=n>; H*(T (n; *)) = S*(R"n)=(Q"|x|x - x2). Thus, as a bigraded algebra, H*(T (n; *)) is a polynomial algebra on the set {Q"Ixn | I is admissible ande(I) < n}, with "QIxn 2 H*(T (n; 2l(I))). Here, if I = (i1; : :;:il), "QI= "Qi1: :":Qil, and e(I), l(I) , and admissibl* *e mean what they did in x1. There is a little wrinkle here however: as "Q0is not the i* *dentity, an admissible sequence can end with 0's. The geometric results of x2 allow us to quickly deduce the behavior of ffi* a* *nd *. Proposition 5.2.ffi* : H*+1(T (n + 1; j)) -!H*(T (n; *)) is determined by (1) ffi*(Q"Ixn+1) = "QIxn, and (2) ffi* is 0 on decomposables. Proof.This follows from Proposition 2.3, and the fact that Dyer-Lashof operatio* *ns commute with the evaluation [CLM , p.6, p.218]. __|_| ____________ 5The relation "Q1"Q2= "Q3"Q0illustrates this. 18 KUHN Proposition 5.3.* : H*(T (n; *)) -!H*(T (n; *)) is determined by (1) When n = 0, *(x2j0) = xj0: (2) *(Q"sx) = "Qs(*x) if s > |x| - n: 0 (3) Whenever the iterated operation Q"Ixn is defined, *(Q"Ixn) = Q"Ixn if I = (I0; 0), and is 0 otherwise. (4) When n 1, * is an algebra map (with the second grading in the domain of * doubled). Proof.This follows from Theorem 2.4 and the last proposition. As (2j)!=j!2j is always odd, statement (1) of Theorem 2.4 implies that statement (1) here is tru* *e. Statement (2) here is implied by statement (5) of Theorem 2.4. To see that stat* *e- ment (3) is true, we first prove this in the special case when I consists only * *of 0's. Note that (1) includes the n = 0 subcase of this special case, and then the sta* *te- ment for general n follows by combining the last proposition with statement (2)* * of Theorem 2.4 (which implies that * and ffi* commute). Now use (2) to deduce (3) for general I from the special case already established. Finally, (4) follows * *from statements (3) and (4) of Theorem 2.4. __|_| Note that as a corollary of Proposition 5.2, we have partially proved Theorem* * 1.5. Corollary 5.4. (1) T (1; j) ' * unless j is a power of 2. (2) H*((T (1; 2k)) = "R[k], where "R[k] =: 6. New Nishida relations In the last section, we determined H*(T (n; *)) in terms of dual Dyer-Lashof operations. Here we describe the Steenrod algebra action. The standard Nishida relations [CLM , p.6, p.214] tell us how (Sqr)* commutes with Qs in H*(Dn+1;*S-n ). Since O(Sqr)6 acting on H*(T (n; *)) corresponds to (Sqr)* acting on H-*(Dn+1;*S-n ), we immediately have the following formula. Lemma 6.1. X -r - s O(Sqr)Q"sx = Q"r+s-iO(Sqi)x: i r - 2i Though this does completely specify the A module structure on H*(T (n; *)), i* *t is in a form completely unsuitable for proving theorems like those in the introduc* *tion. The point of this section is to prove Theorem 6.2. X s - i - 1 SqrQ"sx = "Qr+s-iSqix: i r - 2i ____________ 6O is the antiautomorphism of the connected Hopf algebra A. LOOPSPACES AND EILENBERG MACLANE SPACES 19 The reader may find it amusing to compare this formula to the Adem relation of the last section, X s - i - 1 "Qr"Qsx = "Qr+s-i"Qix; i r - 2i the Adem relations in A, X s - i - 1 SqrSqsx = Sqr+s-iSqix; i r - 2i and the formula defining the "Singer construction" [Si] X s - i - 1 Sqr(ts-1 x) = tr+s-i-1 Sqix: i r - 2i Proof of Theorem 6.2.With Sq denoting the total square 1 + Sq1 + Sq2 + : :,: to verify the formula, it suffices to check that it is consistent with the iden* *tity Sq(O(Sq)) = 1 and Lemma 6.1 above. Fixing n and s, we compute " # X X X -r - s Sqn-rO(Sqr)Q"sx= Sqn-r Q"r+s-iO(Sqi)x r r " i r - 2i # X X r + s - i - j - 1-r - s = "Qn+s-i-jSqjO(Sqi)x i;j" r n - r - 2j r - 2i # X X i + s - j - 1 + p-2i - s - p = "Qn+s-i-jSqjO(Sqi)x i;j p n - 2i - 2j - p p (letting p = r - 2i) X -(i + j) = "Qn+s-(i+j)SqjO(Sqi)x i;j n - 2(i + j) P b+p a-p a+b+1 (using J. Adem's formula [A , (25.3)]: p c-p p c mod 2) " # X -k X = Q"n+s-k Sqk-iO(Sqi)x k n - 2k i ( "s = 0n "Qn+sx = Q x if n = 0 0 otherwise. __|_| Remark 6.3.Our method of proof also shows that the analogues of the formula in Lemma 6.1, X -r - s O(Sqr)Sqsx = Sqr+s-iO(Sqi)x; i r - 2i 20 KUHN and X -r - s O(Sqr)(ts-1 x) = tr+s-i-1 Sqix; i r - 2i respectively hold in the Steenrod algebra and Singer construction. The formula * *in A already appears in the literature as [BaMi , (4.4)], where it is given a proo* *f in the style of Bullett and MacDonald [BuMacD ]. 7. The proofs of Theorem 1.2 and Theorem 1.5 To prove Theorem 1.2, first recall the description of H*(T (n; *)) given in T* *heo- rem 5.1: H*(T (n; *)) = S*(R"n)=(Q"|x|x - x2); where "Rn==n>: Note that "Rnis closed under both the action of A and *, thanks to our Nishida relations and Proposition 5.3, i.e. (R"n; *) is an object in Uae. Thus Theorem * *1.2 will follow from the next two proposition. Proposition 7.1.(R"n; *) ' Fae(n) as objects in Uae. Proposition 7.2.Let n 1. In S*(R"n), the ideal generated by elements of the form "Q|x|x - x2 equals the ideal generated by elements of the form Sq|y|y - (** *y)2. Both propositions will follow from the next result. Theorem 7.3. SqIQ"Jxn = (*)l(I)(Q"I"QJxn); whenever the iterated operation "QI"QJxn is defined. Proposition 7.1 then follows from ( Q"I(Q"0)k-l(I)if l(I) k, Corollary 7.4.If I is admissible, SqI(Q"0)kxn = 0 if l(I) > k. This same corollary, together with Corollary 5.4 proves Theorem 1.5. Proof of Proposition 7.2.Let F (x) = Q"|x|x - x2 and G(x) = Sq|x|x - (*x)2. Using the fact that R"n is unstable, it is easy to deduce that the two ideals in question are generated by elements of the form F (x) and G(x) respectively, whe* *re x 2 R"n. We claim that the sets of such elements are the same; more precisely, F (Q"Ixn) = G(Q"I"Q0xn) and G(Q"Ixn) = F (*(Q"Ixn)): To see that these hold, we let d = |I| + n and compute: F (Q"Ixn)= "Qd"QIxn - (Q"Ixn)2 = SqdQ"I"Q0xn - (*(Q"I"Q0xn))2; using Theorem 7.3 and Proposition 5.3, = G(Q"I"Q0xn): LOOPSPACES AND EILENBERG MACLANE SPACES 21 Similarly, G(Q"Ixn)= SqdQ"Ixn - (*(Q"Ixn))2 = *(Q"d"QIxn) - (*(Q"I"Q0xn))2; using Theorem 7.3 and Proposition 5.3, = "Qd*(Q"Ixn) - (*(Q"I"Q0xn))2; using part (2) of Proposition 5.3 (since n 1), = F (*(Q"Ixn)): __|_| It remains to prove Theorem 7.3. This will follow from a couple of lemmas. Lemma 7.5. SqrQ"J"Q0xn = "Qr"QJxn; whenever the iterated operation "Qr"QJxn is defined. Proof.This is proved by induction on l(J). The induction is started by using the Nishida relations to verify that SqrQ"0xn = "Qrxn. For the inductive step, suppose J = (j; J0). Then 0 0 SqrQ"J"Q0xn= SqrQ"j"QJ"Qxn X r - j - 1 0 = Q"r+j-iSqi"QJQ"0xn(using the Nishida relations) i r - 2i X r - j - 1 0 = Q"r+j-i"Qi"QJxn (by induction) i r - 2i 0 = "Qr"Qj"QJxn (using the Adem relations) = "Qr"QJxn: __|_| Lemma 7.6. SqIQ"J(Q"0)l(I)xn = "QI"QJxn; whenever the iterated operation "QI"Q* *Jxn is defined. Proof.This is proved by induction on l(I), and the last lemma is the case l(I) * *= 1. Let I = (I0; i). Then 0 i J 0 l(I) SqIQ"J(Q"0)l(I)xn= SqI Sq "Q (Q" ) xn 0 i J 0 l(I)-1 = SqI "Q"Q(Q" ) xn(by the case l(I) = 1) 0 i J = "QI"Q"Qxn (by induction) = "QI"QJxn: __|_| 22 KUHN Proof of Theorem 7.3.Applying (*)l(I)to the formula in the previous lemma yields (*)l(I)(SqIQ"J(Q"0)l(I)xn) = (*)l(I)(Q"I"QJxn): As it has a topological origin, (*)l(I)commutes with Steenrod operations. By Proposition 5.3, (*)l(I)(Q"J(Q"0)l(I)xn) = "QJxn: The theorem follows. __|_| 8.The Whitehead conjecture resolution and Theorem 1.6 In this section, we note that the homotopical equivalence of Theorem 1.6 can * *be deduced from the homological isomorphism of Theorem 1.5, using work of Lannes and Zarati [LZ2 ] to improve previous work of the author [K2 , K3]. Letting Zk = T (1; 2k) in the next theorem, Theorem 1.6 follows from Theo- rem 1.5. Theorem 8.1. Any sequence of 2 complete, connective spectra Z0 -! Z1 -! Z2 -! : : : that realizes the length filtration of A in cohomology is equivalent to the seq* *uence SP 1(S0) -!SP 2(S0) -!SP 4(S0) -!: ::: This is proved in [K2 , K3], assuming the extra geometric condition: -k(Zk=Zk-1) is a wedge summand of a suspension spectrum. We note that this geometric condition is automatically satisfied! Under our co- homological hypothesis, H*(-k(Zk=Zk-1)) is isomorphic to H*(M(k)), where M(k) is the stable wedge summand of B(Z=2)k associated to the Steinberg mod- ule. Now consider the Adams spectral sequence for computing maps from M(k) to -k(Zk=Zk-1). An A-module isomorphism H*(-k(Zk=Zk-1)) ' H*(M(k)) can be regarded as an element in E0;02. The following proposition implies that such* * an element is a permanent cycle, i.e. one can topologically realize this isomorphi* *sm. Proposition 8.2.[LZ2 , Proposition 5.4.7.1] If M is an unstable A-module, and N is a summand of H*(B(Z=2)k), then Exts;tA(M; N) = 0 for all t - s < 0. Lannes and Zarati prove this using ideas of W.Singer. As explained in [HK ], * *this proposition can also be deduced from [BC , Lemma 2.3(i)] (slightly modified) in* * the spirit of Carlsson's work [Ca ]. 9. The proof of Theorem 1.9 This sections contains the details of the proof of Theorem 1.9, which was out* *lined at the end of x1. As in [K6 ], F 2 F is said to be finite if it has a finite length composition* * series with simple subquotients, and is said to be locally finite (written F 2 F!) if * *it is the union of its finite subfunctors. Recall that I 2 F is the injective envelo* *pe of the simple functor F . The I are locally finite [K6 ]. Then the general theory* * of locally Noetherian abelian categories [S, p.92] [P , Theorem 5.8.11] implies th* *at, if J 2 F! is any injective, then there is a decomposition in F LOOPSPACES AND EILENBERG MACLANE SPACES 23 M J ' a(; J)I ; 2 where a(; J) = dimZ=2Hom F(F ; J): Applying this to the case J = -1Sj, and noting [KK ] that kj dimZ=2Hom F(F ; -1Sj) = dimZ=2Hom F(F ; S2 ); fork >> 0; we deduce that M -1Sj ' a(; j)I ; 2 with a(; j) as in the introduction. Recall that r : F -! U is defined by letting r(F )j = Hom F(Sj; F ). The fact* * that Sj is finite implies that r will commute with filtered direct limits. In partic* *ular, we can deduce the decomposition in U M -1r(Sj O Sn) ' a(; j)r(I O Sn): 2 Proposition 9.1.r(I O Sn) ' H*(K(; n); Z=2) as A modules. Momentarily postponing the proof of this, to prove Theorem 1.9, we need to show H*(-1T (n; j); Z=2) ' -1r(Sj O Sn) asA modules: Note that this asserts that a certain inverse limit of finite dimensional modul* *es is isomorphic to a certain direct limit of nilclosed modules (i.e. modules of the * *form r(F )). To show this, observe that -1r(S* O Sn) is N x N[1_2] graded. It is even an object in Kae, using -1 : -1r(S2jO Sn) -!-1r(Sj O Sn) as the restriction. Theorem 9.2. H*(-1T (n; *); Z=2) ' -1r(S* O Sn) as objects in Kae. Returning to the proof of Proposition 9.1, we first note that H*(K(; n); Z=2)* * = H*(K(V ; n); Z=2)e and r(I O Sn) = r(IV O Sn)e , where IW 2 F is the injecti* *ve defined by IW (V ) = (Z=2)Hom(V;W). Thus we need just show that r(IW O Sn) = H*(K(W; n); Z=2): Now one has the classic calculation [S, p.184] H*(K(Z=2; n); Z=2) = U(F (n)), where F (n) = A=E(n) is the free unstable module on an n dimensional class, and where U : U -! K is the free functor, left adjoint to the forgetful functor. Ex* *plicitly, U(M) = S*(M)=(Sq|x|x - x2): Similarly, H*(K(W; n); Z=2) = UW (F (n)) where UW : U -! K is given by UW (M) = U(M W *). A simple calculation reveals that F (n) = r(Sn)7 (see e.g. [K8 , Prop.8.1]), * *so the proof of Proposition 9.1 is completed with Lemma 9.3. [K9 ] There are natural isomorphisms UW (r(F )) ' r(IW O F ), for a* *ll F 2 F!. ____________ 7This is false at odd primes: F(n) is not nilclosed in the odd prime case. 24 KUHN Sketch proof.It is easy to reduce to the case when W = Z=2. Let I = IZ=2. By filtering U(M) one then verifies that if M is nilclosed, so is U(M). Thus to id* *entify U(r(F )) with r(I OF ), it suffices to check that l(U(r(F ))) = I OF , where l * *: U -! F is left adjoint to r. The functor l is exact, preserves tensor products, and ca* *n be regarded as localization away from nilpotent modules [HLS , K6]. Thus it carries S*(r(F ))=(Sq|x|x - x2) to the functor that sends V to S*(l(r(F ))(V ))=(x - x2): Since l(r(F )) = F , and I(V ) = S*(V )=(x - x2) [K6 ], this functor is just I * *O F . __|_| To prove Theorem 9.2, we need to use the main result of [K9 ]. As in [K8 ], let U2 be the category of N x N graded modules over the bigraded algebra A A, unstable in each grading. For M 2 U2, there are natural maps 1 : Mm;* -!M2m;*and 2 : M*;n-! M*;2n8, and we let K2 denote the category of commutative algebras M in U2 satisfying the "restriction" axiom: for all x 2 M; (1 2)(x) = x2. Let U2 : U2 -! K2 be left adjoint to the forgetful functor: explicitly, U2(M) = S*(M)=((1 2)(x) - x2): Given M 2 U, M F (1) is an object in U2. F (1) can be regarded as the module:, with x2k having bidegree (1; 2k). Now define Hom F (S*; F ) F (1) -!Hom F (S*; S* O F ) by sending (Si-ff!F )x2kto the composite Si-ff!F -! S2kOF: Since Hom F(S*; S*O F ) is easily checked to be in K2, this map extends to a natural map in K2: F : U2(Hom F (S*; F ) F (1)) -!Hom F (S*; S* O F ): Theorem 9.4. [K9 ] For all F 2 F!, F is an isomorphism. This is proved in a manner similar to the way Lemma 9.3 is proved. Corollary 9.5.r(S* O Sn) ' U2(F (n) F (1)), as objects in K2. Corollary 9.6.-1r(S* O Sn) ' Uae(F (n) -1F (1)), as objects in Kae. Here -1F (1) = , with the restriction map (part of the Kaestruc- ture), taking x2k to x2k-1. By Theorem 1.2, H*(-1T (n; *); Z=2) ' Uae(Fae(n)^) as objects in Kae, where Fae(n)^ denotes the inverse limit Fae(n) ae-Fae(n) ae-Fae(n) ae-: ::: The following observation completes the proof of Theorem 9.2, and thus the proof of Theorem 1.9. Lemma 9.7. Fae(n)^ = F (n) -1F (1), as objects in Uae. ____________ 8These are the Steenrod squares in the right degree LOOPSPACES AND EILENBERG MACLANE SPACES 25 10. Towards the conjectures In this section we outline some possible approaches to the conjectures of the introduction. We start with a rigorous proof of Proposition 1.11. _ Proof of Proposition 1.11.Let X(j) = a(; j)K(; 1); and recall that we wish 2 to topologically realize an A module isomorphism: H*(-1T (1; j); Z=2) ' H*(X(j); Z=2): But Proposition 8.2 tells us that any such A-module map can be realized: in the Adams spectral sequence for computing maps from X(j) to -1T (1; j), Es;t2= 0 for t - s < 0. __|_| Thus far, we have been unable to find any way in which this proof, or the rel* *ated proofs of Conjecture 1.3 [L1, Goe, HK ] in the n = 1 case, generalize to prove * *the n > 1 cases of the conjectures. These Adams spectral sequence based proofs rely on magical properties of the spectra T (j) and K(V; 1), which, in turn, are (pa* *rtly) due to the fact that H*(T (j); Z=2) and H*(K(V; 1)) are injective in U. A search for similar proofs of the conjectures leads to the following questions. Question 10.1. For n > 1, do H*(T (n; j); Z=2) and H*(K(Z=2; n); Z=2) have any sort of injectivity properties in some well chosen subcategory of U? Question 10.2. Is Exts;tA(H*(K(W; n)); H*(K(V; n))) = 0 if t - s < 0? Question 10.3. Is Z=2[Hom (V; W )] -!Hom A (H*(K(W; n)); H*(K(V; n))) an iso- morphism? Note that an affirmative answer to the second question would allow us to prove Conjecture 1.10 along the lines of the above proof of Proposition 1.11. It is n* *ot hard to show that, if the last question has an affirmative answer, then Conjecture 1* *.10 would follow if one could just construct a family of stable maps (10.1) T (n; 2k) -!K(Z=2; n) nonzero in cohomology in dimension n. Related to the [HK ] proof of Conjecture 1.3 in the n = 1 case, we note that,* * by [HK , Proposition 1.6], if Conjecture 1.10 were true, then one could conclude t* *hat ffl : 1 1 T (n; j) -! T (n; j) is onto in mod 2 homology, which is a weak form * *of Conjecture 1.3. Now we discuss a rather intriguing "conceptual" approach to Conjectures 1.3 and 1.10. The idea would be to start with the n = 0 case (!) of the conjectures, using the concept of S-algebras (a.k.a. E1 -ring spectra). Question 10.4. Let denote a divided power algebra over Z2._Does there exist_an N-graded commutative augmented S-algebra structure on T = T (0; j) = S0 j0 j0 such that (1) ss0(T ) = , (2) : T -! T is a map of S-algebras, and (3) T admits a nonzero S-algebra map to 1 (Z=2)+ ? 26 KUHN An affirmative answer to parts (1) and (3) would presumably yield a construct* *ion of maps as in (10.1) upon applying the "bar construction" n times to map (3). We can refine this question, motivated by work in [K8 ]. The key is to rearra* *nge the untidy right side of the isomorphism _ H*(-1T (n; j); Z=2) ' H*( a(; j)K(; n); Z=2): 2 We know that this module corresponds to the functor (-1Sj)OSn 2 F. The proof in [K8 ] that -1Sj is injective in F! reveals that -1Sj ' lim I(F2s)*[j]; s-!1 where (F2s)* is the F2 linear dual of the finite field F2s, and I(F2s)*[j] is t* *he jth eigenspace of I(F2s)*under the action of Fx2s. Furthermore, if we extend t* *he scalars to the algebraic closure F2, this isomorphism is well behaved with resp* *ect to pairings (between various j's). It follows that H*(-1T (n; *); F2) ' H*cont(K(F2 ; n); F2)[*] as N[1_2] graded algebras in U, where we write H*cont(K(F2 ; n); F2) = lim H*(K(F2s; n); F2): s-!1 Just as one can discuss S-algebras, one can discuss SW (F2 )-algebras, where W (F2 ) are the Witt vectors of F2. Question 10.5. With T the S-algebra as in Question 10.4 above, does there exist an equivalence of N[1_2] graded SW (F2 )-algebras -1T ^S SW (F2 ) ' 1 ((F2 )*)+ ^S SW (F2 )? As before, an affirmative answer to this formidable question would presumably yield a proof of Conjecture 1.10 upon applying the bar construction to the equi* *va- lence n times. We end with a question about the most straightforward way to try to get at these sorts of things. Question 10.6. Does there exist a "naturally occurring" spectrum E, with a group action, such that the group action can be used to establish a splitting _ En ' T1(n; j); j where En is the nthinfinite loop space of the spectrum E, and T1(n; j) is a des* *us- pension of T (n; j)? When n = 1, this would be consistent with [GLM ]. However, anyone searching for such a spectrum should make sure their search is compatible with results in [McG ]. LOOPSPACES AND EILENBERG MACLANE SPACES 27 Appendix A. Connections with work of Arone and Mahowald In this appendix, we explain how our constructions are related to those appea* *ring in [AM ] in their work on the Goodwillie tower of the identity. (Our arguments * *are a bit sketchy as we plan to elaborate on these ideas elsewhere.) Recall our definition: "Dn;j(X) = F (C(n; j)+ ; X[j])j . We begin by rewrit* *ing this in a useful way. Let (n; j) Snj be the singular part of the j-space Snj. Then C(n; j) is equivariantly homotopy equivalent to Snj - (n; j) (the configuration space). Thus, by equivariant Alexander duality [LMMS , Theorem III.4.1], F (C(n; j)+ ; (nX)[j]) ' Snj=(n; j) ^ X[j] as j spectra. Now note that this latter spectrum is clearly j-free, as Snj=(n; * *j) is, thus its fixed point spectrum is naturally equivalent to its orbit spectrum* * [LMMS , Theorem II.7.1]. We have proved Proposition A.1. "Dn;j(nX) is naturally equivalent to ((Snj=(n; j)) ^ X[j])j . Checking definitions reveals Lemma A.2. fi : C(n + 1; j)+ ^ S1 -!C(n; j)+ ^ Sj is equivariantly S-dual to * *the evident diagonal map S1 ^ (Snj=(n; j)) -!S(n+1)j=(n + 1; j). Definition A.3.Let "Dj(X) = hocolimn-n "Dn;j(nX), with the colimit induced by either of the maps in the last lemma. Note that, with this notation, T (1; j) = D"j(S-1). Now let Kj be the j-space introduced in [AM ]: Kj is the unreduced sus- pension of K"j, the classifying space of the poset of the nontrivial partitions* * of a set with cardinality j. (By nontrivial, we mean to exclude the partitions (j) a* *nd (1; 1; : :;:1).) Proposition A.4. [AM , early versions] and [AD , x6] There is a j equivariant map hocolimn-n (Snj=(n; j)) -!Kj that is a nonequivariant equivalence. Corollary A.5. "Dj(X) = (Kj^ X[j])hj . Combining this corollary with Theorem 1.6 yields Theorem A.6. (K2k^ S-2k)h2k ' -2SP 2k(S0): In work in progress, we have established the following. Proposition A.7. Localized at 2, there are cofibration sequences D"2k-1(S2n-1) -!D"2k(Sn-1) -!D"2k(Sn) which are short exact in cohomology. The first map here is constructed with Hopf invariant techniques, and is the generalization of : T (1; 2k-1) -!T (1; 2k). Using these sequences when n = 0 and n = 1, one can deduce 28 KUHN Corollary A.8. Localizedkat 2, therekare-equivalences1 (1) (K2k)h2k ' -1SP 2(S0)=SP 2 (S0); (2) (K2k^ S2k)h2k ' SP 2k(S0)=SP 2k-1(S0): Part (2) of this corollary is due to Arone and Mahowald who sketch the follow* *ing elegant and direct short proof in their early versions of [AM ]. (See also [AD * *].) Lemma A.9. The space Snj=j is homeomorphic to SP j(Sn)=SP j-1(Sn). Lemma A.10. ((n; j) ^ Sj)j is contractible. Sketch proof.The partition filtration of (n; j) induces a filtration of ((n; j)* * ^ Sj)j in which each subquotient has the form SP i(S1)=SP i-1(S1), and so is con- tractible. __|_| Corollary A.11. There are homotopy equivalences of spaces SP j(Sn+1)=SP j-1(Sn+1) ' (Snj=(n; j) ^ Sj)j : Proof.SP j(Sn+1)=SP j-1(Sn+1) ' (Snj ^ Sj)j ' (Snj=(n; j) ^ Sj)j : __|_| Now Corollary A.8(2) follows by letting n go to infinity, and using Proposi- tion A.4. We finish with one last observation. Let Dj(X) = F (Kj; X[j])hj . Arone and Mahowald [AM ] show that 1 Dj(X) is the jth fiber of the Goodwillie tower of the identity applied to a space X. Arone and Dwyer [AD ] show that, if X is an odd dimensional sphere, then 2kD2k(X) ' (K2k^ X[2k])h2k . Thus we have Corollary A.12. If X is an odd dimensional sphere, then D2k(X) ' -2kD"2k(X). Corollary A.13. D2k(S-1) ' -(2k+1)SP 2k(S0): References [A]J. Adem, The relations on Steenrod powers of cohomology classes, Algebraic G* *eometry and Topology (R. H. Fox, et. al., editors), Princeton University Press, 1957, 19* *1-242. [AD]G.Z. Arone and W.G. Dwyer, Partition complexes, Tits buildings, and symmetr* *ic product spectra, preliminary preprint, 1997. 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