THE GOODWILLIE TOWER OF THE IDENTITY FUNCTOR AND THE UNSTABLE PERIODIC HOMOTOPY OF SPHERES. June 9, 1998

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1 THE GOODWILLIE TOWER OF THE IDENTITY FUNCTOR AND THE UNSTABLE PERIODIC HOMOTOPY OF SPHERES GREG ARONE AND MARK MAHOWALD June 9, 1998 Abstract. We investigate Goodwiie s Tayor tower of the identity functor from spaces to spaces. More specificay, we reformuate Johnson s description of the Goodwiie derivatives of the identity, and prove that in the case of an odddimensiona sphere the ony ayers in the tower that are not contractibe are those indexed by powers of a prime. Moreover, in the case of a sphere the tower is finite in v k -periodic homotopy. Contents 0. Introduction 2 1. The poset of partitions of a finite set 5 2. The ayers of the Goodwiie tower of the identity Odd sphere case - the cohomoogy of the ayers The homoogy groups 3.2. Action of the Steenrod agebra 4. The v k -periodic homotopy of the tower The case of an odd-dimensiona sphere 4.2. The case of an even-dimensiona sphere Appendix A. Background on v k -periodic homotopy 36 References Mathematics Subject Cassification. Primary 05A18, 55P47, 55Q40, 55S12; Secondary 55P42, 55R12, 55S10. Key words and phrases. poset of partitions, Dyer-Lashof operations, periodic homotopy, Goodwiie cacuus. The first author was partiay supported by the Aexander von Humbodt foundation. 1

2 2 GREG ARONE AND MARK MAHOWALD 0. Introduction In this paper we anayze the Goodwiie tower of the identity functor evauated at spheres. We find that in the case of spheres the tower exhibits a peasant and surprising behavior. Broady speaking, we find two new facts that are not consequences of the genera theory of cacuus. First, in the case of an odd-dimensiona sphere ocaized at a prime p, the ony ayers (homotopy fibers) in the Goodwiie tower of the identity that are not contractibe are the ones that are indexed by powers of p. Thus the tower converges exponentiay faster in this case than it does in genera. Second, the stabe cohomoogy of the p k -th ayer is free over A[k 1], where A[k 1] is a certain finite sub-hopf agebra of the Steenrod agebra (to be defined in section 3.2). This impies, in particuar, that in our case a the ayers beyond the p k -th one are trivia in v k -periodic homotopy (for any reasonabe definition of the atter). Thus, in v k -periodic homotopy the tower has ony k + 1 non-trivia ayers, namey p 0,p 1,...,p k. The two facts impy that the unstabe v k -periodic homotopy of of an odd dimensiona sphere can be resoved into a tower of fibrations with k + 1 stages, with infinite oop spaces as fibers. As indicated above, the fibers are anayzed here to a considerabe extent. For instance, their stabe cohomoogy is competey cacuated. In the body of the paper we wi assume basic famiiarity with the notion of v k - periodic homotopy of spaces and spectra. For an informa discussion of the concept, together with references to a more compete discussion, see appendix A. We wi aso assume famiiarity with the basic ideas of Goodwiie s Cacuus of Functors. The basic references for this materia are [G90, G92, G3]. We now proceed with a more detaied overview of the paper, its genesis and its goas. The simpest exampe of periodic homotopy is v 0 -periodic homotopy, which is essentiay the same as rationa homotopy. There is an od theorem of Serre on rationa homotopy of spheres, which impies that if X is an odd-dimensiona sphere, then the map X Ω Σ X induces an equivaence in v 0 -periodic homotopy. Thus in the v 0 -periodic word the unstabe homotopy of an odd sphere is the same as its stabe homotopy. In [MT92] Mahowad and Thompson found an anaogue of Serre s theorem for v 1 - periodic homotopy. Roughy speaking, v 1 -periodic homotopy is the homotopy theory one obtains by inverting the maps that induce an isomorphism in K-theory. For a based topoogica space X, etp 2 (X) be the homotopy fiber of the we-known natura map Ω Σ (X) Ω Σ (X X Σ2 EΣ 2+ ) which may be defined, at east up to homotopy, as the adjoint of the composed map Σ Ω Σ (X) Σ (X i Σi EΣ i+ ) Σ (X X Σ2 EΣ 2+ ) i=1 where the first map is given by the Snaith spitting and the second map is coapsing

3 THE GOODWILLIE TOWER OF THE IDENTITY 3 on the factor corresponding to i = 2. Mahowad-Thompson s work impies that if X is an odd sphere (ocaized at 2) then the natura map X P 2 (X) induces an equivaence in v 1 -periodic homotopy. Using this resut, the v 1 -periodic homotopy of spheres has actuay been computed in [M82] at the prime 2 and in [T90] at odd primes. From this information, it is aso possibe to recover the integra v 1 -periodic homotopy. This is done in severa paces [MT92, T90]. From one point of view, our goa here is to extend the work of Mahowad and Thompson cited above to higher order periodicity. The first technica difficuty with it seemed to be that this work used the existence of maps connected with the Snaith spitting. Such maps are constructed by means of configuration space methods. The homotopy fibers of these maps do not have nice configuration space modes and thus do not aow new maps to be constructed in the same way. Instead, we use the Goodwiie tower of the identity functor, which turned out to be the perfect too for attacking this probem. We wi anayze this tower of fibrations in the genera case to some extent and appy this understanding to spheres. The Goodwiie tower of the identity ( the Tayor tower of the identity in Goodwiie s terminoogy) is a sequence of functors (from pointed spaces to pointed spaces) P n (X) and a tower of natura transformations.? X - P n (X) D n (X) Q QQQ f n Q Qs? P n 1 (X) D n 1 (X) f n 1?.? P 1 (X) Q(X) :=Ω Σ (X) The functor D n is the homotopy fiber of the natura transformation P n P n 1 and shoud be thought of as the n-th homogeneous ayer, or the n-th differentia of the identity. It foows from the genera theory of cacuus [G3] that for every n there exists a spectrum C n, endowed with an action of the symmetric group Σ n, such that D n (X) Ω ((C n X n ) hσn ):=Ω ((C n X n EΣ n+ ) Σn ).

4 4 GREG ARONE AND MARK MAHOWALD Here, as we as everywhere ese in the paper, stands for weaky homotopy equivaent. The spectrum C n, considered as a spectrum with an action of Σ n,isthen-th derivative of the identity. We need to investigate this tower, whose existence derives from the genera theory. Some information about it had been avaiabe before. As indicated in the diagram above, it is immediate from the definitions that P 1 (X) Q(X), i.e., the inear part of homotopy theory is stabe homotopy theory. The description of the second stage is sti rather cassica: as was indicated above, the second quadratic approximation, P 2 (X), is the homotopy fiber of the stabe James-Hopf map Q(X) Q(XhΣ 2 2 ). The second ayer of the tower is D 2 (X) ΩQ(XhΣ 2 2 ). For a genera n, B. Johnson was the first one to provide an expicit cosed description of D n (X) in terms of standard constructions of homotopy theory. In [Jo95] certain spaces n are constructed, which have the foowing properties: (i) the group Σ n acts on n, (ii) non-equivarianty, n (n 1)! i=1 S n 1, (iii) the n-th derivative of the identity is Map ( n, Σ S 0 ), the Spanier-Whitehead dua of n, considered as a spectrum with an action of Σ n.equivaenty, D n (X) Ω (Map ( n, Σ X n ) hσn ). The description of the space n is a geometric one, it is defined as a quotient of the n(n 1)-dimensiona unit cube by a certain subcompex. In section 1 we reformuate the description of n. Thus we construct a certain combinatoriay defined compex K n. K n has a natura action of Σ n, and we show that for our purposes the suspension of K n is equivaent to n.thuswemaywrite D n (X) Ω (Map (SK n, Σ X n ) hσn ). By the spectra sequence for the homoogy of Bore construction, the stabe homoogy of D n (X), i.e. the homoogy of the spectrum Map (SK n, Σ X n ) hσn is essentiay given by the homoogy of the symmetric group with coefficients in the homoogy modue of the (dua of) K n tensored with the homoogy of X n. Thus, the simper the homoogy of X n is, the simper one may expect the ayers to be. This, of course, suggests the spheres as candidates for investigation. In the case of an even-dimensiona sphere, one is ed to investigating H (Σ n ;H ( K n )), where K n is the dua of K n. In the case of an odd-dimensiona sphere, one is ed to study H (Σ n ;H ( K n ) Z[ 1]), where Z[ 1] is the sign representation. Not surprisingy, odd-dimensiona spheres turn out to be the more basic case. In section 3, we carry out the homoogy cacuations for the odd sphere case. The foowing theorem summarizes some of the resuts in section 3.

5 THE GOODWILLIE TOWER OF THE IDENTITY 5 Theorem 0.1. Let X be an odd-dimensiona sphere. If n is not a power of a prime, then D n (X). If n = p k,thend n (X) has ony p-primary torsion. For a spectrum E, eth s (Ω E)=H ( (E) be the stabe homoogy of E. Insection3 we write an expicit basis for H s Dp k (S 2s+1 );Z/pZ ) and investigate the action of the ( Steenrod agebra on the stabe cohomoogy H s Dp k (S 2s+1 );Z/pZ ). We prove that the stabe cohomoogy of D p k(s 2s+1 )isa[k 1] free, where A[k 1] is a certain finite subagebra of the Steenrod agebra. In section 4 we feed this resut into the vanishing ine theorems of Anderson-Davis [AD73] and Mier-Wikerson [MW81] to concude that the v k 1 periodic homotopy of D p j(s 2s+1 ) is zero for j k and moreover that the Goodwiie tower converges in v k periodic homotopy. This impies the main theorem of the paper which is the foowing: Theorem 4.1 Let X be an odd-dimensiona sphere ocaized at a prime p. The map X P p k(x) is a v j -periodic equivaence for a k 0 and for a 0 j k. In the ast subsection we formuate and prove the anaogue of theorem 4.1 for even dimensiona spheres. Basicay, the tower is sti finite, but it is twice as ong. Acknowedgements. The first named author woud ike to offer a specia thanks to T. Kashiwabara for many hepfu conversations on this materia. We aso are gratefu to N. Kuhn for many in-depth comments on some of the eary versions of the manuscript. 1. The poset of partitions of a finite set Let n be an integer, n>1. Let n = {1,...,n}. Apartition λ of n is an equivaence reation on n (simiary, one defines partitions of an arbitrary finite set). Partitions are ordered by refinements, and may be considered as a category. Let k n be the category of partitions of n. Thus, for two partitions λ 1, λ 2, there is a morphism λ 1 λ 2 iff λ 1 is a refinement of λ 2. It is cear that k n has an initia and a fina object. Denote these ˆ0 andˆ1 respectivey. Let N k n be the simpicia nerve of k n. Since k n has an initia and a fina object, the geometric reaization of N k n is contractibe. Let K n denote the subcompex of the reaization of N k n, whose simpices are those which do not contain the morphism ˆ0 ˆ1 as a face. Thus the zero-simpices of K n are partitions of n and i-simpices are increasing chains of partitions (ˆ0 =λ 1 λ 0 <λ 1 < <λ i λ i+1 = ˆ1) such that not both inequaities ˆ0 λ 0 and λ i ˆ1 are equaities. Let k n be the fu subcategory obtained from k n by deeting ˆ0 andˆ1. Let N k n be the simpicia nerve

6 6 GREG ARONE AND MARK MAHOWALD of k n and et K n be its reaization. It is easy to see that K n is homeomorphic to the unreduced suspension of K n. Equivaenty, there is a cofibration sequence K n+ S 0 K n. Here by cofibration sequence we mean that K n is homeomorphic to the homotopy cofiber of the map K n+ S 0. The + subscript stands for an added disjoint basepoint. For a partition λ et r(λ) be the number of its components. Let S =(S 0,S 1,...,S i ) be a sequence of integers such that n S 0 S 1 S i 1, et K S n N i k n be defined as foows K S n = {(ˆ0 λ 0 λ i ˆ1) N i k n r(λ j )=S j for j =0,...,i} Let Kn i be the set of non-degenerate i-simpices of k n.thus N i k n = Kn S. N i k n = K i n = n S 0 S 1... S i 1 n>s 0 S 1... S i >1 n>s 0 >S 1 >...>S i >1 Notice that if i>n 3thenKn i =. Therefore, K n is n 3-dimensiona. Notice aso that Kn S is defined even if S is empty, and therefore the sets N 1 k n and Kn 1 are defined and have one eement each. By our convention, N 2 k n = Kn 2 =. Definition 1.1. T n is the foowing based simpicia set: The set of i-simpices, Tn,is i Tn i = N i 2 k n + i 0. in particuar, Tn 0 = + = and Tn 1 = S0. The face maps are defined as foows: If 0 <j<ithen d j : Tn i T i 1 n is given by: d j (λ 0,...,λ i 2 )=(λ 0,...,ˆλ j 1,...,λ i 2 ). For j =0,i the formuas are { (λ1,...,λ d 0 (λ 0,...,λ i 2 )= i 2 ) if λ 0 = ˆ0 otherwise { (λ0,...,λ d i (λ 0,...,λ i 2 )= i 3 ) if λ i 2 = ˆ1 otherwise The degeneracy maps are defined simiary. If 0 j i then s j : T i n T i+1 n is determined by s j (ˆ0 =λ 1,λ 0,...,λ i 2,λ i 1 = ˆ1) = (ˆ0 =λ 1,λ 0,...,λ j 1,λ j 1...,λ i 2,λ i 1 = ˆ1). K S n. K S n.

7 THE GOODWILLIE TOWER OF THE IDENTITY 7 It is easy to check that T n is indeed a simpicia set and that its reaization is SK n = SΣ K n,wheres and Σ denote reduced and unreduced suspension respectivey. (This is, essentiay, Minor s suspension construction [Mi72, page 120], appied to K n twice). The symmetric group Σ n acts on k n, and therefore on Kn S, Kn, i K n etc. The action of Σ n on Kn S is not, in genera, transitive. We need to write KS n as a union of Σ n- orbits. The orbits of zero-simpices (partitions of n) are, simpy, partitions of positive integers. A partition P of a positive integer n is a coection n 1,...,n k of positive integers such that n 1 n k and n i = n. We ca {n i } the components of P. Such a partition of n is not trivia if 1 <k<n. We denote the set of partitions of n by Q(n). Proposition 1.2. The quotient set Kn 0/Σ n is naturay isomorphic to the set of nontrivia partitions of n. Proof. Every partition λ of n induces a partition P of the integer n: the components of P are cardinaities of the components of λ. It is eementary to show that this assignment is surjective and that λ 1 and λ 2 induce the same partition of n if and ony if they are in the same orbit of Σ n. If P is as above, we wi ca P the type of λ. We wi sometimes use forma sums n i i to describe partitions of integers. A forma sum as above stands for a partition of n = n i i with n i components of cardinaity i. Proposition 1.3. Let λ be a partition of type P = n i i. The set of partitions of type P is Σ n -equivarianty isomorphic to the set of cosets Σ n /Σ λ,whereσ λ is the stabiizer group of λ. There is an isomorphism Σ λ = Σ ni Σ i. Proof. Easy. We need to cassify orbits of Kn i, i>0, in a manner simiar to the one we have for the orbits of Kn 0. It is sometimes convenient to represent orbits with certain abeed trees. A tree wi aways have a root r. Distance wi mean the number of edges in the unique path between two nodes. Let v be a node. Definition 1.4. A tree is baanced if a its eaves have the same distance from the root. Given a tree, we define a height function on its nodes, which we denote h(v), by etting h(v) be the minima distance from v to a eaf. Define the height of a tree to be h(r) 1. Definition 1.5. A tree is abeed if to every node v there is assigned a positive integer (v).

8 8 GREG ARONE AND MARK MAHOWALD We wi make free use of such expressions as sibing nodes, a singe chid, the subtree spanned by a node, etc. We say that a baanced tree has no forking on eve j if a the nodes of height j have ony one chid. We say that two abeed trees T and T are isomorphic as abeed trees (or just isomorphic) if there is an isomorphism of unabeed trees ψ : T T such that for any node v of T except possiby the root, (v) =(ψ(v)). Definition 1.6. A abeed tree is standard if 1) it is baanced, 2) (r) =1, 3) no two sibing nodes span isomorphic abeed subtrees. Condition (3) impies that every node of height 1 has exacty one chid. In other words, in a standard tree there is no forking on eve 1. Given a standard tree, we define the degree function of its nodes as foows: If h(v) =0thendeg(v) =(v). If h(v) > 0, et u 1,...,u k be the chidren of v, then deg(v) =(v)(deg(u 1 )+ +deg(u k )). The degree of a tree is the degree of its root. Proposition 1.7. There is a bijective correspondence between orbits of Kn i and standard abeed trees of height i +1and degree n. L Proof. For i =0,etP = n j j be an orbit. Then P is represented by the foowing tree: =1 1 n j1 R n jl?? j 1 j L For i>0, the assignment of trees to orbits is constructed inductivey. But before we describe it, we need some more definitions. For a finite set S, etk S be the category of partitions of S. Let S 1 and S 2 be two finite sets. Let Λ 1 =(λ 1 0 λ 1 i )and Λ 2 =(λ 2 0 λ 2 i )bei-simpices of N k S1 and N k S2 respectivey. A morphism ρ :Λ 1 Λ 2 is a map of sets S 1 S 2 which for every 0 j i maps every component of λ 1 j into a component of λ2 j. ρ is an isomorphism if it has a two-sided inverse. Λ 1 and Λ 2 are isomorphic if there exists an isomorphism Λ 1 Λ 2. Definition 1.8. Let Λ = (λ 0 λ j ) be a chain of partitions. Let S be a component of λ j.thenλ 0,...,λ j 1 determine a (j 1)-simpex of N k S. We ca it the restriction of Λ to S and denote it Λ S. Let Λ = (λ 0 λ i )beani-simpex of N k n.thusλ i is the coarsest partition in the chain. Let S 1, S 2 be two components of λ i. We say that S 1 and S 2 induce

9 THE GOODWILLIE TOWER OF THE IDENTITY 9 isomorphic bocks if Λ S1 and Λ S2 are isomorphic. Of course, a necessary condition for S 1 and S 2 to induce isomorphic bocks is that S 1 and S 2 are isomorphic sets. The property of inducing isomorphic bocks defines an equivaence reation on the components of λ i.weconsiderλ S1 as an eement of N i 1 K S1 and the orbit of Λ S1 under the action of Σ S1 as an eement of (N i 1 K S1 ) ΣS1. Two i-simpices Λ 1 and Λ 2 are in the same orbit of Σ n if and ony if they have the same isomorphism casses of bocks, counting with mutipicities. Thus, every orbit B of Kn i can be written uniquey as a forma sum B = n B where B are eements of ( K i 1 k )Σ k for some k 1,...,k m such that n k = n and B 1,B 2,...,B m are pairwise distinct. Assume by induction that we have assigned to B pairwise non-isomorphic standard abeed trees T of height i. NowforeveryT repace the abe 1 at the root with n and join a roots to a common new root. Thus we have constructed a standard tree of height i + 1, which is the tree assigned to B. It s easy to check that the construction is we-defined, i.e., that two i-simpices are in the same orbit if and ony if the above procedure associates to them isomorphic trees. To describe the orbit of a given i-simpex Λ of type B as above, we notice, as we did in the case of 0-simpices, that the stabiizer group Σ Λ of Λ has, up to an isomorphism, the foowing form: Σ Λ = (Σn1 Σ B1 ) (Σ n2 Σ B2 ) (Σ nm Σ Bm ) where Σ B are stabiizers of representatives of B. Notice that a groups in sight are naturay subgroups of Σ n, and the set of partitions of type B can be identified equivarianty with the cosets Σ n /Σ Λ. The stabiizer groups of two representatives of a given orbit are conjugate. In the course of the paper we wi sometimes confuse between the set of orbits and a set of arbitrariy chosen representatives of orbits. The group Σ Λ is isomorphic to a semi-direct product Σ Λ = (Σn1 Σ n2 Σ nm ) ( ) Σ n 1 B 1 Σ n 2 B 2 Σ nm B m where there is an obvious action of Σ n1 Σ nm on Σ n 1 B 1 onemaywriteσ Λ in the form G i+1 (G i ( G 0 )) Σ nm B m. Inductivey, where each G is a product of symmetric groups and there is a wreath product type action of G on G 1 (G 2 G 0 ). As a matter of fact, G is isomorphic to

10 10 GREG ARONE AND MARK MAHOWALD the product of powers of symmetric groups indexed by nodes on eve in the tree corresponding to the type B of Λ. The size of each symmetric group is given by the corresponding abe and the power to which it is raised is given by the abe of the father node. From here unti the end of the section, et p be a fixed prime number. Definition 1.9. Let Λ = (ˆ0 =λ 1 λ 0 λ j ) be a chain of partitions. We say that λ j is a p-coarsening of ˆ0 =λ 1 λ 0 λ j 1 if for every component S of λ j the foowing hods: 1) The number of components of λ j 1 contained in S is a power of p. 2) Any two components of λ j 1 contained is S induce isomorphic bocks. We wi say that λ is a p-partition if it is a p-coarsening of ˆ0. Obviousy, the property of being a p-partition is invariant under the action of Σ n and therefore we may speak about p-partitions of numbers. A p-partition is simpy a partition whose components a have cardinaity which is a power of p. We et P (n) denote the set of ordered p-partitions of n and P (n) denote the set of unordered p-partitions of n, which is the same as the set of p-partitions of n. We use the foowing ogarithmic notation for eements of P (n): a sequence (n 0,n 1,...) denotes the partition with n j components of cardinaity p j for a j 0. Thus n = j n j p j. Definition Let Λ = (λ 0 λ i )beani-simpex of N k n. An ordered p-ramification of Λ is a chain of partitions (ˆ0 =λ 1 δ 0 λ 0 δ 1 λ 1 λ i δ i+1 λ i+1 = ˆ1) such that for a j =0, 1,...,i+1, δ j is a p-coarsening of (ˆ0 =λ 1 δ 0 λ 0 δ 1 λ 1 λ j 1 ). Reca that we denote by Σ Λ the subgroup of Σ n which stabiizes Λ. Σ Λ acts on the set of ordered p-ramifications of Λ. We define an unordered p-ramification of Λ to be an orbit of an ordered p-ramification under the action of Σ Λ. It is cear that if Λ 1 and Λ 2 are two i-simpices in the same orbit of Σ n then the set of unordered p-ramifications of Λ 1 is isomorphic to the set of unordered p-ramifications of Λ 2. Let Ψ be an unordered p-ramification of Λ. Consider Ψ as a 2i+2-simpex of N k n and consider the orbit of Ψ under the action of Σ n. This orbit is represented by a standard tree of height 2i + 3. It foows easiy from the definitions that a the nodes of even height in this tree are abeed by powers of p and that there is no forking on odd eves. It is aso easy to see that the set of orbits of ordered p-ramifications of Λ under the action of Σ n is isomorphic to the set of orbits under the action of Σ Λ, which is the set of unordered p-ramifications of Λ. We denote the set of ordered p-ramifications of Λ by P (Λ) and the set of unordered p-ramifications of Λ by P (Λ). Thus P (Λ) = P (Λ) ΣΛ. Wedenoteby P (B ) k P (B ) the fibered product of k copies of P (B )overp(b ). Thus a point in P (B ) k P (B ) is a k-tupe of eements of P (B )which

11 THE GOODWILLIE TOWER OF THE IDENTITY 11 are a in the same orbit of Σ Λ. We wi need inductive formuae for P (Λ) and P (Λ). Suppose that λ i, the coarsest partition in Λ, has n bocks of type B for =1,...,L L where B are pairwise non-isomorphic. We denote a generic eement of P (n )by (n 0 1,...,nj 1 1 ), (n 0 2,...,nj 2 2 ),...,(n 0 L,...,nj L L ) Proposition There is an isomorphism of Σ Λ -equivariant sets P (Λ) = P (n ) Σ n j=1...j Σ n j Σ p j j ( P (B ) pj P (B ) =1 ) n j Proof. Fix a p-coarsening δ i+1 of Λ. By definition, every component of δ i+1 contains apowerofp of components of λ i+1 of type B for some. Let us say that δ i+1 has n j components containing p j components of type B. A p-ramification of Λ whose coarsest partition is δ i+1 is determined by a coection of p-ramifications of the bocks B such that any two bocks which are in the same component of δ i+1 have isomorphic p-ramifications. This set is isomorphic to j ( P (B ) pj P (B ) On the other hand, the set of p-coarsenings of Λ which have n j components containing p j components of type B is ceary isomorphic to Σ n Σ n j Σ p j j=1...j The proposition foows by taking union over the set of types of p-coarsenings which is isomorphic to the set P (n ). ) n j. Coroary There is an isomorphism of sets P (Λ) = (P (B )) nj j P (n ) Proof. Reca that Σ n j P (Λ) = P (Λ) ΣΛ = P (Λ) (Σn1 Σ B1 Σ nl Σ B ).

12 12 GREG ARONE AND MARK MAHOWALD Appying proposition 1.11 one readiy sees that P (Λ) = j P (n ) ( (P ) ) n j (B ) pj P (B ) Σ p j Σ n j. But ( P (B ) pj P (B )) = P (B ) where the right hand side can be considered as a set with a trivia action of Σ p j. The coroary foows. 2. The ayers of the Goodwiie tower of the identity In this section we wi describe D n (X), the n-th ayer of the Goodwiie tower of the identity in terms of the compexes K n of the previous section. This amounts, basicay, to a reformuation of the main resut of Johnson in [Jo95]. In [AK97] a different way to derive our description of D n (X) ispresented. Theorem 2.1. D n (X) Ω Map (SK n, Σ X n ) hσn. Proof. By [Jo95, coroary 2.3] D n (X) Ω Map ( n, Σ X n ) hσn where n is defined in [Jo95, definition 4.7]. We reca the definition. Let I n2 = {t =(t 11,t 12,...,t 1n,t 21,...,t nn ) R n2 0 t ij 1} be an n 2 -dimensiona cube. Let I n(n 1) be the subspace of I n2 defined by t ii =0for i =1,...,n.ThusI n(n 1) is an n(n 1)-dimensiona cube. For 1 i<j n define W ij = {t I n(n 1) t ik = t jk for 1 k n}. Define aso Z = {t I n(n 1) t ij =1forsome1 i, j n}. Then n = I n(n 1) /{Z W ij }. i<j Thus to prove the theorem it is enough to prove that there is a Σ n -equivariant map n SK n which is a non-equivariant homotopy equivaence. Since I n(n 1) is a Σ n -equivarianty contractibe space, it foows that n is equivarianty equivaent to the suspension of Z i<j W ij. Therefore, it is enough to show that there is an equivariant equivaence K n Z i<j W ij.

13 THE GOODWILLIE TOWER OF THE IDENTITY 13 Reca that K n is itsef an unreduced suspension of K n, the geometric reaization of the category of non-trivia partitions of n. On the other hand, we caim that i<j W ij and Z are both equivarianty contractibe. Indeed, i<j W ij is contractibe by radia projection on (0,...,0) and Z is contractibe by radia projection on the point t defined by t ii =0fori =1,...,n and t ij =1fori j. It foows that Z i<j W ij is equivarianty equivaent to the unreduced suspension of Z i<j W ij. Thus it is enough to prove that there is an equivariant map K n Z W ij i<j which is a non-equivariant equivaence. For 1 i<j n, etu ij = Z W ij. The assertion foows from the fact that the spaces U ij cover Z i<j W ij, a possibe intersections of U ij are contractibe, and the poset associated with this covering is isomorphic to k op n. We state it in two propositions. Proposition 2.2. Z i<j W ij = U ij. Proof. Obvious. Let A = {(i 1,j 1 ), (i 2,j 2 ),...,(i L,j L )} be a coection of pairs 1 i <j n. Let U A = (i,j ) A U i,j. We associate with A agraphonn vertices, abeed 1,...,n,as foows: There is an edge (i, j) iff(i, j) A. The connected components of this graph determine a partition of n. Proposition 2.3. U A depends ony on the partition associated with A. Moreover, U A is empty if the partition associated with A is ˆ1 and is contractibe otherwise. Proof. In fact, it is easy to see that U A = {t =(t 11,t 12,...,t 1n,t 21,...,t nn ) R n2 0 t ij 1 t ij =0ifi and j are in the same component of the partition associated with A, and t ij =1forsome(i, j)} If the partition associated with A is ˆ1 thent ij =0fora(i, j), contradicting the requirement that t ij =1forsome(i, j), so U A =. If the partition is not ˆ1 thenu A is contractibe by radia projection on the point given by t ij =0ifi and j are in the same component and t ij =1otherwise. This competes the proof of the theorem.

14 14 GREG ARONE AND MARK MAHOWALD D n (X) is the infinite oop space associated with the spectrum D n (X) Map (SK n, Σ X n ) hσn. Since SK n is the geometric reaization of the simpicia set T n, it foows that D n (X) is the tota spectrum of a cosimpicia spectrum, which we denote D n (X). The spectrum of i cosimpices of D n (X) ismap (Tn i, Σ X n ) hσn. Moreover, reca that SK n has no non-degenerate simpices in dimensions higher than n 1. It foows that the tower of fibrations associated with D n(x) hasn stages, which we denote Tot i n (D n (X)). In fact, Tot i n (D n (X)) = Map (sk i SK n, Σ X n ) hσn where sk i stands for the i-th skeeton. The homotopy fiber of the map Tot i n (D n(x)) Tot i 1 n (D n(x)) is homotopy equivaent to Map (S i Kn i 2 +, Σ X n ) hσn Since this is a tower of fibrations of spectra, it yieds a spectra sequence cacuating the homoogy of D n (X), which is the stabe homoogy of D n (X). In the next section we wi use this spectra sequence to cacuate the stabe homoogy of D n (X) in some interesting specia cases. 3. Odd sphere case - the cohomoogy of the ayers 3.1. The homoogy groups. We now focus our attention on the odd sphere case. Our goa in this section is to study the stabe homoogy of D n (X), the ayers of the Goodwiie tower, in this case. Thus, we want to study the homoogy of the spectra D n (X) Map (SK n, Σ X n ) hσn where X is an odd-dimensiona sphere. We begin with a proposition which enabes us to focus on the torsion part of homoogy. Proposition 3.1. Let X be an odd-dimensiona sphere. Let n > 1. Rationay D n (X) =Map (SK n, Σ X n ) hσn Proof. We saw in the previous section that there exists a tower of fibrations with n stages converging to D n (X), in which the fibers are of the form Map (S i Kn i 2 +, Σ X n ) hσn where K i 2 n is the set of non-degenerate i 2-chains of partitions. Thus Map (K i 2 n +, Σ X n ) hσn Λ Σ X n hσ Λ

15 THE GOODWILLIE TOWER OF THE IDENTITY 15 where the wedge sum on the right hand side in indexed by representatives of orbits of Kn i 2 +. Thus each stabiizer group Σ Λ can be written as a semi-direct product G i 1 (G i 2 ( G 0 )) where each G is a product of symmetric groups. Since we consider ony nondegenerate orbits, none of the G s is trivia. In particuar, G 0 is not trivia. The proposition foows since for k>1andx an odd-dimensiona sphere, XhΣ k k is rationay trivia. Notice that proposition 3.1 impies the theorem of Serre that the map X Q(X) is a rationa equivaence for an odd sphere X. From now on a spaces considered wi be ocaized at a fixed prime p. A homoogy groups are taken with Z/pZ coefficients. We wi cacuate H (D n (X)) expicity. The case n = 1 is trivia. Assume, ti the end of the section, that n>1. The pan is to use the homoogy spectra sequence associated with the tower of fibrations Tot i n (D n (X)) (as defined on page 14). To see that such a spectra sequence exists, note that since we are deaing with spectra, smashing with a fixed spectrum preserves fibration sequences and finite towers of fibrations. The homoogy spectra sequence is obtained by smashing our tower of fibrations with the Eienberg-MacLane spectrum HZ/pZ and considering the homotopy spectra sequence of the resuting (finite) tower (see [BK72, page 259] for a reference on the spectra sequence of a tower of fibrations). The first term of the spectra sequence has the foowing form: E i,t 1 =H t i ( Map (S i K i 2 n +, Σ X n ) hσn ) =Ht ( Map (K i 2 n +, Σ X n ) hσn ) with a differentia d 1 : E i,t 1 E i+1,t 1. We may view this E 1 term as a cochain compex C of graded Z/pZ-modues. The modue of i-cochains is C i =H ( Map (K i 2 n +, Σ X n ) hσn ). The differentia i : C i C i+1 is given by the aternating sum j( 1) j d j,where d j is induced by the face map d j in N k n. If we write, as we did in the proof of proposition 3.1, Map (Kn i 2 +, Σ X n ) hσn Σ XhΣ n Λ, Λ then for j = 0 and j = i, d i j is the zero homomorphism, and for 1 j i 1, d i j is a direct sum of the transfer maps associated with the incusion of stabiizers of representatives of orbits of chains of the form (λ 0,...,λ i 2 ) into stabiizers of representatives of orbits of chains of the form (λ 0,...,ˆλ j 1,...,λ i 2 ). The incusions are we-defined up to conjugation, and therefore the transfer maps are we-defined.

16 16 GREG ARONE AND MARK MAHOWALD To study this spectra sequence we wi need to study the homoogy of (reduced) Bore constructions on X with respect to certain subgroups of Σ n. We reca a few standard facts about the homoogy of XhΣ n n = X n Σn EΣ n+ as described in terms of Dyer-Lashof operations. Let H be a graded Z/pZ modue. Let (H )bethefree graded Z/pZ modue generated by aowabe Dyer-Lashof words of ength over H (see [CLM76, I.2] and [BMMS86, page 298] for detais). Thus, if >0, then (H ) is generated by the foowing set {β ɛ 1 Q s1 β ɛ Q s u u H,ɛ i {0, 1}, s i > 0, ps i ɛ i s i 1 2s 1 if p>2 i=2 [2s i (p 1) ɛ i ] u } {Q s1 Q s u u H, s i > 0, 2s i s i 1 s 1 if p =2 i=2 s i u } where u H. By convention, 0 (H )=H. The operations Q s come from eements in the mod p homoogy of symmetric groups. Q s raises degree by 2(p 1)s if p>2 and by s if p = 2 (the βs in the odd-primary case are homoogy Böcksteins, and thus ower the degree by 1). Q s u =0ifs< u for p>2and u even (if s< u for p =2) 2 and Q u 2 u = u p for p>2(q u u = u u for p = 2). Thus we incude powers of eements of H in. The convenience of this wi become cear ater - its purpose is to make the negigibe summands in the proof of emma 3.11 negigibe. We wi sometimes abbreviate (H )as when H is cear from the context. Given a graded vector space D, etv (D) be the augmentation idea of the free symmetric agebra generated by D. ThusV (D) is the quotient of k=1 D k by the idea generated by the reations a b ( 1) a b b a where a, b are homogeneous eements. Let E(D) be the quotient of V (D) by the idea generated by a p.sincethe reations are homogeneous, we may write E(D) = E k (D). By abuse of notation, we wi write E k (D) asdσ k k. From now on, whenever we write DΣ k k, where D is a graded Z/pZ modue, we mean E k (D). The mod p homoogy of XhΣ n n is described in terms of the Dyer-Lashof operations. For any based space X, the foowing is true (the homoogy is taken with Z/pZ coefficients) (1) H (X n hσ n )= ( (H (X)) n ) Σn (n 0... ) P (n) 0 Because of our choice to suppress the p-th powers of eements of H (X), the spitting in (1) is not natura, but depends on a choice of basis of H (X). The point is that the projection map V (D) E(D) spits, but not naturay. However, it is easy to

17 THE GOODWILLIE TOWER OF THE IDENTITY 17 seethatifwefiterp (n) by the number of components, then the spitting is natura up to eements of ower fitration. Thus we get a (more or ess natura) expansion of H (XhΣ n n ) into a direct sum indexed by p-partitions of n. We wi ca the terms in the expansion the standard summands (or just summands) ofh (XhΣ n n ). It is we known that the standard summands are detected by certain eementary Abeian subgroups of Σ n. We proceed to reca the basic facts about this. Let us begin with k as a summand of H (Σ p k), or more generay of H (X pk Σ p ). For k =0, 1,... k et A k = (Z/p) k. A k = p k, therefore the action of A k on itsef defines an incusion (up to conjugation) of A k Σ p k. We wi consider A k as a subgroup of Σ p k via this incusion (this is the subgroup that is defined as k in [KaP78, page 95]). Thus A k acts transitivey on p k. A k is very usefu for detecting eements in the homoogy of Σ p k: the pure part of H (Σ p k) is detected on A k. This was probaby first proved by Kahn and Priddy in [KaP78]. For a more detaied account we recommend [AdMi95]. We need a sighty more genera version of this: Proposition 3.2. Let X be a based space. Let H =H (X). Reca that k is a summand of H (X pk hσ ). Write H p k (X pk hσ ) p = k k A. Consider the homomorphism H (X BA k+ ) H (X pk hσ p k ) induced by incusion of subgroups and the diagona map X X n. This map is onto the summand k and zero on the summand A. Proof. For X = S 0 (a zero-dimensiona sphere) this is precisey [KaP78, Proposition 3.4]. The proof generaizes straightforwardy. The idea is to reduce the question from Σ p k to its p-syow subgroup and then proceed by direct cacuation. Now consider a summand on the right hand side of (1) corresponding to a p- partition (n o...,n,...) P (n). This summand is detected, in a suitabe sense, by the eementary Abeian group A = A n. Consider the space X Σ n as a space with a trivia action of A. It is easy to see that there is a diagona map X Σ n X n that is equivariant with respect to the subgroup incusion A Σ n. We have the foowing proposition Proposition 3.3. With notation as above, consider the homomorphism H (X Σ n BA + ) H (X n hσ n ) induced by group incusion A Σ n and the diagona map X Σ n X n.thishomomorphism is onto the summand corresponding to to the p-partition (n o...,n,...) and zero on the other summands of the same fitration and the summands of higher fitration.

18 18 GREG ARONE AND MARK MAHOWALD Proof. The case X = S 0 is we-known. It is argey proved in [KaP78] and in more detai in [AdMi95]. It is a ongish, but straightforward, exercise to extend the resut to a genera X. The reason that we need proposition 3.3 is that we have to study the transfer map in the homoogy of Bore construction. Let n 0 p 0 + n 1 p 1 + n k p k be a p-partition of n. Let Σ P =Σ n0 Σ p 0 Σ nk Σ p k and Σ P =Σ n 0 p 0 Σ n k pk. Let X be any based space. The homoogy groups H (XhΣ n n ), H (XhΣ n P )andh (XhΣ n ) each P have a summand isomorphic to n 0 0 Σ n0 n k k Σ nk whichwedenotesimpy. Write H (XhΣ n n )= A, H (XhΣ n P )= B and H (XhΣ n )= B. Consider the P homomorphisms A B and A B induced by the appropriate transfers. These homomorphisms can be represented as two by two matrices of maps ( ) ( ) B B and A A B A A B Proposition 3.4. The map in both matrices is an isomorphism. Proof. The proof is simiar to the proof of the main theorem of [KaP78]. To prove that the homomorphism is an isomorphism it is enough to prove that it is surjective. To do this for the case of the first matrix, it is enough to show that the composite homomorphism H (X Σ n BA n 0 0 A n k k + ) i H (XhΣ n n ) tr H (XhΣ n P ) is surjective onto. This composed homomorphism can be anayzed by means of a suitabe version of the doube coset formua. It is not hard to show that the composed map above is the same as the homomorphism induced by the group incusion A n 0 0 A n k k Σ P, essentiay because of two reasons: the normaizer of A n 0 0 A n k k in Σ n isthesameasinσ P and the transfer from an eementary Abeian group to a proper subgroup is zero (see [KaP78]). The argument for the second matrix is simiar. We wi need to consider a sighty more genera situation. Let n = i 0 p 0 +i 1 p 1 + i k p k as before. Let K 1,K 2,...,K j be disjoint subsets of {0, 1,...,k} whose union is {0, 1,...,k}. for 1 j, etm = t K i t p t. Consider the group Σ m1 Σ mj as a subgroup of Σ n. It is easy to see that is a summand of H (XhΣ n m1 Σ mj ). Write H (XhΣ n m1 Σ mj )= C and consider the matrix ( C ) A A C describing the transfer map. We have the foowing proposition Proposition 3.5. In the matrix above the map is an isomorphism.

19 THE GOODWILLIE TOWER OF THE IDENTITY 19 Proof. Simiar to the proof of the previous proposition. Next we need to generaize the formua (1) and proposition 3.3 to H (XhΣ n Λ ), where Σ Λ is the stabiizer of a chain of partitions of n. More precisey, we wi show that H (XhΣ n Λ ) spits as a certain direct sum indexed by unordered p-ramifications of Λ. We first show how to associate a graded Z/pZ-vector space to a p-ramification of Λ. Indeed, et Λ = (ˆ0 =λ 1 λ 0 λ i λ i+1 = ˆ1) be an i-chain of partitions of n and et ɛ be an unordered p-ramification of Λ. Reca that we associate with ɛ a standard tree of height 2i + 3 in which a the nodes of even height are abeed by powers of p and there is no forking on odd eves. Given such a tree, a node v in the tree and a graded Z/pZ-vector space H =H (X), we construct a graded Z/pZ-vector space H v and, for future use, a detecting eementary abeian group A v as foows: if the height of v is zero then it is abeed by p k for some k (since 0 is even) and we define H v = k (H )anda v = A k. Assume now that we defined H v and A v for a v of height j 1 or ess. Let v be a node of height j. Let be the abe of v. Assume, first, that j is even. Then = p k for some k. Let u 1,...,u m be the chidren of v. We define H v = k(h u 1 H um ) and A v = A k (A u1 A uk ) (A v shoud be thought of as the diagona subgroup of Σ p k (A u1 A uk )). Now assume j is odd. Then v has ony one chid u and we define H v =(Hu ) Σ and A v =(A u ). Let H ɛ be the modue associated with the root of the tree and simiary et A ɛ be the eementary abeian group associated with the root of the tree. A ɛ is in fact a subgroup of Σ n (determined up to conjugation). Lemma 3.6. Let Λ=(ˆ0 =λ 1 λ 0 λ i λ i+1 = ˆ1) be an i-chain of partitions of n. Reca that P (Λ) is the set of unordered p-ramifications of Λ. There is an isomorphism H (XhΣ n Λ ) = H ɛ. ɛ P (Λ) Proof. We wi prove it by induction on i. The induction starts with i = 1. In this case Λ = (ˆ0, ˆ1) and the emma is given precisey by (1) and proposition 3.3. Assume the emma hods for i 1. Let Λ = (ˆ0 =λ 1 λ 0 λ i λ i+1 = ˆ1) be an i-chain of partitions. Consider λ i, the coarsest partition in the chain. The reation of inducing isomorphic bocks is an equivaence reation on the components of λ i. Let λ i have n components of type B, where the B s are pairwise distinct orbits of

20 20 GREG ARONE AND MARK MAHOWALD i 1 chains of partitions of a set with k eements and =1,...,L. Thus, in the tree corresponding to the orbit of Λ, the root has L chidren abeed n 1,...,n L.The stabiizer group of Λ has the foowing form Thus H (X n hσ Λ )= Σ Λ = Σn1 Σ B1 Σ nl Σ BL. L J ( ( j H (X k hσ B ) )) n j =1 (n 0,n 1,... Σ,n J ) P (n ) j=0 n j which impies H (XhΣ n Λ )= L J ( ( j H (X k hσ B ) )) n j. Σ L =1 j=0 n j P (n ) =1 We see that H (XhΣ n Λ ) spits as a direct sum indexed by assumption, H (X k hσ B ) = H ɛ ɛ P (B ) L P (n ). By the induction =1 Obviousy, if H 1 and H 2 are two graded modues then (H 1 H ) 2 = (H ) 1 (H ), 2 therefore j H (X k hσ B ) = j H ɛ Thus H (X n hσ Λ )= ɛ P (B ) L J j H ɛ L =1 j=0 ɛ P (B ) P (n ) =1 n j Σ n j. By mutipying out, we see that H (XhΣ n Λ ) spits as a direct sum indexed by the set (P (B )) nj Σ j n j P (n ) which, by coroary 1.12 is isomorphic to P (Λ). It is tedious, but entirey straightforward to verify that the summand corresponding to ɛ P (Λ) is indeed H ɛ.

21 THE GOODWILLIE TOWER OF THE IDENTITY 21 Remark 3.7. Reca that given an unordered p-ramification ɛ of Λ we constructed an eementary abeian group A ɛ (just before emma 3.6). It is not hard to show that A ɛ detects the summand H ɛ of the homoogy of XhΣ n Λ in the sense of proposition 3.3. If ɛ 1 and ɛ 2 are different p-ramifications of Λ then A ɛ1 and A ɛ2 are non-conjugate in Σ Λ. The map on the homoogy of Bore constructions induced by the incusion (defined up to conjugation) A ɛ Σ n is non-zero ony on the summand H ɛ and summands corresponding to eementary Abeian groups with stricty fewer components (the number of components of a subgroup G of Σ n is the number of components of the induced partition of n). Definition 3.8. An integer n is pure if n = p k for some integer k. If n is pure, an i-chain of partitions of n, Λ=(ˆ0 =λ 1 λ 0 λ i λ i+1 = ˆ1), is pure if λ j is a p-coarsening of λ 1 λ 0 λ j 1 for a j =0,...,i+1. Ceary, purity is preserved by the action of Σ n, hence we may speak about pure orbits of chains of partitions. It is easy to see that an orbit is pure iff the corresponding tree has one branch and has a abes powers of p. Given a pure Λ, the corresponding stabiizer group has the form Σ Λ = Σp k 0 Σ p k 1 Σ p k i for some (k 0,...,k i )such that k j = k. Aso, consider the chain of partitions (ˆ0 =λ 1 δ 0 λ 0 δ i λ i δ i+1 λ i+1 = ˆ1) in which δ j = λ j for a j =0,...,i+ 1. Since Λ is pure, it is easy to see from definition that it is a p-ramification of Λ. The corresponding summand of H (XhΣ n Λ )isoftheform k0 ( k1...( ki )). We ca it the pure summand associated with Λ. A other summands are impure. Let C i be the graded Z/pZ-modue of i-cochains in the compex C defined above. C i can be written as a direct sum C i = P i I i where P i and I i are the pure and impure summands of C i (P i is often trivia). The coboundary map i : P i I i P i+1 I i+1 can be represented by a matrix of matrices as foows ( P i P i+1 P i I i+1 I i P i+1 I i I i+1 Proposition 3.9. P i I i+1 are zero matrices for a i Proof. It is not hard to show (simiar to proposition 3.2) that the pure summands are detected by the eementary Abeian group A k (the transitive eementary Abeian subgroup of Σ p k). More precisey, the homomorphism H (X BA k+ ) H (X pk hσ p k 0 Σ p k 1 Σ p k i ) ). is onto the pure summand and zero on the impure summands. foows, using the doube coset formua. The proposition

22 22 GREG ARONE AND MARK MAHOWALD Coroary The pure summands span a subcompex of C, which we denote P. P is non-trivia ony if n is a power of p. There is a short exact sequence of cochain compexes 0 P C I 0 where I is the compex of impure summands. The foowing emma is important: Lemma I is acycic. Proof. We wi use the foowing evident proposition Proposition Let C 1 0 C 2 0 C 1 1 C 2 1 C 1 2 C 2 2 C 1 j C 2 j be a cochain compex of graded Z/pZ-vector spaces, where C0 1 is the trivia modue. Suppose that for a j 0 the differentia Cj 1 C2 j C1 j+1 C2 j+1 is given by a matrix ( C 1 j Cj+1 1 Cj 1 Cj+1 2 ) C 2 j C1 j+1 C 2 j C2 j+1 where C 2 j C1 j+1 is an isomorphism. Then the compex is acycic. Consider now the cochain compex I. Obviousy, I 0 is the trivia modue (since n>1). For j 1 we wi write I j as a direct sum of two modues I j 1 I2,which j we now proceed to define. Reca that I j is the direct sum of the impure summands H (XhΣ n Λ ) where Λ ranges through a set of representatives of orbits of non- of Λ degenerate (j 2)-chains of partitions (ˆ0 = λ 1 < λ 0 < < λ j 2 < λ j 1 = ˆ1). Moreover by emma 3.6 we know that given Λ = (ˆ0 = λ 1 < λ 0 < < λ j 2 <λ j 1 = ˆ1), the impure summands of H (X n hσ Λ ) are indexed by unordered p-ramifications ɛ =(ˆ0 =λ 1 δ 0 λ 0 < δ j 2 λ j 2 δ j 1 λ j 1 = ˆ1) such that not for a iδ i = λ i. We say that an unordered p-ramification ɛ of Λ is admissibe if there exists 0 j 2 such that δ m = λ m for a 0 m and λ m = δ m+1. If ɛ is not admissibe than we say it is unadmissibe. Let P a (Λ) and P u (Λ) be the set of admissibe, impure and unadmissibe, impure p-ramifications of Λ. We define I j 1 = Λ (Kn j 2 ) Σn H ɛ ɛ P a(λ)

23 and THE GOODWILLIE TOWER OF THE IDENTITY 23 I j 2 = H ɛ. Λ (Kn j 2 ) Σn ɛ P u(λ) By emma 3.6, I j = I j 1 I j 2 for a j. Ceary, I1 1 is the trivia modue. It remains to prove that the map I j 2 I j+1 1 induced by the coboundary map in I is an isomorphism for a j. Then the emma wi foow from proposition First we estabish that I j 2 and I j+1 1 are abstracty isomorphic. Let be a (j 2)-chain and et Λ=(ˆ0 =λ 1 <λ 0 < <λ j 2 <λ j 1 = ˆ1) ɛ =(ˆ0 =λ 1 δ 0 λ 0 < δ j 2 λ j 2 δ j 1 λ j 1 = ˆ1) be an unadmissibe p-ramification of Λ. Let be the smaest index such that δ +1 λ +1. Such an exists because otherwise ɛ woud be pure. Ca the eve of ɛ. Suppose first that = 1. We caim that if λ 1 = δ 0 then H ɛ is the trivia modue. Indeed, in this case it is easy to see that the tree corresponding to ɛ has a the nodes on eve 0 abeed by 1, but not a the nodes on eve 1 abeed by 1, since λ 1 λ 0. It foows that H ɛ has a tensor factor of the form ( 0(H )) k Σ k,wherek>1. But it is easy to see that ( 0 (H )) k Σ k is the trivia modue if H has exacty one generator of odd degree, which it does if X is an odd-dimensiona sphere. Thus if = 1 and λ 1 = ɛ 0 we say that ɛ is a negigibe p-ramification and H ɛ is a negigibe summand of Ij 2. We denote the set of non-negigibe unadmissibe impure p-ramifications of Λ by Pu n (Λ). We proceed to estabish an isomorphism between the sum of non-negigibe summands of Ij 2 and I1 j+1. We may assume now that λ δ +1 because otherwise ɛ is either negigibe (if = 1) or admissibe (if > 1). It foows that the sequence Λ =(ˆ0 =λ 1 <λ 0 < <λ <δ +1 <λ +1 < <λ j 2 <λ j 1 = ˆ1) is a non-degenerate j 1-chain of partitions. It is obvious by inspection that ɛ =(ˆ0 =λ 1,δ 0,λ 0,δ 1,...,δ,λ,δ +1,δ +1,δ +1,λ +1,...,λ j 2,δ j 1,λ j 1 = ˆ1) is an admissibe, impure p-ramification of Λ and thus H ɛ is a summand of Ij+1. 1 Itis aso obvious by inspection that H ɛ is isomorphic to H ɛ and that the above procedure estabishes an abstract isomorphism between the sum of non-negigibe summands of I j 2 and I j+1 1. It remains to prove that the coboundary homomorphism of I induces an isomorphism between the two. We now may write this map as foows: Λ (Kn j 2 ) Σn ɛ P n u (Λ) H ɛ Λ (Kn j 2 ) Σn H ɛ ɛ Pu n(λ)

24 24 GREG ARONE AND MARK MAHOWALD where ɛ is obtained from ɛ by the procedure described above. This map can be described as a matrix M of maps H ɛ 1 Hɛ 2. To show that this map is an isomorphism it is enough to show that the matrix is bock upper trianguar with respect to a certain ordering of the indexing set and that a the diagona bocks are isomorphisms. To show that a the diagona bocks are isomorphisms we need to show that for any (j 2)-chain Λ as above and for any ɛ Pu n(λ) the map Hɛ Hɛ, induced by the transfer map from Σ Λ to Σ Λ, is an isomorphism. We may write Σ Λ = Gj 1 ( G +1 (G ( G 0 ))) where a G i are products of symmetric groups. Now consider Σ Λ. It is not difficut to see that for i = +1,...,j 1 Σ Λ = G j 1 ( G +1 G +1 (G ( G 0 )) ) where G +1 G +1 is a subgroup of G +1 of the form i Σ mi Σ p i and G i is a subgroup of G i of the form required for coroary 3.5. The fact that the map H ɛ H ɛ is an isomorphism foows from propositions 3.4 and 3.5. It remains to show that the matrix M is equivaent to a bock upper trianguar one with respect to some ordering of the indexing set. It is easy to see, using remark 3.7 and the doube coset formua, that if Λ is a j 2-chain of partitions, and ɛ is an unadmissibe p-ramification of Λ (so H ɛ I2), j then the ony summands of I j+1 1 that H ɛ maps non-triviay on are Hɛ and summands whose detecting eementary Abeian group has stricty fewer components than the eementary Abeian group detecting H ɛ. This competes the proof of emma An immediate consequence of emma 3.11 is the foowing theorem: Theorem Let X be an odd-dimensiona sphere ocaized at a prime p. Assume n is not a power of p. Then D n (X) Ω Map (SK n, Σ X n ) hσn. Proof. Let E 1 be the first term of the spectra sequence associated with the skeeta fitration of K n abutting to H (Map (SK n, Σ X n ) hσn ). We saw that E 1 can be identified with the cochain compex C of graded Z/pZ-vector spaces. Moreover, there is a short exact sequence P C I.Itisobviousthat since n is not a power of a prime, P is trivia. By emma 3.11, I is acycic. It foows that E 2 is zero. Therefore, H (Map (SK n, Σ X n ) hσn ) is zero and the proposition foows.

25 THE GOODWILLIE TOWER OF THE IDENTITY 25 Thus if X is an odd-dimensiona sphere ocaized at a prime p then the ony interesting vaues of n are powers of p. If n = p k then the E 2 term of the spectra sequence computing H (Map (SK n, Σ X n ) hσn ) may be identified with the cohomoogy of the cochain compex P. So we proceed to anayze the compex P. Definition An ordered partition of a positive integer k is an ordered sequence K =(k 1,...,k j ) of positive integers with k k j = k. For future use, we denote by 2 the ordered partition (1,...,1, 2, 1,...,1). }{{} 2 at pace Ordered partitions of k are partiay ordered by refinement, we write K J if K is a refinement of J. Moreover, ordered partitions form a attice: any coection of partitions {K m } m has we behaved greatest common refinement and east common coarsening (denoted by m (K m )and m (K m ) respectivey). In fact, the attice of ordered partitions of k is isomorphic to the Booean attice of subsets of k 1 ordered by incusion. Given K =(k 1,...,k j ), et Σ K be the group Σ p k 1 Σ p k j and et K be the summand k1 k2 kj of H (Σ K ) or more generay of H (X pk hσ K ) depending on the context. We denote by N j (k) the set of ordered partitions of k with j components. The foowing is obvious by inspection: For j>0 P j = P 0 = 0 K N j (k) In particuar, if j>kthen P j = 0. In the foowing definition, the underying assumption is that X is a 2s +1- dimensiona sphere and u is a generator of H 2s+1 (X). Definition For a fixed k, et CU be the free graded Z/pZ modue on the foowing generators: if p>2 {β ɛ 1 Q s1 β ɛ k Q s k u s k s, s i >ps i+1 ɛ i+1 i}, if p =2 { Q s1 Q s k u s k 2s +1,s i > 2s i+1}. K. Thus CU is generated by the competey unadmissibe words of ength k (hence the notation).