GOODWILLIE TOWERS AND CHROMATIC HOMOTOPY: AN OVERVIEW NICHOLAS J. KUHN Abstract. This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie's work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic ho- motopy as in my work and the work of Arone-Mahowald. 1. Introduction About two decades ago, Tom Goodwillie began formulating his calculus of homotopy functors as a way to organize and understand arguments being used by him and others in algebraic K-theory. Though it was clear early on that his general theory offered a new approach to the concerns of classical homotopy, and often shed light on older approaches, it is relatively recently that its promise has been begun to be realized. This has been helped by the recent publication of the last of Goodwillie's series [G1 , G2 , G3 ], and by t* *he support of many timely new results in homotopical algebra and localization theory allowing his ideas to be applied more widely. At the Workshop in Algebraic Topology held in Nagoya in August 2002, I gave a series of three talks entitled `Goodwillie towers: key features and examples', in which I reviewed the aspects of Goodwillie's work that I find most compelling for homotopy theory. A first goal of this paper is to offer a written account of my talks. As in my talks, I focus on towers associated to functors, i.e. the material of [G3 ]. As `added value' in this written version, I include some fairly extensive comments about the general model category requirements for running Goodwillie's arguments. At the Conference on Algebraic Topology held in Kinosaki just previous to the workshop, I discussed a result of mine [K5 ] that says that Goodwillie towers of functors of spectra split after periodic localization. This is one of a number of ways discovered so far in which Goodwillie calculus interacts beautifully with homotopy as organized by the chromatic point of view; another is the theorem of Greg Arone and Mark Mahowald [AM ]. A second goal is to survey these results as well, and point to directions for the future. ____________ Date: October 13, 2004. 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 The paper is organized as follows. In x2, I describe the major properties of Goodwillie towers associated to continuous functors from one topological model category to another. In x3, I discuss model category prerequisites. The basic facts about cubical diagrams and polynomial functors are reviewed in x4. The construction of the Goodwillie tower of a functor is given in x5, and I sketch the main ideas behind the proofs that towers have the properties described in x2. In x6, I discuss some of my favorite examples: Arone's model for the tower of the functor sending a space X to 1 Map (K, Z) [A ], the tower for the functor sending a spectrum X to 1 1 X, the tower of the identity functor on the category of commutative augmented S-algebras, and tower for the identity functor on the category of topological spaces as analyzed by Brenda Johnson, Arone, Mahowald, and Bill Dwyer [J, AM , AD ]. Besides organizing these in a way that I hope readers will find helpful, I have also included some remarks that haven't appeared elsewhere, e.g. I note (in Example 6.3) that the bottom of the tower for 1 1 X can be used to prove the Kahn-Priddy Theorem, `up to one loop'. The long x7 begins with a discussion of how Goodwillie towers interact with Bousfield localization. Included is a simple example (see Example 7.4) that shows that the composite of homogeneous functors between spectra need not again be homogeneous. In the remainder of the section, I survey three striking results in which the Goodwillie towers discussed in x6 interact with chromatic homotopy theory: my theorems on splitting localized tow- ers [K5 ] and calculating the Morava K-theories of infinite loopspaces [K4 ], and Arone and Mahowald's work on calculating the unstable vn-periodic homotopy groups of spheres [AM ]. All of these relate to telescopic functors n from spaces to spectra constructed a while ago by Pete Bousfield and me [B1 , K2 , B3] using the Nilpotence and Periodicity Theorems [DHS , HS ]. This suggests that Goodwillie calculus can be used to further explore these curious functors. Included in this section, as an application of my work in [K4 ], is an outline of a new way to possibly find a counterexample to the Telescope Conjecture. The Kinosaki conference was on the occasion of Professor Nishida's 60th birthday, and I wish to both offer him my hearty congratulations, and thank him for his kind interest in my research over the years. Many thanks also to Noriko Minami and the other conference organizers for their hospitality. 2. Properties of Goodwillie towers The basic problem that Goodwillie calculus is designed to attack is as follows. One has a homotopy functor F : C ! D between two categories in which one can do homotopy. One wishes to un- derstand the homotopy type of F (X), perhaps for some particular X 2 C. CALCULUS AND CHROMATIC HOMOTOPY 3 Goodwillie's key idea is to use the functoriality as X varies, to construct a canonical polynomial resolution of F (X) as a functor of X. The first thing to specify is what is meant by categories in which one can do homotopy theory. In Goodwillie's papers, these are T , the category of pointed topological spaces, or S, an associated category of spectra (e.g. the S-modules of [EKMM ]), or variants of these, e.g. TY , the category of spaces over and under a fixed space Y . But the arguments and constructions of [G3 ] are written in a such a manner that they apply to situations in which C and D are suitably nice based model categories: in x3, we will spell out precisely what we mean. Among all functors F : C ! D, some will be d-excisive (or polynomial of degree at most d). This will be carefully explained in x4.2; we note that a 0- excisive functor is one that is homotopically constant, a functor is 1-excisive if it sends homotopy pushout squares to homotopy pullback squares, and a (d - 1)-excisive functor is also d-excisive. Goodwillie's first theorem says that any F admits a canonical polynomial resolution. Theorem 2.1. [G3 , Thm.1.8] Given a homotopy functor F : C ! D there exists a natural tower of fibrations under F (X), .. . | | fflffl| P2F:(X): tt tttt |p2| e2 tttt fflffl| tttt jP1F4(X)4 ttt jjjj tttjjje1jjjjjt |p1| tjjjjjjtte0 fflffl| F (X) ________________//_P0F (X), such that (1) PdF is d-excisive, and (2) ed : F ! PdF is the universal weak natural transformation to a d- excisive functor. Let us explain what we mean by property (2). By a weak natural transfor- mation f : F ! G, we mean a pair of natural transformation F -g!H- h G such that H(X)- h G(X) is a weak equivalence for all X. Note that a weak natural transformation induces a well defined natural transformation be- tween functors taking values in the associated homotopy category. Property (2) means that, given any d-excisive functor G, and natural transformation 4 KUHN f : F ! G, there exists a weak natural transformation g : PdF ! G such that, in the homotopy category of D, ed(X) F (X) _____//PdF (X) tt f(X) || ttttt fflffl|g(X)yytt G(X) commutes for all X 2 C, and any two such g agree. A very useful property of the Pd construction is the following. Lemma 2.2. Given natural transformations F ! G ! H, if F (X) ! G(X) ! H(X) is a fiber sequence for all X, then so is PdF (X) ! PdG(X) ! PdH(X). Let DdF : C ! D be defined by letting DdF (X) be the homotopy fiber of PdF (X) ! Pd-1F (X). The lemma and theorem formally imply that DdF is homogeneous of degree d: it is d-excisive, and Pd-1DdF (X) ' * for all X. When D is T , Goodwillie discovered a remarkable fact: these fibers are canonically infinite loopspaces. For a general D, we let S(D) be the associ- ated category of `D-spectra' (see x3), and Goodwillie's second theorem then goes as follows. Theorem 2.3. [G3 , Thm.2.1] Let F : C ! D be homogeneous of degree d. Then there is a naturally defined homogeneous degree d functor F st: C ! S(D), such that, for all X 2 C, there is a weak equivalence F (X) ' 1 (F st(X)). The category S(D) is an example of a stable model category. In a manner similar to results in the algebra literature, Goodwillie relates homogenous degree d functors landing in a stable model category to symmetric multilin- ear ones. A functor L : Cd ! D is d-linear if it is homogeneous of degree 1 in each variable, and is symmetric if L is invariant under permutations of the coordinates of Cd. Goodwillie's third theorem goes as follows. Theorem 2.4. [G3 , Thm.3.5] Let F : C ! D be a homogeneous functor of degree d with D a stable model category. Then there is a naturally defined symmetric d-linear functor LF : Cd ! D, and a weak natural equivalence (LF (X, . .,.X))h d ' F (X). If F : C ! D is a homotopy functor with C and D either T or S, let CF (d) = L(DdF )st(S, . .,.S), a spectrum with d-action. Goodwillie refers to CF (d) as the dth Taylor coefficient of F due the following corollary of the last theorem. CALCULUS AND CHROMATIC HOMOTOPY 5 Corollary 2.5. In this situation, there is a weak natural transformation (CF (d) ^ X^d)h d ! (DdF )st(X) that is an equivalence if either X is a finite complex, or F commutes with directed homotopy colimits up to weak equivalence. As will be illustrated in the examples, these equivariant spectra have often been identified. The theorems above are the ones I wish to stress in these notes, but I should say a little about convergence. In [G2 ], Goodwillie carefully proves a generalized Blakers-Massey theorem, and uses it to study questions that are equivalent to the convergence of these towers in the cases when C is T or S. In particular, many functors can be shown to be `analytic', and an analytic functor F admits a `radius of convergence' r(F ) with the property that the tower for F (X) converges strongly for all r(F )-connected objects X. The number r(F ) is often known, as will be illustrated in the examples. A nice result from [G2 ] reads as follows. Proposition 2.6. [G2 , Prop.5.1] Let F ! G be a natural transformation between analytic functors, and let r be the maximum of r(F ) and r(G). If F (X) ! G(X) is an equivalence for all X that are equivalent to high suspensions then it is an equivalence for all r-connected X. 3.Model category prerequisites References for model categories include Quillen's orginal 1967 lecture notes [Q ], Dwyer and Spalinski's 1995 survey article [DS ], and the more recent books by Hovey and Hirschhorn [H1 , Hi]. 3.1. Nice model categories. We will assume that C and D are either simplicial or topological based model categories. `Based' means that the initial and final object are the same: we denote will this object by *. As part of the structure of a based topological (or simplicial) model cat- egory C, given K 2 T and X 2 C, one has new objects in C, X K and Map (K, X) satisfying standard properties. This implies that C supports canonical homotopy limits and colimits: given a functor X : J ! C from a small category J , hocolimJ X and holimJ X are defined as appropriate coends and ends: hocolim X = X(j) j2J EJ (j)+ , and J Z holim X = Map(EJ (j)+ , X(j)) J j2J With such canonical homotopy limits and colimits, C will support a sen- sible theory of homotopy Cartesian and coCartesian cubes, as discussed in [G2 ]: see x4.1 below. To know that certain explicit cubes in C are homotopy coCartesian, one also needs that C be left proper, and it seems prudent to 6 KUHN require both C and D to be proper: the pushout of a weak equivalence by a cofibration is a weak equivalence, and dually for pullbacks. D then needs a further axiom ensuring that the sequential homotopy colimit of homotopy Cartesian cubes is again homotopy Cartesian: assuming that D admits the (sequential) small object argument does the job: see [Sch, x1.3]. Examples 3.1. The following categories satisfy our hypotheses: o TY , the category of spaces over and under Y , o R - Mod , the category of R-modules, where R is an E1 ring spec- trum, a.k.a. commutative S-algebra [EKMM ], o R - Alg, the category of augmented commutative R-algebras, o simplicial versions of all of these, e.g. spectra as in [BF ]. 3.2. Spectra in model categories. Let D be a model category as above, and let X denote X S1. Trying to force the suspension : D ! D to be `homotopy invertible' leads to a model category of spectra S(D) in the `usual way': this has been studied carefully by Schwede [Sch] (following [BF ]), Hovey [H2 ], and Basterra-Mandell [BMa ]. Roughly put, an object in S(D) will consist of a sequence of objects X0, X1, X2, . .i.n D, together with maps Xn ! Xn+1. The point of this construction is that the model cate- gory structure S(D) has the additional property that it is stable: homotopy cofibration sequences in S(D) agree with the homotopy fibration sequences. The associated homotopy category will be triangulated. As in the familiar case when D = T , there are adjoint functors 1 : D ! S(D) and 1 : S(D) ! D. If D is already stable these functors form a Quillen equivalence. For an arbitrary D, this adjoint pair can take a surprising form, as the following example illustrates. Example 3.2. In [BMa ], the authors show that the category S(R - Alg) is Quillen equivalent to R-Mod so that 1 : R-Alg ! S(R-Alg ) identifies with the Topological Andr'e-Quillen Homology functor1 T AQ : R - Alg ! R-Mod , and 1 : S(R-Alg ) ! R-Alg identifies with the functor sending an R-module M to the trivial augmented R-algebra R _M. (Partial results along these lines were also proved in [BMc , Sch].) 3.3. Functors between model categories. Suppose C and D are nice topological model categories. There are couple of useful properties that a functor F : C ! D might have. ____________ 1To be precise, by TAQ(B) we mean the Topological Andr'e-Quillen Homology of* * B with coefficients in the B-bimodule R. CALCULUS AND CHROMATIC HOMOTOPY 7 Firstly F will usually be continuous: for all X and Y in C, the function F : Map C(X, Y ) ! Map D(F (X), F (Y )) should be continuous. If F is continuous, given X 2 C and K 2 T , there is a natural assembly map (3.1) F (X) K ! F (X K) defined by means of various adjunctions. The existence of these assembly maps implies that F will be a homotopy functor: a weak equivalence between fibrant cofibrant objects in C is carried by F to a weak equivalence in D. The second property that some functors F satisfy is that F commutes with filtered homotopy colimits, up to weak equivalence. A functor having this property has sometimes also been termed `continuous', but Goodwillie [G3 ] more cautiously uses the term finitary and so will we. One implication of being finitary is that the assembly map (3.1) will be an equivalence. Thus there are many interesting functors that are not finitary, as the next example shows. Example 3.3. Let LE : S ! S be Bousfield localization of spectra with respect to a spectrum E. Then LE is finitary exactly when the assembly map LE (S) ^ X ! LE (X) is a weak equivalence for all spectra X. In other words, LE is finitary exactly when it is smashing, a property that many interesting LE 's do not have. Just to confuse the issue, we note that if LE is regarded as taking val- ues in the topological model category LE S, in which equivalences are E*- isomorphisms and fibrant objects are E*-local [EKMM , Chap.VIII], then LE : S ! LE S is finitary. Finally, lets say a word about maps between functors. If C is not small, then it seems a bit daunting (set theoretically) to impose a model cate- gory structure on the class of functors F : C ! D. As an adequate fix for calculus purposes, we use the following terminology. Call a natural transformation f : F ! G a weak equivalence, and write F f-!~G, if f(X) : F (X) ! G(X) is a weak equivalence for all X in C. By a weak natural transformation f : F ! G we mean a pair of natural tranformations of the form F -g~ H -h!G or F -h! H -g~ G. We say that a diagram of weak natural transformations commutes if, after evaluation on any object X, the associated diagram commutes in the homotopy category of D. Fi- nally, we say that a diagram of functors F ! G ! H a fiber sequence if F (X) ! G(X) ! H(X) is a (homotopy) fiber sequence for all X. 8 KUHN 4.Cubical diagrams and polynomial functors 4.1. Cubical diagrams. We review some of the theory of cubical diagrams; a reference is [G2 , x1]. Let S be a finite set. The power set of S, P(S) = {T S} , is a partially ordered set via inclusion, and is thus a small category. Let P0(S) = P(S) - {;} and let P1(S) = P(S) - {S}. Definitions 4.1. (a) A d-cube in C is a functor X : P(S) ! C with |S| = d. (b) X is Cartesian if the natural map X (;) ! holim X (T ) T2P0(S) is a weak equivalence. (c) X is coCartesian if the natural map hocolim X (T ) ! X (S) T2P1(S) is a weak equivalence. (d) X is strongly coCartesian if X |P(T) : P(T ) ! C is coCartesian for all T S with |T | 2. Often S will be the concrete set d = {1, . .,.d}. Example 4.2. A 0-cube X (0) is Cartesian if and only if it is coCartesian if and only if X (0) is acyclic (i.e. weakly equivalent to the initial object ** *). Example 4.3. A 1-cube f : X (0) ! X (1) is Cartesian if and only if it is coCartesian if and only if f is an equivalence. Example 4.4. A 2-cube X (0)_____//_X (1) | | | | fflffl| fflffl| X (2)____//_X (12) is Cartesian if it is a homotopy pullback square, and coCartesian if it is a homotopy pushout square. Example 4.5. Strongly coCartesian d-cubes are equivalent to ones con- structed as follows. Given a family of cofibrations f(t) : X(0) ! X(t) for 1 t d, let X : d ! C be defined by X (T ) = the pushout of{f(t) | t 2 T }. (Note that X (T ) can be interpreted as the coproduct under X(0) of X(t), t 2 T .) Critical to Goodwillie's constructions, is a special case of this last exampl* *e. Definition 4.6. If T is a finite set, andaX is an object in C, let X * T be the homotopy cofiber of the folding map X ! X. T CALCULUS AND CHROMATIC HOMOTOPY 9 For T d, the assignment T 7! X * T is easily seen to define a strongly coCartesian d-cube X : if X ! * factors as X -i!CX -p!*, with i a cofi- bration and p an acyclic fibration, then X agrees with the cube of the last example with f(t) = i : X ! CX for all t. In the special case when C = T , X * T is the (reduced) join of X and T : the union of |T | copies of the cone CX glued together along their common base X. There is a very useful way to inductively identify Cartesian cubes. Note that the fibers of the vertical maps in a Cartesian 2-cube as in Example 4.4 form a Cartesian 1-cube as in Example 4.3. This generalizes to higher di- mensional cubes as we now explain. Regard d as the obvious subset of d + 1. Given an (d + 1)-cube X : P(d + 1) ! C, we define three associated d-cubes Xtop, Xbottom, @X : P(d) ! C as follows. Let Xtop(T ) = X (T ) and Xtop(T ) = X (T [ {n + 1}). Then define @X (T ) by taking homotopy fibers of the evident natural transformation between these: @X (T ) = hofib{Xtop(T ) ! Xbottom(T )}. Lemma 4.7. X is Cartesian if and only if @X is Cartesian. Lemma 4.8. If Xtopand Xbottomare Cartesian, so is X . Remark 4.9. Dual lemmas hold for coCartesian cubes. One application of this is that if C is a stable model category, so that homotopy fibre sequences are the same as homotopy cofiber sequences, then X is Cartesian if and only if X is coCartesian. 4.2. Polynomial functors. Let C and D be topological or simplicial model categories as in x3.1. Definition 4.10. F : C ! D is called d-excisive or said to be polynomial of degree at most d if, whenever X is a strongly coCartesian (d + 1)-cube in C, F (X ) is a Cartesian cube in D. Example 4.11. F has degree 0 if and only if F (X) ! F (*) is an equivalence for all X 2 C, i.e. F is homotopy constant. Example 4.12. F : C ! D is 1-excisive means that F takes pushout squares to pullback squares. In the classical case C and D are spaces or spectra, this implies that the functor sending X to ss*(F (X)) satisfies the Mayer-Vietoris property. If F is also finitary, then Milnor's wedge axiom holds as well. Then there are spectra C0 and C1 such that F (X) ' C0 _ (C1 ^ X) if D = S and F (X) ' 1 (C0 _ (C1 ^ X)) if D = T . 10 KUHN Remark 4.13. Without the finitary hypothesis, classifying 1-excisive func- tors seems very hard. Examples of 1-excisive functors of X from spectra to spectra include the localization functors LE X and functors of the form MapS (C, X) where C 2 S is fixed. The following proposition of Goodwillie constructs d-excisive functors out of d-variable 1-excisive functors. Proposition 4.14. [G2 , Prop.3.4] If L : Cd ! D is 1-excisive in each of the d-variables, then the functor sending X to L(X, . .,.X) is d-excisive. Corollary 4.15. In this situation, if L is symmetric, and D is a stable model category, then, given any subgroup G of the dth symmetric group n, the functor sending X to L(X, . .,.X)hG is d-excisive. The various lemmas about identifying Cartesian cubes can be used to prove the next two useful lemmas. Lemma 4.16. If F is d-excisive, then F is c-excisive for all c d. Lemma 4.17. If F ! G ! H is a fiber sequence of functors, and G and H are both d-excisive, then so is F . 5. Construction of Goodwillie towers and the proof of the main properties 5.1. Construction of the tower and the proof of Theorem 2.1. If one is to construct a d-excisive functor PdF , then PdF (X ) needs to be Cartesian for for all strongly coCartesion (d + 1)-cubes X . The idea behind the construction of PdF is to force this condition to hold for certain strongly coCartesion (d + 1)-cubes X . Fix an object X 2 C. As discussed above, for T d + 1, the assignment T 7! X * T defines a strongly coCartesian d + 1-cube X . For example, when d + 1 = 2, one gets the pushout square X ______//CX | | | | fflffl| fflffl| CX _____// X. Definition 5.1. Let TdF : C ! D be defined by TdF (X) = holim F (X * T ). T2P0(d+1 ) Note that there is an evident natural transformation td(F ) : F ! TdF , and that this is an equivalence if F is d-excisive. Definition 5.2. Let PdF : C ! D be defined by td(F) td(TdF) PdF (X) = hocolim{F (X) ---! TdF (X) -----! TdTdF (X) ! . .}.. CALCULUS AND CHROMATIC HOMOTOPY 11 Example 5.3. T1F (X) is the homotopy pullback of F (CX) | | fflffl| F (CX) _____//F ( X), Suppose that F (*) ' *. Then F (CX) ' *, so that T1F (X) is equivalent to the homotopy pullback of * | | fflffl| * _____//F ( X), which is F ( X). It follows that there is a natural weak equivalence P1F (X) ' hocolimn!1 nF ( nX). Example 5.4. Specializing the last example to the case when F is the identity functor Id : D ! D, we see that P1(Id)(X) ' 1 1 X. If D = T , topological spaces, we see that P1(Id)(X) = QX. If D = R - Alg, we see that P1(Id)(B) ' R _ T AQ(B) for an augmented commutative R-algebra B. The proof of Theorem 2.1 amounts to checking that the Pd construction just defined has the two desired properties: PdF should always be d-excisive, and F ! PdF should be universal. Checking the first of these is by far the more subtle, and follows from the next lemma. Lemma 5.5. [G3 , Lemma 1.9] If F : C ! D is a homotopy functor, and X is strongly coCartesian (d + 1)-cube in C, then there is a Cartesian (d + 1)- cube Y in D, such that F (X ) ! TdF (X ) factors through Y. The construction of Y is very devious. Y is (roughly) constructed to be the homotopy limit of (d + 1)-cubes in D that are each seen to be Cartesian for the following reason: they are constructed by applying F to (d + 1)- cubes in C formed by means of evident objectwise equivalences between two d-cubes. In contrast, proving that F ! PdF is appropriately universal is much easier. Once one knows that PdF is d-excisive, universality amounts to checking the following two things: (a) If F is d-excisive, then ed(F ) : F ! PdF is a weak equivalence, and (b) Pd(ed(F ))) : PdF ! PdPdF is a weak equivalence. 12 KUHN These follow immediately from Td-versions of these statements: (a') If F is d-excisive, then td(F ) : F ! TdF is a weak equivalence, and (b') Pd(td(F ))) : PdF ! PdTdF is a weak equivalence. As was noted above, the first of these is clear. The second admits a fairly simple proof based on the commutativity of iterated homotopy inverse limits. Similar reasoning verifies the next lemma, which in turn implies Lemma 2.2, which said that Pd preserves fiber sequences. Lemma 5.6. Given natural transformations F ! G ! H, if F (X) ! G(X) ! H(X) is a fiber sequence for all X, then so is TdF (X) ! TdG(X) ! TdH(X). 5.2. Delooping homogeneous functors and Theorem 2.3. The most surprising property of Goodwillie towers is stated in Theorem 2.3. This says that, for d > 0, homogeneous d-excisive functors are infinitely deloopable. To show this, Goodwillie proves his beautiful key lemma, which says that PdF (X) ! Pd-1F (X) is always a principal fibration if F is reduced: F (*) ' *. Lemma 5.7. [G3 , Lemma 2.2] Let d > 0, and let F : C ! D be a reduced functor. There exists a homogeneous degree d functor RdF : C ! D fitting into a fiber sequence of functors PdF ! Pd-1F ! RdF. Iteration of the Rd construction leads to Theorem 2.3: if F is homogeneous of degree d, then we can let F st(X) be the spectrum with rthspace RrdF (X). The proof of Lemma 5.7 is yet another clever manipulation of categories related to cubes. As an indication of how this might work, we sketch how one can construct a homotopy pullback square TdF (X) ______//KdF (X) | | | | fflffl| fflffl| Td-1F (X) _____//QdF (X) with KdF (X) ' *, in the case when d = 2. One needs to look at how one passes from P0(2) to P0(3). In pictures, P0(2) looks like 1 """ 2____12, CALCULUS AND CHROMATIC HOMOTOPY 13 while P0(3) looks like v1 vvv | 2 ______12 | | | | | 3 ___|____13 | """ | vvv 23 _____123. Now we decompose the poset P0(3) as (5.1) A \ B ___________//B | | | | fflffl| fflffl| A _______//A [ B = P0(3), where A is 3 ________13 """ www 23 _____123, and B is v 1 vvv | 2 _____ 12 | | | | | | w13 | | ww 23 ____123, so that A \ B is 13 vvv 23 ____123. The decomposition of posets (5.1) induces a homotopy pullback diagram holimT2P0(3)F (X * T )_____//_holimT2AF (X * T ) | | | | fflffl| fflffl| holim T2BF (X * T )______//_holimT2A\BF (X * T ). The top left corner is T3F (X), by definition. As P0(2) is cofinal in B, the bottom left corner is equivalent to T2F (X). Finally A has initial object {3}, so that the upper left corner is contractible: holimF (X * T ) ' F (X * {3}) = F (CX) ' *. T2A 5.3. Cross effects and the proof of Theorem 2.4. Definition 5.8. Let F : C ! D be a functor. We define crdF : Cd ! D, the dth cross effect of F , to be the the functor of d variables given by ` ` (crdF )(X1, . .,.Xd) = hofib{F ( Xi) ! holim F ( Xi)}. i2d T P0(d) i2d-T 14 KUHN ` The d-cube sending T to Xi is easily seen to be strongly coCarte- i2d-T sian; letting d = 2 for example, the square X1 _ X2 ____//_X2 | | | | fflffl| fflffl| X1 ________//* is weakly equivalent to the evidently coCartesian square X1 _ X2 _______//CX1 _ X2 | | | | fflffl| fflffl| X1 _ CX2 _____//CX1 _ CX2. It follows that if F is (d - 1)-excisive, then crdF (X1, . .,.Xd) ' * for all Xi. A similar argument [G3 , Lemma 3.3] shows that if F is d-excisive, then crdF is 1-excisive in each of its variables. Another property of crdF : Cd ! D that is easy to see is that it is reduced: crdF (X1, . .,.Xd) ' * if Xi' * for some i. A permutation of d, oe 2 d, induces an evident isomorphism oe* : crdF (X1, . .,.Xd) ! crdF (Xoe(1), . .,.Xoe(d)), satisfying (oe O o)* = oe* O o*: a functor of d variables with this structure is called symmetric. Definition 5.9. Let LdF : Cd ! D be the functor obtained from crdF by applying P1 to each variable. Thus we have LdF (X1, . .,.Xd) ' hocolimn n1+...+ndcrdF ( n1X1, . .,. ndXd). i!1 LdF will always be symmetric and d-linear, and if F is d-excisive, then the natural map crdF ! LdF is an equivalence. If G is a finite group, let G - D denote the category of objects in D with a G-action. Given Y 2 G - D, we let YhG and Y hG denote the associated homotopy quotient and fixed point objects in D. Definition 5.10. Let dF : C ! d - D be defined by dF (X) = LdF (X, . .,.X). A more precise version of Theorem 2.4 is the following. Theorem 5.11. Let F : C ! D be a homotopy functor, with D a stable model category. Then there is a natural weak equivalence dF (X)h d ' DdF (X). If F is d-excisive then dF (X) can be identified with (crdF )(X, . .,.X). In this case, one gets a natural transformation ffd(X) : ( dF )(X)h d ! F (X) CALCULUS AND CHROMATIC HOMOTOPY 15 defined to be the composite `d ( dF )(X)h d ! F ( X)h d ! F (X). i=1 W d Here the second map is induced by the fold map i=1X ! X. The theorem is then proved by verifying that crd(ffd) is an equivalence. We indicate how Corollary 2.5 follows from Theorem 5.11. The assembly map for F induces an assembly map ( dF (X) K^d)h d ! dF (X K)h d, for X 2 C and K 2 T . If C is T , D is S, and X = S, then this reads (CF (d) ^ K^d)h d ! dF (K)h d, where CF (d) = dF (S). By construction, this map is the identity if K = S, and it follows that it will be an equivalence for all finite K, or all K under the additional hypothesis that F is finitary. A similar argument holds if both C and D are S: here the assembly map can be constructed for all K 2 S. When the domain category C is also stable, and D = S, there is an elegant addendum to Theorem 5.11 essentially due to R.McCarthy [McC ]. Given Y 2 G - S, there is a natural norm map N(Y ) : YhG ! Y hG satisfying the property that N(Y ) is an equivalence if Y is a finite free G- CW spectrum. As in [K5 ], we let the Tate spectrum of Y , TG (Y ), be the cofiber. Proposition 5.12. [K5 ] Let F : C ! S be any homotopy functor, with C stable. For all d 1, there is a homotopy pullback diagram PdF (X) ______//_( dF (X))h d | | | | fflffl| fflffl| Pd-1F (X) _____//T d( dF (X)). 6. Examples 6.1. Suspension spectra of mapping spaces. Fix a finite C.W. complex K. Let MapT (K, X) be the space of based continuous maps from K to a space X. Similarly, given a spectrum Y , let MapS (K, Y ) be the evident function spectrum. In [G2 ], Goodwillie proved that the functor from spaces to spectra sending X to 1 MapT (K, X) is analytic with radius of convergence equal to the dimension of K. In [A ], Arone gave a very concrete model for the associated Goodwillie tower {P*K(X)}. The paper [AK ] includes further details about Arone's construction while building in extra structure. 16 KUHN Let E be the category with objects the finite sets d, d 1, and with morphisms the epic functions. Ed will denote the full subcategory with objects c with c d. Given a based space X, let X^ : Eop ! T be the functor sending d to X^d. Then Arone's model for PdK : T ! S is given by PdK(X) = MapS Ed(K^ , 1 X^ ), the spectrum of natural transformations between the two contravariant func- tors of Ed. The natural transformation 1 MapT (K, X) ! PdK(X) is induced by sending f : K ! X to f^ : K^ ! X^ and then stablizing. A by product of this construction is that there is a homotopy pullback square of S-modules which has some of the same flavor as Proposition 5.12: PdK(X) ______//_MapS d(K^d, 1 X^d) | | | | fflffl| fflffl| PdK-1(X) ____//_MapS d(ffid(K), 1 X^d), where ffid(K) K^d denotes the fat diagonal. Thus the dth fiber, DKd(X), can be described as follows. Let K(d)denote K^d=ffid(K). Then we have DKd(X) = MapS d(K(d), 1 X^d) ' MapS (K(d), 1 X^d)h d ' (D(K(d)) ^ X^d)h d. Here D(K(d)) denotes the equivariant S-dual of K(d), and the equivalences follow from the fact that K(d) is both finite and d-free away from the basepoint. It follows that the dth Taylor coefficient of the functor sending X to 1 MapT (K, X) is D(K(d)). Remark 6.1. In [K3 ], we observed that, when X is also a finite complex, the tower P*K(X) also arises as by taking the S-dual of a natural filtration on the nonunital commutative S-algebra D(X) K. By Alexander duality, D(K(d)) can be identified an appropriate equi- variant desuspension of the suspension spectrum of a configuration space. Specializing to the case when K = Sn, this takes the following concrete form. Let C(n, d) denote the space of d distinct little n-cubes in a big n- cube [May ]. Via a Thom-Pontryagin collapse, there is a very explicit duality map of d spaces [AK ] C(n, d)+ ^ Sn(d)! Snd CALCULUS AND CHROMATIC HOMOTOPY 17 One proof that Arone's model works when K = Sn goes roughly as fol- lows. Suppose X = nY . One has the usual filtered configuration space model Cn(Y ) for n nY [May ]. Thus one has maps n 1 FdCn(Y ) ! 1 n nY ! PdS (Y ). The nontriviality of the second map is proved by showing that the composite is an equivalence. By induction on d, is suffices to show that crd applied to this composite is an equivalence, and the verification of that leads back to the above explicit duality map. A bonus corollary of this proof is that one also establishes a rather nice version of `Snaith splitting': the tower strongly splits when X = nY . Example 6.2. One application of this comes from applying mod p coho- mology to the tower. One obtains a spectral sequence of differential graded algebras {Es,tr(Sn, X)} with Ed,*1(Sn, X) = H*((C(n, d)+ ^ ( -dX)^d)h d; Z=p)) and converging strongly to H*( nX; Z=p) if X is n-connected. This E1 term is a known functor of H*(X; Z=p). The differentials have not been fully explored, but seem to be partly determined by derived functors of destablization of unstable modules over the Steenrod algebra, as applied to the A-module -n H*(X; Z=p). 6.2. Suspension spectra of infinite loopspaces.1The previous example can be used to determine the tower {PdS } for the functor from spectra to spectra sending a spectrum X to 1 1 X. Let Xn denote the nth space of the spectrum X. Then we have that nXn ' 1 X for all n, and the natural map hocolimn!1 -n 1 Xn ! X is an equivalence. From this and the last example, one can deduce that the tower converges for 0-connected spectra X and that n S1 hocolimn!1 -n PdS (Xn) ' Pd (X). As hocolimn!1C(n, d)+ is a model for E d+, and this is weakly equivalent to S0, it follows that the formula for the dth fiber is 1 ^d DSd (X) ' Xh d, and thus the dth Taylor coefficient of the functor sending a spectrum X to 1 1 X is the sphere spectrum S for all d > 0. 18 KUHN Finally, Proposition 5.12 specializes to say that for each d > 0 there is a pullback square PdS1(X) ______//(X^d)h d | | | | 1fflffl| fflffl| PdS-1(X)_____//T d(X^d). Example 6.3. The tower begins P2S1(X)88 qqq e2qqqq p1|| qqqqe1 fflffl| 1 1 X ________//X, where e1 : 1 1 X ! X is adjoint to the identity on 1 X. A formal consequence is that 1 1 1 p1 : 1 P2S (X) ! X admits a natural section. The map p1 fits into a natural cofibration sequence 1 p1 P2S (X) -! X ! (X ^ X)h 2. Specializing to the case when X = S-1 , this can be identified [K5 , Appendix] with the cofibration sequence -1RP01 t-!S-1 ! RP-11, where t is one desuspension of the Kahn-Priddy transfer, and RPk1 denotes the Thom spectrum of k copies of the canonical line bundle over RP 1. Letting QZ denote 1 1 Z, we conclude that 1 tr : QRP+1 ! QS0 admits a section: a result `one loop' away from the full strength of the Kahn- Priddy Theorem [KaP ] at the prime 2. The odd prime version1admits a similar proof using that, localized at a prime p, PdS (S-1 ) ' * for 1 < d < p. 6.3. The identity functor for Alg. Let Alg be the category of commuta- tive augmented S-algebras. This is a model category in which weak equiva- lences and fibrations are determined by forgetting down to S-modules. More curious is that the coproduct of A and B is A ^ B. Let {Pdalg} denote the tower associated to the identity I : Alg ! Alg . Given B 2 Alg , it is not too hard to deduce that the tower {Pdalg(B)} will strongly converge to B if I(B) is 0-connected, where I(B) denotes the `augmentation ideal': the homotopy fiber of the augmentation B ! S. Let Dalgd(B) be the fiber of Pdalg(B) ! Pdalg-1(B). As already discussed in Example 5.4, Dalg1(B) can be identified with T AQ(B), the Topological Andr'e-Quillen Homology of B with coefficients in the B-bimodule S. CALCULUS AND CHROMATIC HOMOTOPY 19 The fact that coproducts in Alg correspond to smash products of S- modules leads to a simple calculation of the dth cross effect of I: crd(I)(B1, . .,.Bd) ' I(B1) ^ . .^.I(Bd). From this, one gets a formula for Dalgd(B): Theorem 6.4. Dalgd(B) ' T AQ(B)^dh.d A proof of this in the spirit of this paper appears in [K4 ]. See also [Min ]. A nice corollary of this formula says the following. Corollary 6.5. If A and B in Alg have 0-connected augmentation ideals, then an algebra map f : A ! B is an equivalence if T AQ(f) is. The converse of this corollary - that T AQ(f) is an equivalence if f is - is true even without connectivity hypotheses: see, e.g. [K3 ]. Without any hypotheses implying convergence, one has that if T AQ(f) is an equivalence, so is ^f: ^A! ^B, where A^denotes the homotopy inverse limit of the tower for A [K4 ]. Example 6.6. This tower overlaps in an interesting way with the one for 1 1 X discussed above, and the corollary leads to a simple proof of a highly stuctured version of the classical stable splitting [Ka ] of QZ, for a connected space Z. It is well known that 1 ( 1 X)+ is an E1 ring spectrum. Otherwise put, we can regard 1 ( 1 X)+ as an object in Alg. It is not hard to see that T AQ( 1 ( 1 X)+ ) is equivalent to the connective cover of X, and there is an equivalence 1 alg 1 1 PdS (X) ' Pd ( ( X)+ ) for connective spectra X. Another object in Alg is P(X), the free commutative S-algebra generated by X. As an S-module, 1` P(X) ' X^dh,d d=0 and it is not hard to compute that T AQ(P(X)) ' X. The stable splitting of QZ gets proved as follows. The inclusion j(Z) : Z ! QZ induces a natural map in Alg s(Z) : P( 1 Z) ! 1 (QZ)+ . The construction of s makes it quite easy to verify that T AQ(s(Z)) : 1 Z ! 1 Z is the identity. The above corollary then implies that s(Z) is an equivalence in Alg, and thus in S, for connected spaces Z. A more detailed discussion of this appears in [K4 ]. 20 KUHN 6.4. The identity functor for T . Let {Pd} denote the tower of the iden- tity functor on T . Given a space Z, let Dd(Z) denote the fiber of Pd(Z) ! Pd-1(Z), and then let Dstd(Z) 2 S be the infinite delooping provided by Theorem 2.3. Goodwillie's estimates [G2 ] show that the tower {Pd(Z)} will strongly con- verge to Z when Z is connected. Thus applying ss*, one gets a strongly con- vergent 2nd quadrant spectral sequence converging to ss*(Z), with E1-d,*= ss*(Dstd(Z)). Johnson [J], and Arone with collaborators Mahowald and Dwyer [AM , AD ], have identified the spectra Dstd(Z): Dstd(Z) ' (D( Kd) ^ Z^d)h d, where Kd is the unreduced suspension of the classifying space of the poset of nontrivial partitions of d, and D denotes the equivariant S-dual, as before. Using this model, the discovery of Arone and Mahowald [AM ] is that when Z is an odd dimensional sphere, these spectra are very special spectra that were known previously. To state the theorem, we need some notation. Let p be a prime. Let maek denote the direct sum of m copies of the reduced real regular representation of Vk = (Z=p)k. Then GLk(Z=p) acts on the Thom space (BVk)maek. Let ek 2 Z(p)[GLk(Z=p)] be any idempotent in the group ring representing the Steinberg module, and then let L(k, m) be the associated stable summand of (BVk)maek: L(k, m) = ek(BVk)maek. The spectra L(k, 0) and L(k, 1) agree with spectra called M(k) and L(k) in the literature: see e.g. [MP ], [K1 , KuP ]. Collecting results from [AM ] and [AD , Thm.1.9, Cor.9.6], one has the following theorem. Theorem 6.7. Let m be an odd natural number. (1) Dstd(Sm ) ' * if d is not a power of a prime. (2) Let p be a prime. Dstpk(Sm ) ' m-k L(k, m), and thus has p-torsion homotopy if k > 0. (3) H*(L(k, m); Z=p) is free over the subalgebra A(k - 1) of the Steenrod algebra. As a function of k, the connectivity of L(k, m) has a growth rate like pk. Thus the associated spectral sequences for computing the unstable ho- motopy groups of odd spheres coverges exponentially quickly, and begins from stable information about spectra of roughly the same complexity as the suspension spectra of classifying spaces of elementary abelian p-groups. Remark 6.8. When m = 1, one gets a spectral sequence converging to the known graded group ss*(S1), with E1-k,*= ss*(L(k)). Comparison with my CALCULUS AND CHROMATIC HOMOTOPY 21 work on the Whitehead Conjecture [K1 , KuP ] suggests that E2 = E1 . Greg Arone and I certainly believe this, but a rigorous proof has yet to be nailed down. As discussed near the end of the next section, the properties listed in the theorem have particularly beautiful consequences for computing the periodic unstable homotopy groups of odd dimensional spheres. 7. Interactions with periodic homotopy For topologists who study classical unstable and stable homotopy theory, a major development of the past two decades has been the organization of these subjects via the chromatic filtration associated to the Morava K- theories. One of the most unexpected aspects of Goodwillie towers is that they interact with the chromatic aspects of homotopy in striking ways. In this section, I survey, in inverse order of when they were proved, three different theorems of this sort. 7.1. Goodwillie towers and homology isomorphisms. There are a couple of useful general facts about how Bousfield localization relates to Goodwillie towers. Let E* be a generalized homology theory. A map f : X ! Y of spaces or spectra is called an E*-isomorphism if E*(f) is an isomorphism. A nat- ural transformation f : F ! G between functors F, G : C ! S is an E*- isomorphism if f(X) is for all X 2 C. Then we have Proposition 7.1. [K6 , Cor. 2.4] If F : C ! S is finitary and f : X ! Y is an E*-isomorphism then so are DdF (f) : DdF (X) ! DdF (Y ) and PdF (f) : DdF (X) ! PdF (Y ) for all d. Proposition 7.2. [K5 , Lemma 6.1] If a natural transformation f : F ! G between functors F, G : C ! S is an E*-isomorphism then so are Ddf : DdF ! DdG and Pdf : PdF ! PdG for all d. Both of these follow by observing that the various constructions defining Pd and Dd preserve E*-isomorphisms. The next example illustrates that the finitary hypothesis in Proposi- tion 7.1 is needed. Example 7.3. Consider LHZ=p : S ! S, a homogeneous functor of degree 1. Then HZ=p1 is HQ*-acyclic (i.e. HZ=p1 ! * is an HQ*-equivalence), but LHZ=p(HZ=p1 ) = HZp is not. For an application of Proposition 7.1 to the homology of mapping spaces, see [K6 ]. Proposition 7.2 is crucially used in proofs of the two theorems discussed in the next two subsections. We end this subsection with some observations related to the phenomenon illustrated in the last example. 22 KUHN If F : C ! S is homogeneous of degree d, the functor LE F : C ! S will always again be d-excisive, but need no longer be homogeneous. Example 7.4. Let F : S ! S be defined by F (X) = (X ^ X)h 2. The composite functor LE F will be 2-excisive, but need no be longer homoge- nous, even when restricted to finite spectra. Indeed, a simple calculation shows that P1(LE F )(S) = hocofib{LE S ^ RP 1 ! LE RP 1}. This can easily be nonzero. For example, when E is mod 2 K-theory, one has that LE RP 1 = LE S, as the transfer RP 1 ! S is an KZ=2*-isomorphism. It follows that P1(LE F )(S) has nonzero rational homology. As a fix for this problem, we have the next lemma, which follows from Proposition 7.2. Lemma 7.5. If F : C ! S is homogeneous of degree d, then Pd-1(LE F ) is E*-acyclic. Otherwise said, Dd(LE F ) ! LE F is an E*-isomorphism. 7.2. Goodwillie towers and periodic localization. We will consider two families of periodic homology theories. Fixing a prime p, K(n)* is the nth Morava K-theory. To define the second family, recall that a p-local finite complex M is of type n if M is K(m)*-acyclic for m < n, but is not K(n)*-acyclic. If M is of type n, then M admits a vn-self map, a K(n)*-isomorphism v : dM ! M. We let T (n) denote the telescope of v. A consequence of the Nilpotence and Periodicity Theorems of Devanitz, Hopkins, and J.Smith [DHS , HS , R] is that the associated Bousfield localization functor LT (n) : S ! S is independent of the choice of both the complex and self map. Also, we recall that T (n)*-acyclics are K(n)*-acyclic; thus the associated localization functors are related by LK(n) ' LK(n)LT(n). The main theorem of [K5 ] says that Goodwillie towers of functors from spectra to spectra always split after applying LT(n). Theorem 7.6. [K5 , Thm.1.1] Let F : S ! S be any homotopy functor. For all primes p, n 1, and d 1, the natural map pd(X) : PdF (X) ! Pd-1F (X) admits a natural homotopy section after applying LT(n). Corollary 7.7. Let F : S ! S be polynomial of degree less than d and G : S ! S homogeneous of degree d. Then any natural transformation f : F ! LT(n)G will be null. The corollary follows formally from the theorem using Lemma 7.5: we leave verifying this as an exercise for the reader. The theorem is proved by CALCULUS AND CHROMATIC HOMOTOPY 23 combining Proposition 5.12 and Proposition 7.2 with the following vanishing theorem about Tate homology. Theorem 7.8. [K5 , Thm.1.5] For all finite groups G, primes p, and n 1, LT(n)TG (LT(n)S) ' *. In [K5 ], I manage to first reduce the proof of the theorem to the case when G = p. There are familiar `inverse limits of Thom spectra' models for LT(n)T p(LT(n)S). Using these, the equivalence LT(n)T p(LT(n)S) ' * can be shown to be equivalent to the case when X = S of the following statement about the Goodwillie tower of 1 1 X. 1 -k Theorem 7.9. [K5 , Thm.3.7] holim kLT(n)PpS ( X) ! LT(n)X admits k a homotopy section. This theorem follows immediately from the existence of the natural section jn(X) of LT(n)e1(X) : LT(n) 1 1 X ! LT(n)X to be discussed in the next subsection. Remark 7.10. A weaker version of Theorem 7.8 with K(n) replacing T (n) appears in work by Greenlees, Hovey, and Sadofsky [GS , HSa ], and certainly inspired my thinking, if not my proof. Theorem 7.8 when G = Z=2 is equivalent to the main theorem of [MS ]. 7.3. The periodic homology of infinite loopspaces. Using the full strength of the Periodicity Theorem, Bousfield and I have constructed `tele- scopic functors' as in the next theorem. Theorem 7.11. [B1 , K2 , B3 ] For all p and n > 0, there exists a functor n : T ! S such that n( 1 X) ' LT(n)X. Furthermore n(Z) is always T (n)-local. Some further nice properties of n will be discussed in the next subsection: see Proposition 7.16. Here we note the following corollary. Corollary 7.12. After applying LT(n), the natural transformation e1(X) : 1 1 X ! X admits a section jn(X) : LT(n)X ! LT(n) 1 1 X. The section is defined by applying n to the natural map j( 1 X) : 1 X ! Q 1 X. Remark 7.13. jn is unique up to `tower phantom' behavior in the following sense: for all d, the composite jn(X) 1 1 LT(n)ed(X) S1 LT(n)X ----! LT(n) X -------! LT(n)Pd (X) LT(n)pd(X) is the unique natural section of LT(n)PdS1(X) -------! LT(n)X. Here uniqueness is a consequence of Corollary 7.7. 24 KUHN In [K4 ], I use jn to prove a splitting result in a manner similar to Exam- ple 6.6. The natural transformation jn(X) : X ! LT(n) 1 1 X induces a map of commutative augmented LT(n)S-algebras sn(X) : LT(n)P(X) ! LT(n) 1 ( 1 X)+ . As Example 6.6, sn(X) has been constructed so that it is easy to see that T AQ(sn(X)) : LT(n)X ! LT(n)X is homotopic to the identity, and one learns that sn(X) induces an equivalence of localized Goodwillie towers. Because the towers have been localized with respect to a nonconnected ho- mology theory, the convergence of localized towers is problematic. However, one can easily deduce the first statement of the next theorem, and starting from this, I was able to establish the rest. Theorem 7.14. [K4 ] For all X 2 S, sn(X)* : K(n)*(P(X)) ! K(n)*( 1 X) is monic, and fits into a chain complex of commutative K(n)*-Hopf algebras n+1O K(n)*(P(X)) sn*--!K(n)*( 1 X) ! K(n)*(K(ssj(X), j)). j=0 This sequence of Hopf algebras is exact if X is T (m)*-acyclic for all 0 < m < n, and only if X is K(m)*-acyclic for all 0 < m < n. Note that the two acyclicity conditions on X are empty if n = 1. They agree if n = 2, by the truth of the Telescope Conjecture when n = 1. Recall that the Telescope Conjecture asserts that a K(n)*-acyclic spec- trum will always be T (n)*-acyclic, and is believed to be not true for n 2. Peter May has remarked that maybe the theorem could be used to disprove it. I end this subsection by describing one way this might go. If Z is a connected space, let jn(Z) : LT(n)P(Z) ! LT(n)P(Z) be the composite sn( 1 Z) 1 LT(n)s(Z)-1 LT(n)P(Z) ------! LT(n) QZ --------! LT(n)P(Z). Here we have written P(Z) for P( 1 Z). The theorem says that if Z is T (m)*-acyclic for all 0 < m < n, and only if Z is K(m)*-acyclic for all 0 < m < n, there is a short exact sequence of K(n)*-Hopf algebras n+1O K(n)*(P(Z)) jn*--!K(n)*(P(Z)) ! K(n)*(K(ssSj(Z), j)). j=0 It appears that for some Z, a calculation of both K(n)*(P(Z)) and jn* may be accessible. If one could find a K(n - 1)*-acyclic space2 Z, and ____________ 2If a space Z is K(n - 1)*-acyclic, then it is K(m)*-acyclic for all m < n b* *y [B2]. CALCULUS AND CHROMATIC HOMOTOPY 25 explicit calculation showed that the above sequence is not exact, it would follow that Z would not be T (m)*-acyclic for some 0 < m < n. The first example to check is when p = 2, n = 3, and Z = K(Z=2, 3): it is known that this space is K(2)*-acyclic, but it is unknown whether or not it is T (2)*-acyclic. 7.4. The periodic homotopy groups of odd dimensional spheres. Let v : dM ! M be a K(n)*-isomorphism of a space M whose suspension spectrum is a finite compex of type n. If Z is a space, one can use v to define periodic homotopy groups by letting v-1 ss*(Z; M) = colimr[ rdM, Z]*. It is clear that these behave well with respect to fibration sequences in the Z variable. These can be similarly defined for spectra, and it is evident that there is an isomorphism v-1 ss*( 1 X; M) = v-1 ss*(X; M). The direct limit appearing in the definition suggests that these functors of spaces do not necessarily commute with holimits of towers. However Arone and Mahowald note that the properties listed in Theorem 6.7 imply that the tower for an odd dimensional sphere leads to a convergent spectral sequence with only a finite number of infinite loop fibers for computing periodic homotopy. More precisely, they show prove the following. Theorem 7.15. Let m be odd. With (M, v) as above, the natural map v-1 ss*(Sm ; M) ! v-1 ss*(PpnSm ; M) is an isomorphism. Bousfield notes that periodic homotopy can be computed using the tele- scopic functor n (at least for the version of n defined in [B3 ]). Proposition 7.16. [B3 , Thm.5.3(ii) and Cor.5.10(ii)] There are natural isomorphisms v-1 ss*(Z; M) ' [M, n(Z)]*. Furthermore, given f : Y ! Z, v-1 ss*(f; M) is an isomorphism if and only if n(f) : n(Y ) ! n(Z) is a weak equivalence. Assembling all of the various results, one deduces the following theorems. Theorem 7.17. When m is odd, there is a spectral sequence converging to v-1 ss*(Sm ; M) with ( [M, LT(n)L(k, m)]*+k-m for0 k n E1-k,*= 0 otherwise. Theorem 7.18. Let m be odd. The spectrum n(Sm ) admits a finite de- creasing filtration with fibers LT(n) m-k L(k, m) for k = 0, . .,.n. 26 KUHN Example 7.19. Using that L(0, m) = S0, L(1, m) = RP 1=RP m-1, and some naturality properties of Goodwillie towers, one can quite easily deduce that there is a weak equivalence 1(S2k+1) ' 2k+1LK(1)RP 2k. One can define periodic homotopy groups with integral coefficients. Ob- serve that v-1 ss*(Z; M) is really only dependent on the spectrum 1 M. As in [K2 ], one can construct a sequence under S of finite spectra of type n, C1 ! C2 ! . . . so that the induced map C(n) ! S is a K(n)*-isomorphism, where C(n) = hocolimrCr. One defines v-1 ss*(Z) by letting v-1 ss*(Z) = colimrv-1 ss*(Z; D(Cr)). The results above show that alternatively this can be computed as v-1 ss*(Z) = C(n)*( n(Z)). 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