LOCALIZATION OF ANDR'E-QUILLEN-GOODWILLIE TOWERS, AND THE PERIODIC HOMOLOGY OF INFINITE LOOPSPACES NICHOLAS J. KUHN Abstract. Let K(n) be the nthMorava K-theory at a prime p, and let T(n) be the telescope of a vn-self 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 S-algebra 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 n-connected, ßn+1(X) is torsion, and T(i)*(X) = 0 for 1 i n-1. 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'e-Quillen 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'e-Quillen 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 S-modules as in [EKMM ], and so, e.g., commutative S-algebras 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 S-module, 1 ( 1 X)+ is naturally a commutative S-algebra. 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 S-algebras 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 K-theory spectrum at a fixed prime p, and T (n) its `telescopic' variant: the telescope of a vn-self 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)S-algebras 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 S-module 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 p-complete integral height n complex oriented commutative S-algebra 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'e-Quillen 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'e-Quillen-Goodwillie 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 S-algebras 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 S-module, 1 ( 1 X)+ is naturally a commutative S-algebra 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'e-Quillen tower in Alg, and can be viewed as the I(A)-adic completion of A. The second is an S-module taq(A), a form of Topological Andr'e-Quillen homology, and can be viewed as I(A)=I(A)2. There is a convergence result: if A is 0-connected then the canonical map A ! Abis an equivalence. Applied to the examples above, if A is either PX or 1 ( 1 X)+ , with X -1-connected 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 0-connected then all of these maps are weak equivalences. 6 KUHN In more down-to-earth 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'e-Quillen towers. Now we mix Bousfield localization with the general theory. If E is any S-module, LE S will be a commutative S-algebra, and we define LE (Alg ) to be the category of commutative LE S-algebras, which are also augmented over LE S, and are E-local. 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 S-module 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 non-connected 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 S-module 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 vn-self 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 down-to-earth 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)S-algebras, such that all maps in the diagram are LT(n)S-algebra 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 S-modules 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 H-space 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*n-algebras 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*n-algebras. 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 S-modules 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 Eilenberg-MacLane 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 d-connected cover of an S-module X. Our results about Sn and SKn are most pleasantly described by first in- troducing two more classes of S-modules. 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 Lfn-1denote localiza- tion with respect to T (0) _ . ._.T (n - 1). We recall that this is smashing: Lfn-1X ' Lfn-1S ^ X. Armed with this terminology, we have the following theorem and propo- sition. Theorem 2.10. (1) ~Sn= {X 2 S | Lfn-1X is strongly T (n)*-acyclic}, and Sn ~Sn. (2) ~SKn= {X 2 S | Lfn-1X 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) Lfn-1X is strongly T (n)*-acyclic. (3) Lfn-1X 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 p-divisible 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 p-divisible for 1 j n, and also ßn+1(X)=(torsion) is uniquely p-divisible. 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)S-algebras LK(1)P(X) ' LK(1) 1 ( 1 X)+ for all 1-connected 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 S-modules described by the second statement of The- orem 2.12. In particular, there is a natural equivalence of commutative augmented LK(2)S-algebras LK(2)P(X) ' LK(2) 1 ( 1 X)+ for all K(1)*-acyclic, 2-connected X, with torsion ß3. Example 2.15. These results tell us precisely which p-local 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 Eilenberg-MacLane 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 S-module 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)* ( Ln-1X). 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)*(Lfn-1X) = 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 Lfn-1X 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*n-isomorphisms. The next theorem has Theorem 2.12(2) as a consequence. Theorem 2.21. Let X 2 ~SKn, i.e. Lfn-1X is strongly K(n)*-acyclic. Then each of the maps K(n)*( 1 X) ! K(n)*( 1 X) is an inclusion of a normal sub-K(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 Lfn-1X is strongly K(n)*-acyclic, and also X is n-connected 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 n-2 v 2p k(n) -! k(n) ! HZ=p induces a short exact sequence of commutative, cocommutative K(n)*-Hopf algebras K(n)*(k(n)2pn-2+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 0-connected S-modules 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 `pro-free' as an E*n-module, 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 -1-connected 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 0-connected spectra X, there is a natural factorization by weak S-module maps (2.4) LHQ771 1 XO ''0(X)pppp OOOLHQffl(X)O pppp OOO ppp OOO'' LHQ X _________________________LHQ X. For X just -1-connected, 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 0-connected X. Proposition 2.3 combines with the lemma to prove: Theorem 2.28. For all -1-connected 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 0-connected X, and, in this case, all three maps are equivalences. 16 KUHN As a corollary, one recovers the known theorem: for 0-connected 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'e-Quillen towers associated to commutative augmented S-algebras, 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 James-Hopf invariants. 3.The Andr'e-Quillen tower of commutative algebras 3.1. Categories of commutative S-algebras. We work always within the topological model category of S-modules as in [EKMM ]. This is a symmetric monoidal category with unit the sphere spectrum S, and we recall that associative, commutative, unital S-algebras are modern day versions of E1 -algebras. Given such an algebra R, we let R - Alg denote the category of associative, commutative, unital, augmented R-algebras. When R = S we simplify the notation to Alg. Given A 2 R-Alg , 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 R-algebras, 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 R-modules. 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 R-algebra 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 n-connected, then the natural map I ! (S1 I) is 2n + 1-connected. We feature two families of examples. Example 3.2. Given an S-module X, we let PX denote the free commuta- tive S-algebra 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 S-modules 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 fflffl|II$$(f~) I(PX) _____//I(A) commutes. Given a commutative S-algebra R, and an R-module 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 S-module 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 S-modules induces an equivalence in Alg: 1 ( 1 X<-1>)+ -~! 1 ( 1 X)+ . 18 KUHN 3.2. Topological Andr'e-Quillen 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 ! R-modules 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 R-Alg 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 R-modules. Proof.As will be elaborated on in x3.6, the functor taqR , defined via sta- blization as above, will be 1-excisive 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 S-modules 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 S-modules 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 S-module 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 -1-connected S-module, 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 -1-connected S-module 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'e-Quillen tower of an augmented R-algebra. It seems that various people have noted the existence of an `Andr'e-Quillen 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 R-modules, 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,r-1A 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 S-algebra, there is a natural weak equivalence of towers under R ^ A: R ^ Pr(A) ' PR,r(R ^ A). (4) If I(A) is 0-connected, then er is r-connected. The uniqueness statement means up to natural weak equivalence. The weak equivalence in the second property is as R-modules.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 0-connected, 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 S-modules 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 R-modules. Proof.Both ^R and ( )h r (homotopy r-orbits) 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 R-algebra 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'e-Quillen 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 S-modules to Alg. Proposition 3.17. The functor P satisfies the following properties. (1) P takes homotopy colimits in S-modulesWto 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 S-modules, 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 S-modules. 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 S-modules to filtered homotopy colimits in Alg. (2) 1 ( 1 )+ takes coproducts in S-modules to coproducts in Alg. (3) If X ! Y ! Z is a cofibration sequence S-modules, with X and Y -1-connected and Z 0-connected, 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 S-modules. (Com- pare with [EKMM , xII.7].) But, as a functor to S-modules, 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 S-modules 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 -1-connected, 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 0-connected (so that ß0(f) is onto), one can deduce that both A and 1 ( 1 Z)+ have 0-connected 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 S-module, 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 S-algebra, 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)=Fd-1taqR (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'e-Quillen 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 E-local 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 R-algebras 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 E-local if and only if it is weakly equivalent LE A, for some A 2 Alg. More precisely, B is E-local 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'e-Quillen 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 S-algebra 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 S-module, 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'e-Quillen 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 R-modules. 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 r-excisive 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 r-excisive 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 1-disks 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 S-algebras to strongly homotopy cocartesion cubes of R-algebras, and homotopy cartesion cubes of S-algebras to homo- topy cartesion cubes of R-algebras (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 r-connected if I(A) is 0-connected, 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) ! Pr-1F (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 i2r-T 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 i2r-T 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 R-modules 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 S-modules 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 0-connected 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: 1|ZCXXXXXX___________________________________________51|Z5l | 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 | fflffl|vvnn 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 James-Hopf 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 0-connected, 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 p-locally. For n 1, let K(n) be the nth Morava K- theory, and T (n) be the telescope of a vn-self map of a finite complex of type n. It is known that the Bousfield class of T (n) is independent of choice of vn-self 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 ! S-modules 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 S-modules 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 S-module 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 S-module, 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 S-modules ''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________________________________________LX|55jjj | @@@ 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 | fflffl|wwpp 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 0-connected spectra X, there is a natural factorization by weak S-module maps LHQ771 1 XO ''0(X)pppp OOOLHQffl(X)O pppp OOO ppp OOO'' LHQ X _________________________LHQ X. For X just -1-connected, 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 -1-connected S-module, 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 -1-connected 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 0-connected 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 HQ-local S-modules. 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 0-connected, 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 -1-connected, one can still choose a Q-linear 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 Q-vector space, would define a natural section to the augmentation Q[V ] ! V defined on the category of Q-vector 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 0-connected 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 0-connected X. 5. When sn(X) is an equivalence, and related matters Recall that Sn = {S-modules X | sn(X) is an equivalence} and that SKn = {S-modules 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 S-module such that Lfn-1X ' *. (1) X 2 Sn if X is c(n)-connected. (2) X 2 SKn if X is n + 1-connected. 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 S-module of type n, then dF 2 Sn for d 0. Step 2 Assuming Step 1, we show that if F is a 0-connected 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 S-modules 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)S-algebras 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)S-modules 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 0-connected. 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 S-modules 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 -1-connected finite S-module 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 0-connected finite S-module of type n. (1) c(n)F 2 Sn. (2) n+1F 2 SKn. PERIODIC HOMOLOGY OF INFINITE LOOPSPACES 37 Proof.Let F be 0-connected and finite of type n. Let v : dF ! F be a vn-map, 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 Atiyah-Hirzebruch-Serre 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 -1-connected spectrum X has p-torsion 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 -1-connected 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 S-modules ~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 Eilenberg-MacLane spaces of 38 KUHN type K(A, c) where A is p-torsion 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 S-modules with X and Y -1-connected and Z 0-connected. Then X, Y 2 S0 implies that Z 2 S0. In particular, if X 2 S0 is -1-connected, then X 2 S0. (4) Let X be the filtered homotopy colimit of S-modules 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 Eilenberg-MacLane 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 S-modules. 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 S-modules in C, and, for all i 0, Xi+1 is the cofiber of a map Wi! Xi with Wi a wedge of S-modules in C. Note that properties (3) and (4) of Proposition 5.7 imply that if C is any subset of 0-connected S-modules in Sn or SKn, then any S-module 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. Lfn-1X ' * and X is c-connected if and only if X can be built from c-connected finite S-modules of type n. To prove this we first need a lemma. Lemma 5.10. Suppose Lfn-1X ' *. Then any f : Y ! X, with Y finite of type at most n and c-connected, can be factored as a composite Y ! F ! X such that F is a finite S-module that is both c-connected 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 c-connected of type at least i+1. Let f : Y ! X, where Y has type i and is c-connected. By [HS ], there exists a vi-self 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 c-connected, 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 c-connected finite S-modules of type at least n, then Lfn-1X ' * and X is c-connected. Conversely, suppose that Lfn-1X ' * and X is c-connected. We describe how to construct a diagram j0 j1 j2 X0 _____//X1____//_X2____//m. . . -g mmmm g0||g1-----2mmmmmm fflffl|""--vvmmmmm X 40 KUHN showing that X is built from c-connected finites of type at least n. First choose a wedge of c-connected 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 c-connected 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 c-connected 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 c-connected 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 Xi-ji!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 -1-connected 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 -1-connected. 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 c-1P 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 0-connected 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 Cn-1,d(X) and fd be defined so that Cn-1,d(X) fd-!X ! (Lfn-1X) is a fibration sequence of S-modules. As Lfn-1is smashing, and T (i)*(T (n)) = 0 for 0 i n - 1, it follows that Lfn-1X 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, Cn-1,d(X), cCn-1,-1(X) 2 Sn. Now consider the diagram T (n)*(P(Cn-1,d(X)))____//_T (n)*( 1 Cn-1,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 cf-1) is an isomorphism for large c. The last lemma and Lemma 2.17 now combine to say that X 2 ~Sn, X 2 ~Sn, Lfn-1X 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 Lfn-1X ! LQ X is an equivalence. Since rational spectra are certainly strongly T (n)*-acyclic, condition (2) holds: Lfn-1X is strongly T (n)*-acyclic. Condition (2) obviously implies condition (3). Suppose condition (3) holds: Lfn-1X 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 Lfn-1X is strongly K(n - 1)*-acyclic. With notation as in the last proof, Theorem 5.1 implies that Cn-1,-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 Bousfield-Wilson theorem implies that X is K(i)*-acyclic for 1 i n-1, 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 0-connected 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=p-Hopf 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 Cn-1,d(X) fd-!X ! (Lfn-1X), and that Cn-1,d(X) 2 SKn if d is large. Now consider the diagram used in the proof of Theorem 2.10: K(n)*(P(Cn-1,d(X))) _____//K(n)*( 1 Cn-1,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 Cn-1,d(X)) -----! K(n)*( X) ! K(n)*( (Ln-1X)). 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)* ( Ln-1X). 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)*| (|1|f)* 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 0-connected 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 p-torsion, [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 p-divisible, (3) j = c(n) and A=(torsion)is uniquely p-divisible, or (4) j > c(n) + 1. Proof of Theorem 2.12(1).Given an S-module 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 p-divisible, (3) j = n + 1 and A=(torsion)is uniquely p-divisible, or (4) j > n + 1. Proof of Theorem 2.12(2) and Theorem 2.21. We can assume that X 2 ~SKn is 0-connected. 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 p-divisible for 1 j n, and also ßn+1(X)=(torsion) is uniquely p-divisible. 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) = Cn-1,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) fflffl|sn(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 S-modules 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 S-modules 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 fflffl|ff(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 S-modules) 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 S-modules 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 S-modules 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 zig-zag of LT(n)-equivalences relating any S-module 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 : S-modules ! Spaces that is a left adjoint of sorts to the telescopic functor n : Spaces ! S-modules . 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 ! Lfn-1X. Thus Lfn-1 1 n(X) ' *, and there is a natural T (n)*-equivalence 1 n(X) ! LfnX. Furthermore, Bousfield observes that n(X) is always dn-connected, 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 zig-zag 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 Lfn-equivalences be- tween dn-connected 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 Lfn-1X ' *, 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 o|LT(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 zig-zag 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 o|fl(A)| |R1''2(L1A)| fflffl|R1R2fi(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 dn-connected 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 zig-zag 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 James-Hopf 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 James-Hopf 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) ! 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