffl
T (n)*(P(X)) _______________//_T (n)*( 1 X)
both vertical maps are equivalences and (1) follows.
For (2), consider the commutative square the square
sn*(X<0>)
T (n)*(P(X<0>))___________//T (n)*( 1 (X<0>))
 
 
fflffl sn*(X) fflffl
T (n)*(P(X)) ______________//T (n)*( 1 X).
The left vertical map is an isomorphism, and the horizontal maps are monic
by Theorem 2.5. As 1 (X<0>) is just one of the path components of 1 X,
the right vertical map is only epic if ß0(X) = 0, and (2) follows.
Properties (3) and (4) follow by combining Proposition 3.17, Proposi
tion 3.18, and Proposition 3.24.
5.4. Proof of Theorem 5.1: step 4. It is convenient to make the follow
ing definition, a variant on similar notions in the literature.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 39
Definition 5.8. Let C be any collection of Smodules. Say that an S
module X is built from C if X ' hocolimi Xi, for some sequence X0 !
X1 ! . . .such that X0 is a wedge of Smodules in C, and, for all i 0,
Xi+1 is the cofiber of a map Wi! Xi with Wi a wedge of Smodules in C.
Note that properties (3) and (4) of Proposition 5.7 imply that if C is any
subset of 0connected Smodules in Sn or SKn, then any Smodule built
from C is also in Sn or SKn.
The following proposition is a variant of a well known consequence (as in
[Mil]) of the Nilpotence Theorems. I would like to thank Pete Bousfield for
suggesting the simple proof.
Proposition 5.9. Lfn1X ' * and X is cconnected if and only if X can
be built from cconnected finite Smodules of type n.
To prove this we first need a lemma.
Lemma 5.10. Suppose Lfn1X ' *. Then any f : Y ! X, with Y finite of
type at most n and cconnected, can be factored as a composite Y ! F ! X
such that F is a finite Smodule that is both cconnected and of type n.
Proof.We prove this by downwards induction on the type of Y . The in
duction is begun by noting that there is nothing to prove if Y has type
n.
So suppose the lemma has been established for all g : Z ! X, where Z
is cconnected of type at least i+1. Let f : Y ! X, where Y has type
i and is cconnected. By [HS ], there exists a viself map v : dY ! Y .
N f
Since T (i)*(X) = 0, there exists an N such that Nd Y v!Y ! X is
null. Letting Z be the cofiber of vN , it follows that f factors as a composite
Y ! Z g!X. Since Z is of type i + 1 and is still cconnected, g, and thus
f, factors as needed.
Proof of Proposition 5.9.The Nilpotence Theorem implies that T (i)*(F ) =
0 whenever F is finite of type greater than i. Thus one implication is clear:
if X can be built from cconnected finite Smodules of type at least n, then
Lfn1X ' * and X is cconnected.
Conversely, suppose that Lfn1X ' * and X is cconnected. We describe
how to construct a diagram
j0 j1 j2
X0 _____//X1____//_X2____//m. . .
g mmmm
g0g12mmmmmm
fflffl""vvmmmmm
X
40 KUHN
showing that X is built from cconnected finites of type at least n.
First choose a wedge of cconnected spheres T and a map f : T ! X that
is onto in ß*. By the last lemma, this factors as a composite T ! X0 g0!X
with X0 a wedge of cconnected finites of type n.
Assume gi : Xi ! X has been constructed with ß*(gi) onto, and Xi c
connected. Let Yi be the fiber of gi. Choose a wedge of cconnected spheres
T and a map f : T ! Yi that is onto in ß*. By the last lemma, this factors
as a composite T ! Wi g0!Yi with Wi a wedge of cconnected finites of
type n. If we then let Xi+1 be the cofiber of the composite Wi ! Yi ! Xi,
gi+1
it follows that gi will factor as a composite Xiji!Xi+1 ! X.
By construction, ß*(gi) is onto and ker(ß*(ji)) = ker(ß*(gi)) for all i. It
follows that hocolimiXi' X as needed.
5.5. Virtual homology and the proof of Lemma 2.17. We need a
variant of Proposition 5.6.
If Eilenberg MacLane spectra are strongly E*acyclic, let c(E) be the
smallest c such that ~E*(K(Z, c)) = 0.
Proposition 5.11. If a 1connected spectrum X is strongly E*acyclic,
then ~E*( 1 c(E)X) = 0.
The proof of Proposition 5.6 goes through without change.
Proof of Lemma 2.17. Suppose X is 1connected. We need to show that
~E*( 1 cX) = 0 for large c if and only if ~E*( 1 X) = 0 for large d.
Let P dX denote the dth Postnikov section of X, so there is a cofibration
sequence of spectra
X ! X ! P dX.
Then, for all c 1, there is a fibration sequence of spaces
1 c1P dX ! 1 c(X) ! 1 cX.
If c > c(E), this fiber will be E*acyclic, and thus there will be an isomor
phism
E*( 1 c(X)) ~!E*( 1 cX).
Suppose E~*( 1 X) = 0. Then E~*( 1 c(X)) = 0 for all c 0.
Thus by our remarks above, ~E*( 1 cX) = 0 for c > c(E).
Conversely, suppose ~E*( 1 cX) = 0 for all large c. Then, for all d 0,
~E*( 1 c(X)) = 0 for all large c, i.e. X is strongly E*acyclic. If
d c(E)  1, we can apply Proposition 5.11 to the spectrum c(E)(X)
to conclude that ~E*( 1 X) = 0.
5.6. Proof of Theorem 2.10 and Proposition 2.11. We start with a
general lemma.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 41
Lemma 5.12. Suppose f : X ! Y is a map between 0connected spectra
with cofiber C. For any homology theory E*, there are implications (1) )
(2) ) (3).
(1) ~E( 1 1C) = 0.
(2) E*( 1 f) is an isomorphism.
(3) ~E( 1 C) = 0.
Proof.To see that (1) implies (2), note that 1 1C is the fiber of 1 f.
To see that (2) implies (3), note that
1 f
1 ( 1 X)+ ! 1 ( 1 Y )+ ! 1 ( 1 C)+
is a cofibration sequence in Alg. Thus Proposition 3.24 applies to say that
if 1 f is an E*isomorphism, then 1 ( 1 C)+ is E*equivalent to S.
Proof of Theorem 2.10.We prove statement (1); the proof of statement (2)
is similar.
We temporarily introduce a new class of spectra: let
S~n= {X 2 S  c(X<1>) 2 Sn for largec}.
Let Cn1,d(X) and fd be defined so that
Cn1,d(X) fd!X ! (Lfn1X)
is a fibration sequence of Smodules.
As Lfn1is smashing, and T (i)*(T (n)) = 0 for 0 i n  1, it follows
that Lfn1X is always T (n)*acyclic, thus fd is always a T (n)*isomorphism.
Theorem 5.1 then implies that, for all large d and large c,
Cn1,d(X), cCn1,1(X) 2 Sn.
Now consider the diagram
T (n)*(P(Cn1,d(X)))____//_T (n)*( 1 Cn1,d(X))
  1
 T(n)*( fd)
fflffl fflffl
T (n)*(P(X))__________//T (n)*( 1 X)
If d is large, both the top map and the left map are isomorphisms, and we
conclude that X 2 Sn if and only if T (n)*( 1 fd) is an isomorphism.
Thus X 2 ~Snif and only if T (n)*( 1 fd) is an isomorphism for large d.
Similarly X 2 S~nif and only if T (n)*( 1 cf1) is an isomorphism for
large c.
The last lemma and Lemma 2.17 now combine to say that
X 2 ~Sn, X 2 ~Sn, Lfn1X is strongly T (n)*acyclic.
The inclusion Sn ~Snis evident, using Proposition 5.7, thus Sn ~Sn.
42 KUHN
Proof of Proposition 2.11.Suppose condition (1) holds: T (i)*(X) = 0 for
1 i n  1. Then Lfn1X ! LQ X is an equivalence. Since rational
spectra are certainly strongly T (n)*acyclic, condition (2) holds: Lfn1X is
strongly T (n)*acyclic.
Condition (2) obviously implies condition (3).
Suppose condition (3) holds: Lfn1X is strongly K(n)*acyclic. The main
theorem of either [B6 ] and [W ] implies that if a spectrum Y is strongly
K(n)*acyclic, then it is strongly K(i)*acyclic for 1 i n. Applied to
our situation, we deduce that Lfn1X is strongly K(n  1)*acyclic. With
notation as in the last proof, Theorem 5.1 implies that Cn1,1(X) is also
strongly K(n  1)*acyclic. Thus so is X, i.e. condition (4) holds.
Finally, if X is strongly K(n  1)*acyclic, then the BousfieldWilson
theorem implies that X is K(i)*acyclic for 1 i n1, and thus condition
(5) holds.
5.7. Proof of Theorem 2.18 and Theorem 2.26. The fact that the Kun
neth Theorem holds for K(n)* allows for special calculational techniques.
For example, [EKMM , Thm.7.7] applies to show that, if A ! B ! C
is a cofibration sequence in Alg , the bar spectral sequence converging to
K(n)*(C) has
E2p,q= TorK(n)*(A)p,q(K(n)*(B), K(n)*).
This has the following consequence of relevence to us.
Lemma 5.13. Suppose f : X ! Y is a map between 0connected spectra
with cofiber C. If K(n)*( 1 f) is monic, there is a short exact sequence of
K(n)*Hopf algebras
( 1 f)* 1 1
K(n)*( 1 X) ! K(n)*( Y ) ! K(n)*( C).
Proof.K(n)*( 1 X) is in the category of K=pHopf algebras studied by
Bousfield in [B4 , Appendix]. He shows [B4 , Thm.10.8] that objects in this
category are flat over subobjects. It follows that, if K(n)*( 1 f) is monic,
the spectral sequence associated to the cofibration sequence in Alg
1 ( 1 X)+ ! 1 ( 1 Y )+ ! 1 ( 1 C)+
collapses, giving the desired short exact sequence.
Proof of Theorem 2.18.Recall that we have cofibration sequences
Cn1,d(X) fd!X ! (Lfn1X),
and that Cn1,d(X) 2 SKn if d is large.
Now consider the diagram used in the proof of Theorem 2.10:
K(n)*(P(Cn1,d(X))) _____//K(n)*( 1 Cn1,d(X))
  1
 ( fd)*
fflffl sn(X)* fflffl
K(n)*(P(X)) _________//_K(n)*( 1 X).
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 43
The left map is always an isomorphism, as is the top map if d is large. The
bottom map is always monic by Theorem 2.8, thus so is the right map, if d
is large. The previous lemma applies to say that, for all large d, there is a
short exact sequence of K(n)*Hopf algebras
( 1 fd)* 1 1 f
K(n)*( 1 Cn1,d(X)) ! K(n)*( X) ! K(n)*( (Ln1X)).
This rewrites as the short exact sequence of the theorem:
sn(X)* vir 1 vir 1 f
K(n)*(PX) ! K(n)* ( X) ! K(n)* ( Ln1X).
Proof of Theorem 2.26.Suppose given f : X ! Y with X 2 SKn. In the
diagram
K(n)*(PX) _____//K(n)*( 1 X)
P(f)* (1f)*
fflffl fflffl
K(n)*(PY ) _____//K(n)*( 1 Y ),
we then know that the top map is an isomorphism. Since the bottom map
is always monic, if the left map is monic, we deduce that ( 1 f)* is also
monic. If X and Y are 0connected and C is the cofiber of f, the lemma
applies, yielding the short exact sequence of the theorem.
Remark 5.14. This proof makes evident the following T (n)* variant of The
orem 2.26: if f : X ! Y is a T (n)*isomorphism, and X 2 Sn, then
( 1 f)* : T (n)*( 1 X) ! T (n)*( 1 Y ) is monic.
5.8. Proof of Theorem 2.12 and Theorem 2.21. Since T (n)* is all
ptorsion, [B2 , x4] implies
Lemma 5.15. Let A be an abelian group. K(A, j) is T (n)*acyclic if
(1) j = 0 and A = 0,
(2) 1 j c(n)  1 and A is uniquely pdivisible,
(3) j = c(n) and A=(torsion)is uniquely pdivisible, or
(4) j > c(n) + 1.
Proof of Theorem 2.12(1).Given an Smodule X, for each j 0, there is
a fibration sequence of spaces
K(ßj+1(X), j) ! 1 X ! 1 X.
Under the theorem's hypotheses on ß*(X), the fiber is T (n)*acyclic, so the
second map is a T (n)*isomorphism. We deduce that X 2 Sn if and
only if X 2 Sn.
The hypothesis that X 2 S~nmeans that X 2 Sn for all large d. By
downward induction on j, we deduce that X 2 Sn for all j 0. Since
ß0(X) = 0, X<0> 2 Sn implies X 2 Sn.
44 KUHN
For K(n)*, we have sharper results.
Lemma 5.16. Let A be an abelian group. K(A, j) is K(n)*acyclic if and
only if
(1) j = 0 and A = 0,
(2) 1 j n and A is uniquely pdivisible,
(3) j = n + 1 and A=(torsion)is uniquely pdivisible, or
(4) j > n + 1.
Proof of Theorem 2.12(2) and Theorem 2.21. We can assume that X 2 ~SKn
is 0connected. As in our proof of Lemma 2.17, let P dX denote the dth
Postnikov section of X, so there is a cofibration sequence of spectra
X ! X ! P dX.
If d is large, then X 2 SKn. Then Theorem 2.26 applies, and we deduce
that there is a short exact sequence of K(n)*Hopf algebras
K(n)*( 1 X) ! K(n)*( 1 X) ! K(n)*( 1 P dX).
Thus X 2 SKn if and only if 1 P dX is K(n)*acyclic.
The main theorem of [HRW ] says that there is an isomorphism
Od
K(n)*( 1 P dX) ' K(n)*(K(ßj(X), j).
j=1
Theorem 2.21 follows.
By the lemma, this tensor product will be isomorphic to K(n)* if and only
if ßj(X) is uniquely pdivisible for 1 j n, and also ßn+1(X)=(torsion)
is uniquely pdivisible. Theorem 2.12(2) follows.
5.9. sn(X) is universal: proof of Proposition 2.6 and related results.
We prove the first part of Proposition 2.6; the proofs of the other variants,
including Proposition 2.20, are similar and left to the reader.
Suppose F : S ! S is functor preserving T (n)*isomorphisms, and T is
a weak natural transformation of the form
T (X) : F (X) ! LT(n) 1 ( 1 X)+ .
We show it uniquely factors through sn.
Let C(X) = Cn1,c(n)+2(X) defined as in the proof of Theorem 2.10.
Then sn(C(X)) is an equivalence, and C(X) ! X is a T (n)*isomorphism.
We simplify notation; let
P (X) = LT(n)P(X) and L(X) = LT(n) 1 ( 1 X)+ .
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 45
By naturality, we have a commutative diagram
T(C(X)) sn(C(X))
F (C(X)) ___________//L(C(X))oo___~_____P (C(X))
o  o
fflffl T(X) fflfflsn(X) fflffl
F (X) _____________//_L(X)oo____________P (X),
where the left vertical map is an equivalence since F preserves T (n)*
isomorphisms. The canonical factorization of T through sn is evident.
Appendix A. Comparison of Theorem 2.2 with other stable
splittings
Let CZ be the free E1 space generated by a space Z, as in [M1 ]. The
inclusion Z ! QZ then induces the standard approximation map ff(Z) :
CZ ! QZ. The suspension spectrum 1 (CZ)+ has the structure of an
object in Alg such that 1 (ff(Z))+ is an algebra map.
The purpose of this appendix is to make the following two observations.
Firstly, s(Z) : P( 1 Z) ! 1 (QZ)+ refines to a natural map sC (X) :
P( 1 Z) ! 1 (CZ)+ . Secondly, sC (Z) is always an equivalence, and agrees
with the standard `stable splittings' in the literature.
This first point is easily checked. Recall that s(Z) is defined to be the the
natural weak map in Alg induced by the weak natural map of Smodules
1 ''(Z) 1 ~ 1
1 Z ! QZ  I( (QZ)+ ).
Similarly we define sC (Z) to be the natural weak map in Alg induced by
the weak natural map of Smodules
1 ''(Z) 1 ~ 1
1 Z ! CZ  I( (CZ)+ ).
Then there is an evident factorization
P( 1 Z)O
OOOs(Z)O
sC(Z)OOOO
fflfflff(OO''Z)
1 (CZ)+ _____// 1 (QZ)+ .
To check the second point, we begin by observing that sC admits a slightly
different definition. Let a(Z) denote the fiber (in Smodules) of the evident
`augmentation' 1 (Z+ ) ! S. Note that the composite a(Z) ! 1 (Z+ ) !
1 Z is always an equivalence. Then sC (Z) can alternatively be defined as
the natural weak map in Alg induced by the weak natural map of Smodules
1 Z ~ a(Z) ! a(CZ) = I( 1 (CZ)+ ).
Now we need to recall that C can be defined on the spectrum level
[LMMS ]. Let S  S denote the category of diagrams of Smodules of the
46 KUHN
form
~X??AA
~~~~ AAA
~~ AA__
S ______________S.
There is a functor C : S  S ! Alg such that
(1) C( 1 (Z+ )) = 1 (CZ)+ , and
(2) X ! X _ S induces P(X) = C(X _ S).
Using (2), the commutative diagram
oo_~_____
1 Z __________ 1 Z a(Z)

  
  
fflffl~ fflffl~ fflffl
1 Z _ S _____// 1 Z x Soo___ 1 (Z+ ),
induces a diagram in Alg
P( 1 Z) __________P( 1 Z) oo__~____ P(a(Z))
  
  
 ~ fflffl ~ fflffl
C( 1 Z _ S) _____//C( 1 Z x S)oo___ C( 1 (Z+ )).
Now using (1), this shows that sC (Z) is the natural weak equivalence
P( 1 Z) = C( 1 Z _ S) ~!C( 1 Z x S) ~ 1 (CZ)+ .
Defined this way, sC (Z) satisfies the characterization of natural splittings
given in [K4 , Appendix B].
We end this appendix by noting that the proof of Theorem 2.4 generalizes
in a straightforward way to prove the following variant.
Theorem A.1. If a map of spectra f : 1 Z ! X is an E*isomorphism,
then the composite
ff(Z) ( 1 f)* 1
E*(CZ) ! E*(QZ) ! E*( X)
is a monomorphism.
Appendix B. Comparison with recent work of Bousfield
In this appendix, we show how Theorem 2.2 and Theorem 2.5 can be com
pared by using Bousfield's beautiful natural zigzag of LT(n)equivalences
relating any Smodule X to a suspension spectrum determined by X [B7 ].
This allows for an alternative proof of Theorem 5.1, and thus of many of the
results in x2.5 and x2.6.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 47
Bousfield constructs a functor
n : Smodules ! Spaces
that is a left adjoint of sorts to the telescopic functor
n : Spaces ! Smodules .
Using this adjunction, the equivalence
LT(n)X ~! n( 1 X),
corresponds to an equivalence
Lfn 1 n(X) ~!MfnX,
where MfnX is the fiber of LfnX ! Lfn1X. Thus Lfn1 1 n(X) ' *,
and there is a natural T (n)*equivalence 1 n(X) ! LfnX. Furthermore,
Bousfield observes that n(X) is always dnconnected, where dn is defined
in [B7 , x4.3]: one can deduce that dn c(n) + 1 from [B2 , Prop.2.1]. One
also has [B7 , Thm.3.3] that MfnLT(n)' Mfnand LT(n)Mfn' LT(n).
Thus we get a zigzag of T (n)*equivalences:
fi(X) f
1 n(X) ! (LnX) X ! X.
This allows us to consider the following diagram:
sn(X) 1 1
LT(n)P(X) ________________________//_LT(n) X+
OO OO
o 
 sn(X) 1 1
LT(n)P(X) ____________________//_LT(n) X+
o o
fflffl sn((LfnX)) fflffl
LT(n)P((LfnX)) ________________//_LT(n) 1 1 (LfnX)+
OO OO
o 
 LT(n)s( n(X)) 
LT(n)P( 1 n(X)) _________~_________//LT(n) 1 Q n(X)+ .
Below we will show that the diagram commutes. Thus the classical stable
splitting of Q n(X) given by Theorem 2.2, corresponds to the splitting of
LT(n) 1 1 X given by Theorem 2.5.
A crucial point about this diagram is that, as indicated, the middle ver
tical map on the right is an equivalence, as 1 takes Lfnequivalences be
tween dnconnected spectra to T (n)*equivalences, thanks to [B7 , Cor.4.8]8.
Thus, since the diagram commutes, it is clear that sn(X) is an equivalence
on highly connected X if and only if the bottom right vertical map is an
____________
8As dn c(n) + 1, this also follows from Theorem 5.1. However, if one wishe*
*s to offer
an alternative proof of Theorem 5.1, it seems prudent to not argue this way.
48 KUHN
equivalence, i.e. ( 1 fi(X))* : T (n)*(Q n(X)) ! T (n)*( 1 (LfnX))
is an isomorphism. Again appealing to [B7 , Cor.4.8], this will happen if
Lfn1X ' *, and we have reproved Theorem 5.1 using Bousfield's results.
It is illuminating to note that 1 fi(X) is always a T (n)*monomorphism,
by virtue of our Theorem 2.4.
The top two squares of the diagram obviously commute. Checking that
the bottom square commutes quickly reduces to verifying that the following
diagram commutes:
''n((LfnX)) f
LT(n)(LfnX) ________________//_LT(n) 1 1 ((LnX))
OO OO
oLT(n)fi(X) LT(n)1 1 fi(X)
 LT(n) 1 ''( n(X)) 
LT(n) 1 n(X) ____________________//LT(n) 1 Q n(X).
We show this using a variant of a proof which was outlined to us in
email from Pete Bousfield. It is an exercise in using the various adjunctions
constructed in [B7 ], as summarized in [B7 , Thm.5.14].
By the naturality of jn, it suffices to verify the following proposition.
Proposition B.1. jn( 1 n(X)) ' LT(n) 1 j( n(X)).
To prove this, we first observe that 1 n preserves T (n)*equivalences,
and thus so does 1 Q n. Thus the zigzag of T (n)*equivalences
X ! LfnX MfnX
can be used to reduce the proof of the proposition to the case when X =
MfnX, i.e. X 2 Mfnin the notation of [B7 ].
For any space Z, unravelling the definitions reveals that
jn( 1 Z) = n(j(QZ)),
while
LT(n) 1 j(Z) = n(Qj(Z)).
Both of these maps clearly agree after precomposition with
n(j(Z)) : n(Z) ! LT(n) 1 Z.
Thus the next lemma will compete the proof of the proposition.
Lemma B.2. If X 2 Mfn, then
n(j( n(X))) : n( n(X)) ! LT(n) 1 n(X)
is split epic.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 49
To prove this, we will use a very general categorical lemma. Suppose one
has two categories A and B, and two pairs of adjoint functors
__L1_// _L2_//_
A oo___ B oo___A.
R1 R2
Let j1 : 1A ! R1L1 and j2 : 1B ! R2L2 be the units of the adjunctions.
Now suppose we are also given a natural transformation fl : 1A ! R1R2
with adjoint fi : L2L1 ! 1A .
Lemma B.3. fl is an equivalence if and only if fi is an equivalence. In this
case, the map
R1j2(L1A) : R1L1A ! R1R2L2L1A
is split epic for all A.
Proof.The first statement is clear. The second statement then follows from
the commutative diagram
''1(A)
A ________________//R1L1A
ofl(A) R1''2(L1A)
fflfflR1R2fi(A) fflffl
R1R2A oo____~____R1R2L2L1A.
Proof of Lemma B.2. The previous lemma applies to the pair of adjoint
functors appearing in [B7 , Thm.5.14] to say that
Mfn n(j2( n(X))) : Mfn n n(X) ! Mfn n 1 (Lfn 1 n(X))
is split epic, where j2, defined on a certain category of dnconnected spaces,
has the form
j2(Z) : Z ! 1 (Lfn 1 Z).
Applying LT(n), one deduces that
n(j2( n(X))) : n n(X) ! LT(n)((Lfn 1 n(X)))
is split epic.
Using the zigzag of T (n)*equivalences
(Lfn 1 n(X)) ! Lfn 1 n(X) 1 n(X),
it follows that this last map identifies with
n(j( n(X))) : n n(X) ! LT(n) 1 n(X).
50 KUHN
Appendix C. A comparison of sn and s
One might wonder for what Z the two natural maps
sn( 1 Z), LT(n)s(Z) : LT(n)P( 1 Z) ! LT(n) 1 (QZ)+
are homotopic. Here we briefly summarize what we can say about this.
On the positive side, sn( 1 Z) ' LT(n)s(Z) if and only if
jn( 1 Z) ' LT(n)j(Z) : LT(n) 1 Z ! LT(n) 1 QZ.
Thus Proposition B.1 implies
Proposition C.1. If Z ' n(X) then sn( 1 Z) ' LT(n)s(Z).
Lemma 5.2 gave another sufficient condition on Z; we do not know if the
proposition includes this as a special case.
On the negative side, since s(Z) is an equivalence for all connected Z, we
have an obvious necessary condition.
Lemma C.2. If Z is connected and 1 Z 62 Sn then sn( 1 Z) 6' LT(n)s(Z).
Thus, for example, the two maps are distinct for Z = S1, and for all
Z = Sd if n 2. More examples of suspension spectra not in Sn can be
found using the next simple lemma, which doesn't seem to follow immedi
ately from our other results.
Lemma C.3. If X 62 Sn then 1 1 X 62 Sn.
Proof.The evaluation map 1 1 X ! X induces a commutative diagram
sn( 1 1 X)
T (n)*(P 1 1 X) __________//_T (n)*(Q 1 X)
 
 
fflffl sn(X) fflffl
T (n)*(PX)_______________//T (n)*( 1 X).
The horizontal maps are always monic, and the right vertical map is al
ways epic, as it admits an obvious splitting. Thus if the top map is an
isomorphism, so is the bottom.
When jn( 1 Z) and LT(n)j(Z) differ, one can roughly measure the dif
ference by means of JamesHopf invariants. Let
tr(Z) : 1 QZ ! 1 DrZ
be the rth component of t(Z), as given by Theorem 2.2. In the literature,
the adjoint
jr(Z) : QZ ! QDrZ
is usually called the rth JamesHopf invariant.
PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 51
Now consider the two maps
LT(n)''(Z) 1 LT(n)tr(Z) 1
LT(n) 1 Z ! LT(n) QZ ! LT(n) DrZ,
and
''n( 1 Z) 1 LT(n)tr(Z) 1
LT(n) 1 Z ! LT(n) QZ ! LT(n) DrZ.
For r 2, the former is 0, while the latter is n(jr(Z)), as is easily
checked.
Comparison with Proposition C.1 implies the next corollary.
Corollary C.4. If sn( 1 Z) ' LT(n)s(Z), e.g. if Z ' n(X) for some X,
then n(jr(Z)) is null for all r 2.
There are some intriguing open questions regarding the natural transfor
mations
n(jr(Z)) : LT(n) 1 Z ! LT(n) 1 DrZ.
For example, they induces natural transformations
E*n(DrZ) ! E*n(Z),
and one might wonder if these are related to either the constructions in
[HKR ], or, via duality, to total power operations in E*ncohomology.
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Department of Mathematics, University of Virginia, Charlottesville, VA
22903
Email address: njk4x@virginia.edu