MODULI OF SUSPENSION SPECTRA JOHN R. KLEIN Abstract. For a 1-connected spectrum E, we study the mod- uli space of suspension spectra which come equipped with a weak equivalence to E. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Good- willie's calculus of homotopy functors. 1. Introduction It's well known that the singular cohomology of a space has the structure of a commutative graded ring. The corresponding statement for spectra fails to hold. The problem arises from the observation that a spectrum E might not have a "diagonal map" E ! E ^E. Of course, a diagonal map exists when E is a suspension spectrum. This motivates Question. Let E be a spectrum. When can we find a based space X and a weak equivalence of spectra 1 X ' E? In how many ways? Our first result gives a criterion for deciding the existence part of this question in the metastable range. Recall that E is r-connected if its homotopy groups ß*(E) vanish when * r. Write dim E n if E is, up to homotopy, a CW spectrum with cells in dimensions n. Recall that the k-th extended power Dk(E) is the homotopy orbit spectrum of k acting on the k-fold smash product E^k. Theorem A (Existence). There is an obstruction ffiE 2 [E, D2(E)] , which is trivial when E has the homotopy type of a suspension spectrum. Conversely, if ffiE = 0, E is r-connected, r 1 and dim E 3r+ 2, then E has the homotopy type of a suspension spectrum. Before stating our second result, we comment on the relation be- tween Theorem A and one of the early results of Kuhn ([Ku , Th. 1.2]) ____________ Date: October 17, 2002. 2000 MSC. Primary: 55P42, 55P43. Secondary: 55P40, 55P65. The author is partially supported by NSF Grant DMS-0201695. 1 2 JOHN R. KLEIN which says that a connected spectrum E is a retract of a suspension spectrum if and only if there is a weak equivalence ` 1 1 E ' Dk(E) . k 1 Theorem A implies one can remove the word "retract" from Kuhn's result in the metastable range. Next consider the collection of pairs (X, h) in which X is a based space and h : 1 X ! E is a weak equivalence. Equate two such pairs (X, h) and (Y, g) if and only if there is a map of spaces f :X ! Y such that g O 1 f is homotopic to h (in particular, f is a homology isomorphism). This generates an equivalence relation. Let E denote the associated set of equivalence classes. Our second result identifies E in the metastable range. Theorem B (Enumeration). Assume E is nonempty and is equipped with basepoint. Then there is a basepoint preserving function OE : E ! [E, D2(E)] . If E is r-connected, r 1 and dim E 3r + 2, then OE is a surjection. If in addition dim E 3r + 1, OE is a bijection. In particular, E has the structure of an abelian group when dim E 3r + 1. Theorem B leads to the possibility of calculating E in a number of simple cases. For example, table 4.1 appearing in Mahowald's memoir [Mah ] provides extensive calculations of E at the prime 2 in the case of spectra with two cells (see x8 below). In fact, E is the set of path components of a space ME which is defined as follows: let CE be the category whose objects are pairs (Y, h) with Y a based space and h : 1 Y ! E a weak equivalence. A morphism (Y, h) ! (Z, g) consists of a map of based spaces f :Y ! Z such that g O 1 f = h. Then define ME := |CE | , i.e., the geometric realization of (the nerve of) CE .1 We call ME the moduli space of suspension structures on E. Our third result gives a spectral sequence converging to the homo- topy of ME in positive degrees. Formulating it requires some prepara- tion. ____________ 1There are set-theoretic difficulties presented by this definition, since CE* * isn't a small category. This problem can be avoided in several ways (compare [Wa , p. 379]). MODULI OF SUSPENSION SPECTRA 3 For q 2, let Wq be the spectrum with q-action which classifies the q-th layer of the Goodwillie tower of the identity functor from based spaces to based spaces. Unequivariantly, Wq is a wedge of (q- 1) copies of the (1- q)-sphere spectrum (see Johnson [Jo ]). If q 1, we take Wq to be the trivial spectrum. If E is a spectrum, then its q-fold smash power E^q has the structure of a (naive) spectrum with q-action. Give the smash product Wq^E^q the diagonal q-action. We are then entitled to form Wq ^h q E^q , i.e, the homotopy orbit spectrum of q acting on Wq ^ E^q. Let F (E, Wq ^h q E^q) denote the function space of spectrum maps from E to Wq ^h q E^q. With these definitions we are ready to state our third result. Theorem C (Spectral Sequence). Let E be 1-connected. Assume ME is non-empty and equipped with basepoint. Then there is a first quad- rant spectral sequence converging to ßp+ 1(ME ) with E1p,q:= ßp(F (E, Wq ^h q E^q)) and d1-differential of bi-degree (-1, 1). Note when dim E n, we have E1p,q= 0 for p q + 1 - n. Thus for fixed p, E1p,qis non-zero for finitely many q. Our last result says that the obstruction ffiE is, in some sense, 2-local. Theorem D (Localization). For any connected spectrum E, the ho- motopy class ffiE becomes trivial after inverting the prime 2. Furthermore, if f :E ! E0 is a map of spectra which is a 2-local weak equivalence, then ffiE = 0 if and only if ffiE0 = 0. Consequently, if E and E0 satisfy the connectivity assumptions of Theorem A, then E is weak equivalent to a suspension spectrum if and only E0 is weak equivalent to a suspension spectrum. Remarks. Starting with the work with Berstein and Hilton [B-Hi ], the problem of deciding when a space has the homotopy type of a (single or iterated) suspension has been intensively studied. To the best of my knowledge, the literature contains much less in- formation about the desuspension problem for spectra, aside from the trivial stable range case (Freudenthal's theorem) and a p-local version considered by Gray [Gr ]. My original interest in the desuspension question for spectra came from embedding theory. In a future paper, we intend to use the above results to attack certain embedding questions. 4 JOHN R. KLEIN Acknowledgements. This paper would not have come into being had I not had the fortune of discussing mathematics with Greg Arone, Randy McCarthy and Tom Goodwillie in the middle to late 1990s. I am indebted to them for their help. I am also grateful to Bob Bruner for explaining to me how to compute the stable homotopy of stunted projective space in low degrees. I am also grateful to Bruner for drawing my attention to the early work of Nick Kuhn. Outline. x2 consists of preliminary material. The proof of Theorem A is contained in x3. In x4 we prove Theorem B. In x5 we express the connected components of the moduli space ME as the classifying space of a suitable monoid. x6 contains the proof of Theorem C. Theorem D is proved in x7. In x8 we provide examples in connection with Theorem B. In x9 we discuss some loose ends. 2. Preliminaries In this section we give the conventions and tools used in the rest of the paper. We make no claim to completeness. Spaces. All spaces below will be compactly generated, and Top will denote the category of compactly generated spaces. In particular, we make the convention that products are to be retopologized with respect to the compactly generated topology. Let Top* denote the category of based spaces. A weak equivalence of spaces is shorthand for (a chain of) weak homotopy equivalence(s). We use the usual connectivity terminology for spaces: A non-empty space is r-connected if its homotopy vanishes in degrees r for ev- ery choice of basepoint (in particular, every non-empty space is (-1)- connected). A map A ! B of spaces, with B nonempty, is r-connected if for any choice of basepoint in B, the homotopy fiber with respect to this choice of basepoint is an (r- 1)-connected space. A commutative square of spaces A --- ! C ? ? ? ? y y B --- ! D is said to be k-cocartesian if the map hocolim (B A ! C) ! D is k-connected. It is 1-cartesian if it is k-cartesian for all k. Dually, the square is k-cartesian if the map A ! holim (B ! D C) is k- connected. The square is 1-(co)cartesian if it is k-(co)cartesian for all k. MODULI OF SUSPENSION SPECTRA 5 Spectra. A spectrum E will be taken to mean a collection of based spaces {Ei}i2N together with based maps Ei ! Ei+ 1where Ei is the reduced suspension of Ei. A morphism of spectra E ! E0 consists of maps Ei ! E0ithat are compatible with the structure maps. We denote the category of spectra by Sp. A map of spectra is r-connected if it induces a surjection on homo- topy up through degree r and an isomorphism in degrees less than r. A spectrum is r-connected if the map to the trivial spectrum (consisting of the one point space in each degree) is (r+ 1)-connected. A map of spectra is a weak equivalence if it is r-connected for all integers r. This notion of weak equivalence comes from a Quillen model structure on Sp. In this model structure, the fibrant objects are the -spectra (those spectra E such that Ei ! Ei+ 1is a weak equivalence for all i. A cofibrant object is (a retract of) a spectrum which is built up from the zero object by attaching cells. The model structure on Sp comes equipped with functorial factorizations; in particular, fibrant and cofibrant approximation is functorial. For details, see for example Schwede [Sc ]. We typically apply fibrant and/or cofibrant approximation functors to maintain homotopy invariance. To avoid clutter, we usually suppress the application of these approximations in the notation. For example, if E is a fibrant spectrum we write 1 E for the zero space E0 of E. If E isn't fibrant, to get a good construction we first replace E by its associated -spectrum E] and then define 1 E as E]0. If X is a based space, its suspension spectrum 1 X has j-th space jX, the j-fold reduced suspension of X (for this to have the correct homotopy type, we assume that X is a cofibrant space; i.e., the retract of a cell complex). In particular, Q(X) := 1 1 X is the reduced stable homotopy functor. A map of spaces Y ! 1 E is adjoint to a map of spectra 1 Y ! E. As in the introduction, we write dim E n if and only if E is, up to homotopy, obtained from the zero object by attaching cells of dimension n. A spectrum is finite if it is built up from the zero object by attaching a finite number of cells. A spectrum is homotopy finite if it is weak equivalent to a finite spectrum. In this paper, we can get away with a notion of smash product which is associative, commutative and unital up to homotopy (see e.g. Lewis, May and Steinberger [L-M-S , Chap. 2]). We also need to know that a construction of the extended power spectrum Dk(E) = (E^k)h k := (E k)+ ^ k E^k 6 JOHN R. KLEIN exists, is functorial, homotopy invariant and coincides with the usual one in the case of spaces. For details, see [L-M-S , Chap. 6]. Truncation. Let Y be a based space, W be a 1-connected spectrum and f : 1 Y ! W a map of spectra. Assume dim W n and that f is n-connected. Lemma 2.1. If n 0, there exists a space Z, and an (n- 1)-connected map g :Z ! Y such that the composite 1 g 1 f 1 Z ! Y ! W is a weak equivalence. Proof. If n 1, then W is weak equivalent to the zero object, and the proof is trivial in this case. Assume then that n 2. Let hZ be the Eilenberg-Mac Lane spec- trum. Then the induced map hZ ^ 1 Y ! hZ ^ W can be thought of as a map of chain complexes (via the Dold-Kan correspondence). Applying the truncation lemma [Kl , 4.1], We obtain a space Z and an (n- 1)-connected map Z ! Y such that the composite hZ ^ 1 Z ! hZ ^ 1 Y ! hZ ^ W is a weak equivalence. But this composite is obtained by smashing the composite 1 Z ! 1 Y ! W with hZ. The result then follows by application of Whitehead's theo- rem. Corollary 2.2 (Freudenthal). Let E be a an r-connected spectrum, r 1. Assume dim E 2r+ 2. Then E is weak equivalent to a suspension spectrum. Proof. This follows from 2.1 because the map 1 1 E ! E is (2r+ 2)- connected (see 3.2 below). Another description of the moduli space. We can use Lemma 2.1 to give an alternative description of the moduli space when E is 1-connected. Let C0E be the (not full) subcategory of CE having the same objects, but with the property that morphisms (Y, h) ! (Y 0, h0) satisfy the condition that Y ! Y 0is a weak homotopy equivalence. Let M0Edenote the realization of C0E. Proposition 2.3. Assume that E is fibrant, cofibrant and 1-connected. Then the inclusion M0E ME is a homotopy equivalence. MODULI OF SUSPENSION SPECTRA 7 Proof. Let Y 7! Y + denote the plus construction. Applying functorial factorization, we can arrange it so that the natural map Y ! Y + is a cofibration. Observe that if (Y, h) is an object of CE , then the 1-connectedness of E implies Y has trivial first homology. Therefore, Y + is a 1-connected space. So if (Y, h) ! (Z, h0) is a morphism, the induced map of plus constructions Y + ! Z+ is a weak equivalence by Whitehead's theorem. Since E is fibrant and cofibrant, the space 1 E is cofibrant and 1- connected. The natural map 1 E ! ( 1 E)+ is a cofibration which is a weak equivalence. Consequently, there is a retraction r :( 1 E)+ ! 1 E. Let i :C0E! CE denote the inclusion. Define a functor ß :CE ! C0E by the rule ß(Y, h) = (Y +, h[), where h[: 1 (Y +) ! E is adjoint to the composite (^h)+ 1 + r 1 Y + ! ( E) ! E . Then there is an evident natural transformation from the identity func- tor of CE to the composite functor i O ß. There is a similar evident nat- ural transformation from the identity functor of C0Eto ß O i. It follows that i induces a homotopy equivalence on realizations. 3. Proof of Theorem A After defining the obstruction ffiE , the idea of the remainder of the proof will be to construct a highly connected map from a suspension spectrum to E. The proof is then completed by applying Lemma 2.1. Definition of the obstruction. We describe below a certain fibration sequence of spectra D2(E) ! S2(E) ! E in which S2: Sp ! Sp is a certain homotopy functor. Assuming this construction has been specified, we have: Definition 3.1. The class ffiE 2 [E, D2(E)] is the obstruction to splitting the above fibration sequence, i.e., the homotopy class of its connecting map to the right. 8 JOHN R. KLEIN Construction of the fibration. Consider the homotopy functor 1 1 : Sp ! Sp which assigns to a spectrum the suspension spectrum of its zero space. Let Sk(E) denote the k-th stage of the Goodwillie tower of this functor and let Fk(E) := fiber(Sk(E) ! Sk- 1(E)) denote the k-th layer. The following result has been noted by several people, including Goodwillie, Arone, McCarthy, and Ahearn and Kuhn [A-K , Cor. 1.3]. Lemma 3.2. Assume E is r-connected. Then the map 1 1 E ! Sk(E) is ((k+ 1)r + k+ 1)-connected. Consequently, the Goodwillie tower of 1 1 E is convergent if E is 0-connected. Furthermore, there is a natural weak equivalence of functors Fk(E) ' Dk(E) . Applying 3.2, we see that the bottom of the tower yields a fibration sequence of spectra D2(E) ! S2(E) ! E together with a (3r+ 3)-connected map 1 1 E ! S2(E). Corollary 3.3. Assume E is r-connected and dim E 3r + 3. Then ffiE = 0 if and only if the map 1 1 E ! E admits a section up to homotopy. Proof. The class ffiE is trivial if and only if the S2(E) ! E has a section. The dimension constraint on E and the lemma show that S2(E) ! E admits a section if and only if 1 1 E ! E admits a section up to homotopy. Now assume E is r-connected, r 1, dim E 3r + 3 and ffiE = 0. By the above remarks, we are entitled to choose a homotopy section oe :E ! 1 1 E. Let 1 oe : 1 E ! 1 1 1 E be the corresponding map of zero spaces. There is another map c : 1 E ! 1 1 1 E (not an infinite loop map) which is defined by taking the adjoint to the identity map 1 1 E ! 1 1 E. MODULI OF SUSPENSION SPECTRA 9 Let Y be the homotopy pullback of the maps 1 oe and c. Thus we have an 1-cartesian square of spaces j (1) Y ________________// 1 E i || ||1 ff fflffl| fflffl| 1 E _______c__//_ 1 1 1 E (commutative up to preferred homotopy). Lemma 3.4. With respect to the above assumptions, let ^j: 1 Y ! E be the adjoint to the map labeled j in diagram (1). Then ^jis (3r + 2)- connected. Proof. Consider the diagram of spectra 1 j qE (2) 1 Y ________________// 1 1 E__________________//E 1 i|| |1| 1 ff |ff| fflffl| fflffl| fflffl| 1 1 E _____1_c__//_ 1 1 1 1 E _q_1__1_E_//_ 1 1 E in which the maps labeled with q are the counits to the adjunction for E and 1 1 E. The map ^jis therefore given by the composite qE O 1 j, and the composite along the bottom, q 1 1 EO 1 c is clearly the identity. The left square commutes up to a preferred homotopy and is 1-cocartesian. The right square commutes on the nose. The maps 1 oe and c in diagram (1) are both (2r+ 1)-connected. Consequently, by the dual Blakers-Massey theorem (see e.g., [Go2 ]) diagram (1) is a (4r + 2)-cocartesian square of spaces. In particular, the left square in diagram (2) is a (4r+2)-cocartesian square of spectra. Regarding the right square in diagram (2), If C denotes the ho- motopy cofiber of oe, then the evident map C ! D2(E) is (3r+ 3)- connected. Similarly, if C0 denotes the homotopy cofiber of 1 1 oe, an argument using the Blakers-Massey theorem, which we omit, shows that the evident map C0 ! 1 1 D2(E) is also (3r+ 3)-connected. Lemma 3.2 shows that the map 1 1 D2(E) ! D2(E) is (4r+ 4)- connected (because D2(E) is (2r+ 1)-connected). We infer that the map C0 ! C is also (3r+ 3)-connected. Therefore the right square in diagram (2) is (3r+ 3)-cocartesian. Putting both squares together, it follows that diagram (2) is (3r+ 3)- cocartesian. Since the bottom composite is the identity map, we infer that the top composite ^jis (3r+ 2)-connected, as asserted. 10 JOHN R. KLEIN Completion of the proof. Assume in addition to the above that dim E 3r+ 2. Since the map ^j: 1 Y ! E is (3r+ 2)-connected we can apply 2.1. This gives a based space Z and a based map Z ! Y such that the composite 1 Z ! 1 Y ! E is a weak equivalence. This completes the proof of Theorem A. 4. Proof of Theorem B Step 1. Let SE denote the homotopy classes of sections of the fibra- tion S2(E) ! E. Note that the abelian group [E, D2(E)] acts freely and transitively on SE (cf. Lemma 6.1 below). Thus if SE is given a basepoint, it follows that [E, D2(E)] and SE are isomorphic (the iso- morphism is dependent on the choice of basepoint). There is a function OE : E ! SE defined by sending (Y, h) to the class represented by E ' 1 Y ! S2( 1 Y ) ' S2(E) (the map in the middle is the preferred section of S2( 1 Y ) ! 1 Y ). If E ! S2(E) is a representative of SE , and dim E 3r + 2, we can perform the constructions in the previous section to get an element of E (this element is not necessarily unique). It is straightforward to check that this element of E maps to the given element of SE . Thus, when dim E 3r+ 2, the function OE : E ! SE is onto. To complete the proof of Theorem B, it will be sufficient to show that OE is one-to-one when dim E 3r + 1. Choose a basepoint (Y, h) for E , then SE inherits a basepoint and SE becomes identified with [E, D2(E)]. Thus we may rewrite OE as a basepoint preserving function E ! [E, D2(E)] , where the basepoint of the codomain is the zero element. Since Y was chosen arbitrarily, it suffices to show that OE is one-to-one at the inverse image of the basepoint. Step 2. We digress to develop a relative version of Theorem A. Sup- pose that A æ E is a cofibration in the category of spectra. We write dim (E, A) n if E is obtained from A up to homotopy by attach- ing cells of dimension n. Assume that A = 1 Z is a suspension spectrum. Theorem 4.1. There is an obstruction ffi(E,A)2 [E=A, D2(E)] MODULI OF SUSPENSION SPECTRA 11 whose triviality is necessary to finding a cofibration Z ! W and a weak equivalence 1 W ' E extending the identity on 1 Z = A. If E is r-connected (r 1) and dim (E, A) 3r+ 2, then the vanishing of this obstruction is also sufficient. The obstruction ffi(E,A)is defined in the same way as ffiE taking care to notice that the restriction of ffiE to A has a preferred trivialization. The proof of 4.1 is virtually the same as the proof of Theorem A, so we omit it. Step 3. We now return to the proof of Theorem B. We shall apply 4.1 in the following situation: choose a representative (Y, h) for an element in E (call this the basepoint). Suppose that (Y 0, h0) represents an another element in E . Both elements combine to assemble to give a weak equivalence 1 (Y _ Y 0) ~! E _ E . Applying 4.1 to the pair (E ^ I+ , E _ E) we get a necessary obstruction ffi(E^I+,E_E) 2 [ E, D2(E)] = [E, D2(E)] to finding a cofibration Y _ Y 0 æ W of based spaces and a weak equivalence ( 1 W, 1 (Y _ Y 0)) ' (E ^ I+ , E _ E) . It follows from this that the zig-zag Y ! W Y 0equates (Y, h) with (Y 0, h0) in E . Thus, fixing (Y, h) and allowing (Y 0, h0) to vary defines a function E ! [E, D2(E)] , which is just another description of the function OE (we omit the details). Assume that E is r-connected and dim E 3r+ 1. Then 4.1 shows that (Y 0, h0) maps to zero under OE if and only if (Y 0, h0) is the basepoint of E . This completes the proof of Theorem B. 5. ME as a classifying space Fix a cofibrant based space Z. Let Top*=Z denote the category of based spaces over Z. An object y of this category consists of a based space Y together with a based map pY :Y ! Z. A morphism (Y, pY ) ! (Y 0, pY 0) is given by a map of based spaces f :Y ! Y 0such that pY 0O f = pY . Since Top*=Z is an over category of the Quillen model category Top*, it follows that it too has the structure of a Quillen model category (see [Qu ]). A weak equivalence is a morphism y ! y0 whose underlying map of spaces is a weak homotopy equivalence. We say y ! y0 is a 12 JOHN R. KLEIN fibration if its underlying map of spaces is. We say that y ! y0 is a cofibration if it satisfies the left lifting property with respect to the acyclic fibrations. Let wTop*=Z denote the subcategory consisting of the weak equiv- alences. Let w(y)Top*=Z denote the full subcategory of wTop*=Z con- sisting of those objects connected to y by a chain of weak equivalences. The proof of the following proposition, which we attribute to Wald- hausen, is proved by the same method as [Wa , 2.2.5]. We omit the details. Proposition 5.1. There is a weak equivalence of spaces |w(y)Top*=Z| ' BG(y) , where G(y) denotes the topological monoid of self homotopy equiva- lences of y in Top*=Z, and BG(y) is its classifying space. We apply this in the following special case: let E be a 1-connected fibrant and cofibrant spectrum. The category C0E, whose realization is the moduli space M0E(cf. x2), is just the full subcategory of wTop*= 1 E whose objects Y are such that the map pY :Y ! 1 E is adjoint to a weak equivalence of spectra. If we combine Proposition 5.1 with Proposition 2.3, we obtain Corollary 5.2. Let y be an object of CE which is fibrant and cofibrant as an object of Top*= 1 E. Let ME,(y)be the connected component of ME which contains y. Then there is a homotopy equivalence ME,(y)' BG(y) . 6. Proof of Theorem C Outline of the proof. The basepoint y = (Y, h) of ME can be taken as an object of CE . Taking the plus construction if necessary, we can assume without loss in generality that Y is 1-connected (see the argu- ment in the proof of Proposition 2.3). We may also assume that y is fibrant and cofibrant when considered as an object of Top*= 1 E. We will construct a tower of fibrations of based spaces . .!.T3(y) ! T2(y) ! T1(y) such that o T1(y) is contractible; o for k > 1, there is a weak equivalence fiber(Tk(y) ! Tk-1(y)) ' 1 (Wk ^h k Y ^k) , where Wk denotes the spectrum that classifies the k-th layer of the Goodwillie tower of the identity functor on based spaces. MODULI OF SUSPENSION SPECTRA 13 o there is a weak equivalence ME,(y)' limTy(y) . k Assuming this has been done, we can define the spectral sequence {Erp,q} as the homotopy spectral sequence of the tower {Tk(y)}k. We now digress to discuss generalities about section spaces and the basic properties of the Goodwillie tower of the identity functor. Digression. Suppose p :E ! Z is a fibration of based spaces. We say that p is induced if there exists a commutative 1-cartesian square of based spaces E --- ! P ? ? p?y ?y Z --- ! B with the property that P is contractible. Let Sec (p) denote the space of sections of p. Suppose that Z is connected. If p :E ! Z is induced, there is an ä ction" B x E ! E. If there is a section Z ! E one can combine it with this action to produce a map of fibrations B x Z ! E covering the identity map of Z. This implies that p is weak fiber homotopically trivial. In particular, Lemma 6.1. Assume p :E ! Z is induced. Assume that Sec (p) is non-empty and comes equipped with basepoint. Then there is a weak equivalence of based spaces Sec(p) ' F (Z, B) . We next recall for the reader the basic properties of the Goodwillie tower of the identity functor on based spaces (cf. Goodwillie [Go1 ], [Go2 ], [Go3 ], [Go4 ], Johnson [Jo ] and Arone [Ar ]). Theorem 6.2. There is a tower of fibrations of homotopy functors on based spaces . . .! P2(X) ! P1(X) and compatible natural transformations X ! Pk(X) such that o P1(X) = Q(X) is the stable homotopy functor; o the k-the layer Lk(X) is naturally weak equivalent to the functor X 7! 1 (Wk ^h k X^k ) ; 14 JOHN R. KLEIN o if X is 1-connected, then the natural map X ! limPk(X) k is a weak equivalence. An additional fact we will need, but which to my knowledge is not stated in the literature, is Addendum 6.3. For each k 2, the fibration Pk(Y ) ! Pk- 1(Y ) is induced. This is due to Goodwillie (private communication). We will assume 6.3 without providing a proof. Completion of the proof. We are now in a position to define the tower {Tk(y)}k. Definition 6.4. Let Tk(y) be the space of lifts Pk(Y<)< z z | z | zz fflffl| Y _____//P1(Y ) where Y ! P1(Y ) is the natural map. Note that Tk(y) comes equipped with a basepoint defined by the natural map Y ! Pk(Y ). From the definition of Tk(y), there is an evident tower of fibrations of based spaces . . .! T2(y) ! T1(y) with T1(y) = *. Furthermore, the fiber of the map Tk(y) ! Tk-1(y) at the basepoint is identified with the space of lifts Pk(Y;); v v | v | vv fflffl| Y ____//_Pk- 1(Y ) . Note that this is just the space of sections of the pulled back fibration Y xPk-1(Y )Pk(Y ) ! Y . The latter fibration is induced and comes equipped with a preferred section. Applying 6.1, 6.2 and 6.3, we obtain a weak equivalence of based spaces fiber(Tk(y) ! Tk- 1(y)) ' F (Y, 1 (Wk ^h k Y ^k)) . Taking adjunctions, we see that F (Y, 1 (Wk ^h k Y ^k)) is weak equiv- alent to the space F (E, Wk ^h k E^k) (where we are using the identi- fication 1 Y ' E). MODULI OF SUSPENSION SPECTRA 15 To complete the proof of Theorem C, it suffices to identify the in- verse limit of the tower {Tk(y)}k. It is clear that this inverse limit is just the space of lifts limPk(Y ) k;; x x | x | xx fflffl| Y _____//_P1(Y ) . Recall that P1(Y ) = Q(Y ) and since Y is 1-connected, limkPk(Y ) ' Y . With respect to these identifications, we infer that this space of lifts is weak equivalent to the underlying space of the topological monoid G(y). Applying 5.2 and 2.3 completes the proof of Theorem C. 7. Proof of Theorem D Let tr: D2(E) ! E^2 be the transfer and let ß :E^2 ! D2(E) be the projection ([L-M-S , Chap. 4]). It is well known that the composite p O tr:D2(E) ! D2(E) is a weak equivalence after inverting 2. To prove the first part of Theorem D, it is clearly enough to show Lemma 7.1. For a connected spectrum E, the class ( tr) O ffiE 2 [E, E ^ E] is trivial. Proof. Consider the functor from spectra to spectra defined by E 7! E ^ E. This is a homogeneous homotopy functor of degree 2. We will describe a natural transformation : 1 1 E ! E ^ E . By taking adjoints it is enough to describe a natural transformation of spaces 1 E ! 1 (E ^ E). It will suffice to do this when E is a coor- dinate free spectrum, i.e., a spectrum indexed over finite dimensional subspaces of R1 (with induced inner product). If V R1 is a finite dimensional subspace, there is a map V EV ! V V(EV ^ EV ) which assigns to a map f :SV ! EV the map f ^ f :SV ^ SV ! EV ^ EV . Taking the colimit over all V , the domain becomes 1 E, and the codomain becomes identified with 1 (E ^ E) once we choose an isom- etry R1 R1 ! R1 . This describes the natural transformation . 16 JOHN R. KLEIN Since E 7! E^2 is a homogeneous quadratic functor, it coincides with the second stage of its Goodwillie tower and its first stage is trivial. So the map of towers in degrees 2 may be displayed as D2(E) I_____//E ^ EIII III IIIIIIII III IIIIII II$$ III S2(E) _____//E ^ E | | | | fflffl| fflffl| E _________//* . We assert that the induced natural transformation D2(E) ! E ^ E of second layers is in fact homotopic to the transfer.2 Here is a sketch proof: the natural transformation D2(E) ! E ^ E is determined by what it does on the category of sets of cardinality 2 (by identifying a based finite set T with its suspension spectrum). This observation is due to McCarthy [Mc , Th. 4.6], and independently to Arone, Dwyer and Rezk (unpublished). I first learned of the result from Dwyer. In particular, it is enough to check that the map is the transfer when E is a suspension spectrum. But for suspension spectra 1 X, it is not difficult to exhibit a natural weak equivalence S2( 1 X) ' ( 1Z2(X ^ X))Z2 , where the right side is the Z2-fixed points of the equivariant stable homotopy of X ^ X. With respect to this identification, the map D2( 1 X) ! S2( 1 X) corresponds to the norm map. But the norm map followed by the forgetful map ( 1Z2(X ^ X))Z2 ! ( 1 (X ^ X)) is the transfer. This completes the sketch proof of the assertion. Finally, if we take horizontal homotopy cofibers of the maps from stage two to stage one, we obtain a homotopy commutative diagram E ___________//_* ffiE|| || fflffl| |fflffl D2(E) __tr_// E ^ E which shows that ( tr) O ffiE is null homotopic. We now prove the second part of Theorem D. Let f :E ! E0 be a map which is a 2-local weak equivalence. Assume that E and E0 satisfy ____________ 2Randy McCarthy has informed me that the proof of this is implicit in the re* *cent Ph.D. thesis of K. Baxter Bauer [Ba ]. MODULI OF SUSPENSION SPECTRA 17 the connectivity hypotheses of Theorem B. Then the naturality of the obstruction shows that the diagram ffiE E --- ! D2(E) ? ? f?y ?y D2(f) E0 --- ! D2(E0) ffiE0 is homotopy commutative. If we invert 2, then by the first part, the obstructions ffiE and ffi0Evanish. It we localize at 2, then f and D2(f) become weak equivalences and the ffiE and ffiE0 coincide. Thus, inte- grally, ffiE = 0 if and only ffi0E= 0. 8. Illustrations of Theorem B We give two examples. Two cells. For p > 1 and q 3p- 3, consider the suspension spectrum E = 1 (Sp [f eq+ 1) where f :Sq ! Sp is some map. According to Theorem B, we have E ~= [E, D2(E)] ~= ßq+ 1(D2(Sp)) , where the second of these isomorphisms comes from that fact that a homotopy class E ! D2(E) maps the top cell of E into D2(Sp) D2(E). The first non-trivial group occurs when q = 2p- 1. The group ß2p(D2(Sp)) is isomorphic to Z if p is even and Z2 if p is odd. The distinct elements of E are represented by the suspension spectra of Sp [g e2p, with g = f + k[', '] , where k is an integer (mod 2 if p is odd). The group structure on E is given by adding these integers. In fact, at the prime 2, Mahowald has computed ßq+ 1(D2(Sp)) for q min (3p - 3, 2p + 29)(see [Mah , table 4.1], see also Milgram [Mi , table 13.5]). For example, assuming in addition p 1 mod 16, the first few groups are ____________________________________________|||||||| |______j________|0___|_1__|2___|3__|4_|5___|_|p||||||| |ß2p+_j(D2(S_))_|Z2__|Z2__|Z8__|Z2_|0_|Z2__|_. 18 JOHN R. KLEIN Tori. Let E = 1 (Sp x Sp), with p > 1. Then E ~= [E, D2(E)] ~= ß2p(D2(Sp _ Sp)) . When p is even, the group ß2p(D2(Sp_Sp)) is isomorphic to Z 3. When p is odd, it is isomorphic to Z2 Z2 Z. The elements of E can be represented by the suspension spectra of the complexes (Sp _ Sp) [g e2p with attaching map g = k[', '] + `[', '] + m! , where ! is the attaching map for the top cell of Sp x Sp, and k, ` and m are integers (take k and ` modulo 2 if p is odd). Hence the elements of E are specified by triple (k, `, m). The identity element of E is (0, 0, 1) and addition is given by (k, `, m) + (k0, `0, m0) = (k + k0, ` + `0, m + m0- 1) . 9. Loose Ends Relation to the James-Hopf invariant. Another approach to desus- pension questions is to inductively desuspend cell-by-cell (using a suit- able cell decomposition for the spectrum E). We outline here how this relates to our approach. The idea is this: when E can be written as a homotopy cofiber of a map of suspension spectra f : 1 A ! 1 B, it turns out that ffiE is closely related to the James-Hopf invariant H2(f) 2 [ 1 A, D2( 1 B)] . There is a homomorphism _ :[ 1 A, D2( 1 B)] ! [E, D2(E)] given by suspending, precomposing with the connecting map E ! 1 A and postcomposing with the inclusion D2( 1 B) ! D2(E). We assert that _ maps H2(f) to ffiE (we defer the proof to another paper). If we also assume that B is r-connected (r 1), dim B 2r+ 1, A is 2r-connected and dim A 3r+ 1, then we infer that E is r-connected and dim E 3r+ 2. Furthermore, obstruction theory implies that _ is an isomorphism. We conclude that E desuspends if and only if H2(f) = 0 (the `if' part of this statement is well known). MODULI OF SUSPENSION SPECTRA 19 Musings on the spectral sequence. It would be desirable to have a version of the spectral sequence in Theorem C which converges to ß*(ME ) in degree zero. The spectral sequence still lacks a geometric interpretation. Con- jecturally, the spectral sequence should be a packaging machine for the obstructions to equipping a spectrum with the structure of an "E1 -coalgebra over the sphere spectrum." What I have in mind here comes from an observation that S2(E) is a model for (E ^ E)Z2 = the categorical fixed points of Z2 acting on E ^E, and a choice of homotopy section to the map S2(E) ! E can be thought of as commutative (but not necessarily associative) "diagonal" for E. It is tempting to conjecture that the spectral sequence arises from a tower whose k-stage encodes, in some sense, the moduli space of spec- tra equipped with choice of commutative diagonal that is coherently homotopy associative up to order k - 1. References [A-K] Ahearn, S. T., Kuhn, N. J.: Product and other fine structure in polynomi* *al resolutions of mapping spaces. Algebr. Geom. Topol. 2, 1123-1150 (2002) [Ar] Arone, G.: A generalization of the Snaith-type filtration. Trans. Amer. Soc. 351, 591-647 (1999) [B-Hi] Berstein, I., Hilton, P. J.: On suspensions and comultiplications. Topol* *ogy 2, 73-82 (1963) [Ba] Bauer, K. B.: On Hopf algebra type and rational calculus decompositions. Ph.D. thesis, University of Illinois Urbana-Champaign 2001 [Go1] Goodwillie, T. G.: Calculus I, the derivative of a homotopy functor. K- theory 4, 1-27 (1990) [Go2] Goodwillie, T. G.: Calculus II: analytic functors. K-theory 5, 295-332 (1992) [Go3] Goodwillie, T. G.: Calculus III: the Taylor series of a functor. unpubli* *shed [Go4] Goodwillie, T. G.: The differential calculus of homotopy functors. In: P* *ro- ceedings of the International Congress of Mathematicians, (Kyoto, 1990), pp. 621-630. Math. Soc. Japan 1991 [Gr] Gray, B.: Desuspension at an odd prime. In: Algebraic topology, Aarhus 1982, Springer LNM 1051, pp. 360-370. Springer 1984 [Jo] Johnson, B.: The derivatives of homotopy theory. Trans. Amer. Math. Soc. 347, 1295-1321 (1995) [Kl] Klein, J. R.: Poincar'e embeddings and fiberwise homotopy theory. Topol- ogy 38, 597-620 (1999) [Ku] Kuhn, N. J.: Suspension spectra and homology equivalences. Trans. Amer. Math. Soc. 283, 303-313 (1984). [L-M-S]Lewis, L. G., May, J. P., Steinberger, M., McClure, J. E.: Equivariant Stable Homotopy Theory. (LNM, Vol. 1213). Springer 1986 20 JOHN R. KLEIN [Mah] Mahowald, M.: The metastable homotopy of Sn. (Memoirs of the Amer. Math. Soc, No. 72). Amer. Math. Soc. 1967 [Mc] McCarthy, R.: On n-excisive functors of module categories. to appear in Math. Proc. Cambridge Philos. Soc. [Mi] Milgram, R. J.: Unstable Homotopy from the Stable Point of View. (LNM, Vol. 368). Springer 1974 [Qu] Quillen, D.: Homotopical Algebra. (LNM, Vol. 43). Springer 1967 [Sc] Schwede, S.: Spectra in model categories and applications to the algebra* *ic cotangent complex. J. Pure Appl. Algebra 120, 77-104 (1997) [Wa] Waldhausen, F.: Algebraic K-theory of spaces. In: Algebraic and geomet- ric topology, Proceedings Rutgers 1983, LNM 1126, pp. 318-419. Springer 1985 Wayne State University, Detroit, MI 48202 E-mail address: klein@math.wayne.edu