THE MOD 2 HOMOLOGY OF INFINITE LOOPSPACES

arXiv:1109.3694v2 [math.AT] 2 Nov 2011

THE MOD 2 HOMOLOGY OF INFINITE LOOPSPACES NICHOLAS J. KUHN AND JASON B. MCCARTY Abstract. Applying mod 2 homology to the Goodwillie tower of the functor sending a spectrum X to the suspension spectrum of its 0th space, leads to a spectral sequence for computing H∗ (Ω∞ X; Z/2), which converges strongly when X is 0–connected. The E 1 term is the homology of the extended powers of X, and thus is a well known functor of H∗ (X; Z/2), including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra A, and a left module over the Dyer–Lashof operations. This paper is an investigation of how this structure is transformed through the spectral sequence. We use an operad structure on the tower, and the Tate construction, to determine differentials in this spectral sequence that hold for any spectrum X. These universal differentials then lead us to construct, given an A–module M , an algebraic spectral sequence depending functorially on M . The algebraic spectral sequence for H∗ (X; Z/2) agrees with the topological spectral sequence for X for many spectra, including suspension spectra and almost all generalized Eilenberg–MacLane spectra, and seems to give a sort of upper bound in general. The E ∞ term of the algebraic spectral sequence has form and structure similar to E 1 , but now the right A–module structure is unstable. Our explicit formula involves the derived functors of destabilization as studied in the 1980’s by W. Singer, J. Lannes and S. Zarati, and P. Goerss.

1. Introduction and main results An infinite loopspace is a space of the form Ω∞ X, the 0th space of a fibrant (a.k.a. omega) spectrum X. We let all homology be with mod 2 coefficients, and consider the following basic problem. Problem 1.1. How can one compute H∗ (Ω∞ X) from knowledge of H∗ (X)? The graded vector space H∗ (X) has a minimum of extra structure: it is an object in M, the category of locally finite right modules over the mod 2 Steenrod algebra A. By contrast, the structure of H∗ (Ω∞ X) is much, much richer: it is an object in the category HQU of restricted Hopf algebras in the abelian category of left modules over the Dyer–Lashof algebra with compatible unstable right A–module structure. Date: November 2, 2011. 2000 Mathematics Subject Classification. Primary 55P47; Secondary 55S10, 55S12, 55T99. This research was partially supported by National Science Foundation grant 0967649. 1

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An ideal solution to our problem would be to describe a functor from M to HQU whose value on H∗ (X) would be H∗ (Ω∞ X). It is not a surprise that such a functor doesn’t exist, and Example 1.17 illustrates this. However, one punchline of this paper is that one can come surprisingly close. r (X)} associated to the unreIn brief, we study the spectral sequence {E∗,∗ duced Goodwillie tower of the functor X Σ∞ Ω∞ X. This converges ∞ strongly to H∗ (Ω X) when X is 0–connected. We first determine universal differentials. After identifying d1 , we learn about deeper differentials by using the Z/2 Tate construction to reveal how Dyer–Lashof operations are reflected in the spectral sequence associated to a tower acted on by the operad C∞ . For the Goodwillie tower under consideration, such an action was constructed in [AK]. Guided by our formula for universal differentials, we construct, for M ∈ M, an algebraic spectral sequence depending functorially on M . The algebraic spectral sequence for H∗ (X; Z/2) agrees with the topological spectral sequence for X for many spectra X, including suspension spectra and almost all Eilenberg–MacLane spectra, and seems to give a sort of upper bound in alg,∞ general. Our algebraic functor E∗,∗ (M ) takes values in the category HQU , and is built out of the derived functors of ‘destabilization’ which were the subject of much research in the 1980’s by W. Singer [Si], J. Lannes and S. Zarati [LZ1], and P. Goerss [Goe]. We now introduce our cast of characters, then describe our results in more detail. ∞ 1.1. The tower for Σ∞ + Ω X, and the spectral sequence. We let T denote the category of based topological spaces and S the category of S– modules as in [EKMM]. The suspension spectrum functor Σ∞ : T → S and the 0th space functor Ω∞ : S → T induce an adjoint pair on homotopy categories. We use the notation Σ∞ + Z for the suspension spectrum of Z+ , the union of the space Z with a disjoint basepoint. T. Goodwillie’s general theory of the calculus of functors [Goo], applied ∞ to the endofunctor of S sending X to Σ∞ + Ω X, yields a natural tower P (X) of fibrations .. .

P3 (X)

s9 ss s s ss ss e3 sss s P (X) s 2 4 s iiii ss ss ie2iiiiii s s ss iiii ss iiii e1 ∞ / P1 (X). Σ∞ +Ω X

Basic properties include the following.

MOD 2 HOMOLOGY OF INFINITE LOOPSPACES

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• P1 (X) identifies with X × S so that e1 corresponds to the evaluation Σ∞ Ω∞ X → X. • The fiber of the map Pd (X) → Pd−1 (X) is naturally equivalent to the spectrum Dd X = (X ∧d )hΣd , the dth extended power of X. • If X is 0–connected, then ej is j–connected. Applying mod 2 homology to the tower P (X) yields a left half plane r (X)} ⇒ H (Ω∞ X) which converges strongly when spectral sequence {E∗,∗ ∗ X is 0–connected. The Steenrod algebra A acts on the right of the columns of the spectral sequence, and the −d line of E 1 equals Σd H∗ (Dd X). Using G. Arone’s explicit model for this tower [Ar], further properties were explored in [AK]. The spectral sequence is a spectral sequence of Hopf algebras. The product and coproduct on E ∞ are induced by the H-space product and diagonal on Ω∞ X, while the product and coproduct on E 1 are induced by the multiplication maps Db X ∧ Dc X → Db+c X and the transfer maps Db+c X → Db X∧Dc X associated to the subgroup inclusions Σb ×Σc ֒→ Σb+c . Besides all this structure, the action of the little cubes operad C∞ on the infinite loop space Ω∞ X induces a corresponding action on the tower. This 1 (X), leads to Dyer–Lashof operations acting on both H∗ (Ω∞ X) and on E∗,∗ where they take the form Qi : Hj (Dd X) → Hj+i (D2d X), and are defined ∞ (X), and for all i ∈ Z. How these operations correspond on the level of E∗,∗ how they act on the rest of the spectral sequence is part of the content of Theorem 1.3 below. Finally we note that there is also a reduced tower P˜ (X) satisfying P˜d (X)× ˜ ∗ (Ω∞ X) ˜ r (X)} ⇒ H S = Pd (X), yielding a reduced spectral sequence {E ∗,∗ which only differs from the unreduced spectral sequence in that is it missing the unit in bigrading (0, 0). 1.2. Lots of categories and a description of E 1 . We introduce various algebraic categories. • M is the category of locally finite right A–modules. The Steenrod squares go down in degree: given x ∈ M ∈ M, |xSq i | = |x| − i. • U is the full subcategory of M consisting of modules satisfying the unstable condition: xSq i = 0 whenever 2i > |x|. • Q is the category of graded vector spaces M acted on by Dyer–Lashof operations Qi : Md → Md+i , for i ∈ Z, satisfying the Adem relations and the unstable relation: Qi x = 0 whenever i < |x|. • QM is the full subcategory of M ∩ Q consisting of objects whose Dyer–Lashof structure is intertwined with the Steenrod structure via the Nishida relations. • QU = QM ∩ U . All these categories admit tensor products, via the Cartan formula for both Steenrod and Dyer–Lashof operations. Then we define various categories of Hopf algebras.

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• HM is the category of bicommutative Hopf algebras in M. • HQM is the category of bicommutative Hopf algebras in QM satisfying the Dyer–Lashof restriction axiom: Q|x| x = x2 . • HQU = HQM ∩ U . We also need two ‘free’ functors. • R∗ : M L → QM is left adjoint to the forgetful functor. Explicitly, R∗ M = ∞ s=0 Rs M where Rs : M → M is given by Rs M = hQI x | l(I) = s, x ∈ M i/(unstable and Adem relations).

Here, if I = (i1 , . . . , is ), QI x = Qi1 · · · Qis x, and l(I) = s. • UQ : QM → HQM is left adjoint to the functor taking an object H ∈ HQM to its module P H of primitives. Explicitly, UQ (M ) = S ∗ (M )/(Q|x| x = x2 ), with M ⊂ UQ (M ) primitive. We then begin our study of the spectral sequence knowing the following. • H∗ (Ω∞ X) is an object in HQU . 1 (X) = U (R (H (X))) as an object in HQM. Here, if x ∈ • E∗,∗ Q ∗ ∗ H∗ (X) and I = (i1 , . . . , is ), QI x has bidegree (−2s , 2s + |x| + |I|), where |I| = i1 + · · · + is . Steenrod operations act vertically, while Dyer–Lashof operations double the horizontal degree. 1 (X) = H∗ (Dd X) are easily identi• The individual columns E−d,d+∗ fied, since Rs (H∗ (X)) ⊆ H∗ (D2s X). r (X) is an object in HM, and each dr is A–linear and both • Each E∗,∗ a derivation and coderivation. For references and a bit more detail about the E 1 term, see §2.3. Remark 1.2. The careful reader may note that H∗ (Ω∞ X) ∈ HQU satisfies one more condition than has been described: the dual of the classic restriction axiom for unstable A–algebras, Sq |x| x = x2 . This property likely is not preserved by the filtration on H∗ (Ω∞ X). The good news is then that this extra structure is available to be used to help determine extension problems. 1.3. Universal differentials. Our first theorem identifies universal structure on the spectral sequence. r (X)}. Theorem 1.3. For all spectra X, the following hold in {E∗,∗ X (a) For all x ∈ H∗ (X), d1 (x) = Qi−1 (xSq i ). i≥0

Er,

(b) If y ∈ H∗ (Dd X) lives to and dr (y) is represented by z ∈ H∗ (Dd+r X), i then Q y ∈ H∗ (D2d X) lives to E 2r , and d2r (Qi y) is represented by Qi (z) ∈ H∗ (D2d+2r X). ∞ (X), then Qi y ∈ (c) If y ∈ H∗ (Dd X) represents z ∈ H∗ (Ω∞ X) in E−d,∗ ∞ (X). H∗ (D2d X) represents Qi z ∈ H∗ (Ω∞ X) in E−2d,∗


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A consequence of the first two parts of the theorem is the following identification of universal differentials. Corollary 1.4. For all spectra X, x ∈ H∗ (X), and I of length s, QI x lives 2s to E−2 s ,∗ (X) and X s 2s QI Qi−1 (xSq i ) ∈ E−2 d2 (QI x) = s+1 ,∗ (X). i≥0

To further give context to what Theorem 1.3 says about how Dyer–Lashof operations work in the spectral sequence, let 0 = B1 ⊆ B2 ⊆ · · · ⊆ Br ⊆ · · · ⊆ Z r ⊆ · · · ⊆ Z 2 ⊆ Z 1 = E1 be cycles and boundaries as usual, so that E r = Z r /B r . Then Theorem 1.3(b) implies that for all r, Dyer–Lashof operations on E 1 restrict to maps Qi : Z r → Z 2r and Qi : B r → B 2r−1 . As Z 2r /B 2r−1 both includes into E 2r−1 and projects onto E 2r , one gets Dyer–Lashof operations of two flavors: Qi : E r → E 2r−1 and Qi : E r → E 2r . This discussion holds when r = ∞, leading to the next corollary. ∞ (X) ∈ HQM, with structure induced Corollary 1.5. For all spectra X, E∗,∗ 1 from E . To the extent that the spectral sequence converges, this structure is also induced from H∗ (Ω∞ X).

Remark 1.6. Since H∗ (Ω∞ X) is always an unstable A–module, it follows ∞ (X) ∈ HQU . We believe this is the case that if X is 0–connected, then E∗,∗ for all spectra X. Theorem 1.3 will be proved in §3, supported by the results in the preceding background section. We briefly comment on the proof. Statement (a) amounts to a calculation of δ∗ , where δ : X → ΣD2 X is the connecting map of the cofibration sequence D2 X → P˜2 X → P˜1 X ≃ X. When |x| > 0, this was calculated (in dual formulation) by the first author in [K3] by means of universal example, and it is not too hard to extend this to all x. We give proofs of statements (b) and (c) that show that versions of these statements will hold in the spectral sequence associated to any tower of spectra admitting an action of the operad C∞ . The key idea is to use the (once desuspended) Z/2 Tate construction in place of homotopy orbits. For example, where naively one might hope for maps (Pd (X) ∧ Pd (X))hZ/2 → P2d (X) which do not exist, vanishing results show that one does have maps tZ/2 (Pd (X) ∧ Pd (X)) → P2d (X)

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which do the job. As tZ/2 (X ∧X) ≃ holim Σn D2 (Σ−n X), the colinearization n

of D2 (X) in McCarthy’s sense [McC], our proof channels this idea as well. A technical point is that, at an appropriate moment, we need to pass from towers of S–modules to towers of HZ/2–modules. 1.4. An algebraic spectral sequence. We now build an algebraic spectral sequence using only the differentials given by the formula in Corollary 1.4. Our discovery is that this spectral sequence can be completely described, with an interesting E ∞ term. We need yet more terminology and notation related to the category U . • Let Ω∞ : M → U be right adjoint to the inclusion. Explicitly, Ω∞ M is the largest unstable submodule of M . • Let Ω : U → U be right adjoint to the suspension Σ : U → U . Explicitly, ΩM is the largest unstable submodule of Σ−1 M . • The functor Ω∞ is left exact, and we let Ω∞ s : M → U denote the associated right derived functors. 1−s M . We observe that these funcIt is convenient to let Ls M = ΩΩ∞ s Σ tors have extra structure. Proposition 1.7. There are natural operations Qi : Ls M → Ls+1 M giving L∗ M the structure of an object in QM. Our second theorem then goes as follows. Theorem 1.8. For all M ∈ M, there is a left half plane spectral sequence alg,r {E∗,∗ (M )} described by the following properties. (a) The spectral sequence is a functor of M taking values in HM, with each dr both a derivation and coderivation. alg,1 (b) E∗,∗ (M ) = UQ (R∗ M ) as an object in HQM. s

(c) dr is not zero only when r = 2s , and d2 is determined by the formulae alg,2s in Corollary 1.4: for x ∈ M and I of length s, QI x lives to E−2 s ,∗ (M ), and X s d2 (QI x) = QI Qi−1 (xSq i ). i≥0

alg,r (d) For all r, E∗,∗ (M ) is primitively generated with nonzero primitives alg,r concentrated in the −2s lines. For all r > 2s , P E−2 s ,2s +∗ (M ) ≃ Ls M . alg,∞ (e) E∗,∗ (M ) ≃ UQ (L∗ M ) as an object in HQU.

Combining Corollary 1.4 and Theorem 1.8 yields the next corollary.


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Corollary 1.9. For a spectrum X, the following are equivalent. r (X)} and {E alg,r (H (X))} agree. (a) The spectral sequences {E∗,∗ ∗,∗ ∗ c+1 ∞ (X) for all c. (X) = E−c,∗ (b) E−c,∗

(c) For all s ≥ 0 and t ≥ 2, the differential s (2t −1)

d2

2s (2t −1)

: E−2s ,∗

2s (2t −1)

(X) → E−2s+t ,∗+2s (2t −1)−1 (X)

is zero on Ls H∗ (X). 1 Note that property (b) fails to hold only if there exists x ∈ E−c,∗ (X), and r r r > c such that x lives to E , and d x 6= 0. We call such a differential a rogue differential. The equivalence of (b) and (c) is then a reflection of the fact that the smallest c for which a rogue differential exists will necessarily have the form c = 2s , with a corresponding smallest r of the form r = 2s (2t − 1) with t ≥ 2, and dr will then be nonzero on the primitives. See Remark 5.8 for a bit more discussion. 2 Theorem 1.3(a) implies that E−1,1+∗ (X) = Ω∞ H∗ (X), and it follows that, when X is 0–connected, rogue differentials off of the −1 line measure the failure of the evaluation map ǫ∗ : H∗ (Ω∞ X) → Ω∞ H∗ (X) to be onto. Theorem 1.3(b) and Corollary 1.9 then tell us that in some circumstances this can be the only source of rogue differentials.

Corollary 1.10. Let X be 0–connected. Suppose that Ω∞ H∗ (X) = L0 H∗ (X) generates L∗ H∗ (X) as a module over the Dyer–Lashof algebra. Then ǫ

∗ r alg,r {E∗,∗ (X)} = {E∗,∗ (H∗ (X))} ⇐⇒ H∗ (Ω∞ X) −→ Ω∞ H∗ (X) is onto.

We note that in every example we have calculated so far, L∗ M is generated by L0 M as a module over the Dyer–Lashof algebra. Remark 1.11. It is tempting to hope that Corollary 1.4 and Theorem 1.8 r (X) is a subcan be combined to tell us that, for all spectra X and all r, E∗,∗ alg,r quotient of E∗,∗ (H∗ (X)). Our results do say that the algebraic boundaries, alg,r B , are contained in the topological boundaries B r . To conclude that E r is a subquotient of E alg,r , it would suffice to show that Z r ⊆ B r + Z alg,r . Related to this, in ongoing work, the second author has determined that r E∗,∗ (X) is always primitively generated, with primitives which are subquo1 (X), viewed as A–modules. It follows that, tients of the primitives in E∗,∗ very generally, the only possible nonzero differentials are dr for r of the form 2s (2t − 1). The development of our algebraic spectral sequence, and the proof of its properties as in Theorem 1.8, is the topic of §5. This relies heavily on §4, which is focused on the connection between Rs and Ω∞ s . We say a bit about this connection here.

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We relabel: let Rs = ΣRs Σs−1 : M → M. Let ds : Rs M → Rs+1 M be given by the formula X ds (QI x) = QI Qi−1 (xSq i ), i≥0

where we have repressed some suspensions. The next theorem is a variant of theorems in [Goe] and [P]. All such results go back to work of Singer [Si] for inspiration. Theorem 1.12. For all M ∈ M, d

d

d

0 1 2 R0 M −→ R1 M −→ R2 M −→ ···

is a chain complex with Hs (R∗ M ; d) naturally isomorphic to Ω∞ s M. 1−s M ). Theorem 1.12 says that Remark 1.13. Recall that Ls M = ΩΩ∞ s (Σ ds−1

d

s Rs−1 (Σ−1 M ) −−−→ Rs (M ) −→ Rs+1 (ΣM )

1−s M ) at R (M ). This may make it plausible has homology Σ−1 Ω∞ s s (Σ (though by no means obvious) that there might be an algebraic spectral sequence with E 1 = UQ (R∗ M ) and E ∞ = UQ (L∗ M ).

In §4, we will give a complete presentation of Theorem 1.12, which is much more topologically based and less explicitly computational than similar results in the literature. Also included in this section is a proof of Proposition 1.7. 1.5. Examples. We give some examples showing the strength of our main results and their limitations. More details will be given in §6. Example 1.14. Let A be a Z–graded abelian group, and let HA be the generalized Eilenberg–MacLane spectrum with πn (HA) = An for all n. The evaluation map H∗ (Ω∞ HA) → Ω∞ H∗ (HA) is onto in all cases except when A0 or A−1 has 2–torsion of order at least 4. Ls H∗ (HA) = 0 for all s > 0 in all cases except when A−1 has 2–torsion of order at least 4, or when ha ∈ A0 | 2a = 0i → A0 ⊗ Z/2 is not an isomorphism. If A0 is finitely generated, this last exception means that A0 has a direct summand of the form Z or Z/2r with r ≥ 2. When both properties hold we learn that ∞ alg,∞ E∗,∗ (HA) = E∗,∗ (H∗ (HA)) = Λ∗ (Ω∞ H∗ (HA)).

This agrees with classical calculation, e.g. H ∗ (K(Z/2, n)) = U (F (n)), the free unstable algebra generated by an n–dimensional class. As cohomology is represented by mod 2 Eilenberg–MacLane spectra, this shows that the differentials described in Corollary 1.4 can be regarded as all the universal differentials. The three key exceptional cases are the following. When HA = HZ, there r are no rogue differentials, and when HA = HZ/2r with r ≥ 2, d2 −1 is the only rogue differential. In both cases, E ∞ correctly computes the homology


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of the discrete group. The case when HA = Σ−1 HZ/2r with r ≥ 2 is most s r complicated: there is an infinite family of rogue differentials, d2 (2 −1) for s ≥ 0, leaving a trivial E ∞ term. Example 1.15. Let X = Σ∞ Z, a suspension spectrum, so that H∗ (X) is unstable. The tower is known to split — e.g., when Z is connected, _ Σ∞ Ω∞ Σ∞ Z ≃ Σ∞ Dd Z d

— and so the spectral sequence collapses at E 1 . Thus ∞ (X) ≃ UQ (R∗ (H∗ (X))). E∗,∗

As we clearly have no rogue differentials, our works says that ∞ (X) ≃ UQ (L∗ (H∗ (X)). E∗,∗

This is in agreement with the main theorem of [LZ1], which says (in 1−s M ), so that R M ≃ dual form) that, if M ∈ U , then ΣRs M ≃ Ω∞ s s (Σ 1−s M ) = L M . (In our context, see Theorem 4.34.) Indeed, it was ΩΩ∞ s s (Σ interpreting the work of Lannes and Zarati this way that led us to the correct formulation of Theorem 1.8. Example 1.16. Partially published work of Lannes and Zarati [LZ2] from the 1980’s suggests that one might often be able to ‘mix and match’ the last two examples. We offer one such example. Let S 1 h1i be the cofiber of S → HZ. By dimension shifting, one can easily compute that, for all s ≥ 0, Ls H∗ (S 1 h1i) = Ls+1 H∗ (S 1 ) = Rs+1 H∗ (S 1 ). There is a well known stable map ΣRP ∞ = D2 S 1 → S 1 h1i whose image in homology is exactly Ω∞ H∗ (S 1 h1i), and one easily concludes that H∗ (Ω∞ S 1 h1i) → Ω∞ H∗ (S 1 h1i) is onto. ∞ (S 1 h1i) = U (R 1 Corollary 1.10 now applies to say that E∗,∗ Q ∗>0 H∗ (S )). ∞ 1 This is in agreement with known calculation: Ω S h1i is the fiber of the split fibration Ω∞ Σ∞ S 1 → S 1 , and one can deduce that, localized at 2, _ D2d S 1 . Σ∞ Ω∞ S 1 h1i ≃ d

Example 1.17. Here is perhaps the simplest example of a rogue differential occurring in the spectral sequence of a 0–connected spectrum X. Let X be the cofiber of 4 : RP 4 → RP 4 . It is easy to see that H∗ (X) ≃ H∗ (RP 4 ∨ΣRP 4 ) as right A–modules, and so is unstable. Then the algebraic spectral sequence collapses, and the hypothesis of Corollary 1.10 holds. We conclude that the topological spectral sequence collapses at E 1 if and ǫ∗ only if H∗ (Ω∞ X) −→ Ω∞ H∗ (X) is onto. As will be explained in more detail

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in §6.5, results about Brown–Gitler spectra apply to say that ǫ∗ cannot be onto. There is only one possibility for a first rogue differential: d3 b = a4 , where 1 1 (X) are the bottom and top (X) and b ∈ H5 (X) = E−1,6 a ∈ H1 (X) = E−1,2 1 nonzero classes in E−1,∗ . This example illustrates that even the additive structure of H∗ (Ω∞ X) can not be determined by just knowing the A–module H∗ (X). It also illustrates that if X is a spectrum built as the cofiber of a map of Adams filtration s between spectra with no rogue differentials, one expects s d2 −1 to be the first rogue differential for X. The intriguing comparison between our algebraic E ∞ term and formulae in [HM], as well as results in [LZ2], and a number of our examples, suggest the following slightly vague query. alg,∞ Question 1.18. Does our E∗,∗ (H∗ (X)) arise via an unstable Adams spectral sequence converging to H∗ (Ω∞ X)?

2. Preliminaries 2.1. Prerequisites on spectra. T will be the category of pointed topological spaces, and S the category of S–modules as in [EKMM]. An S–module X is a spectrum of the classic sort (as in [LMMS]) equipped with extra structure, and we let Xn denote its nth space. Thus Ω∞ X = X0 . By a weak natural transformation F → G between two functors with values in a model category, we mean a zig-zig of natural transformations F ← H → G (or F → H ← G) for which the backwards arrow is a weak equivalence (on any object). We say that a diagram of such weak natural transformations commutes if it induces a commutative diagram in the homotopy category (on each object). Though we will try to not dwell too deeply on the details of the model, studied in [AK], for our Goodwillie tower, the following proposition summarizes the formal properties of S–modules that are needed to make the arguments in [AK] work. Proposition 2.1. The category S of S–modules has the following structure. • S is a topological category tensored and cotensored over T : given K ∈ T and X ∈ S, there are spectra K ∧ X and MapS (K, X), natural in both variables, satisfying standard adjunction properties. • There are natural maps η : MapS (K, X) → MapS (L ∧ K, L ∧ X). • There are natural maps MapS (K, X) ∧ MapS (L, Y ) → MapS (K ∧ L, X ∧ Y ), which are weak equivalences if K and L are finite CW complexes. • The suspension spectrum functor Σ∞ : T → S commutes with smash product. • There are natural maps e : Σ∞ MapT (K, Z) → MapS (K, Σ∞ Z).


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• There is a weak natural equivalence hocolim Σ−n Σ∞ Xn → X. n

Σ−n X

Here and elsewhere we write for MapS (S n , X). It is only the last item that really needs comment. See Appendix A for more discussion of this point. We end this subsection by describing the setting for the ‘evaluation/diagonal’ natural transformations ǫ : ΣF (X) → F (ΣX) which play a significant role in our work. As S is a topological category, HomS (X, Y ) has the structure of a based topological space. A functor F : S → S is said to be continuous if F : HomS (X, Y ) → HomS (F (X), F (Y )) is a continuous function. If F is also reduced, i.e. F (∗) = ∗, then this continuous function is also based. Definition 2.2. Given a continuous reduced functor F : S → S, and K ∈ T , we let ǫ : K ∧ F (X) → F (K ∧ X) be adjoint to the composite of continuous functions F

K → HomS (X, K ∧ X) − → HomS (F (X), F (K ∧ X)), where the first map is the unit of the adjunction HomS (K ∧ X, Y ) ≃ HomT (K, HomS (X, Y )). 2.2. The Tate construction. If G is a finite group, we let G–S denote the category of S–modules with a G–action: the category of ‘naive’ G–spectra. More generally, if R is a commutative S–algebra, we let G–R–mod be the category of R–modules with G–action. (For us, R will eventually be HZ/2.) Given Y ∈ G–R–mod, we let YhG and Y hG respectively denote associated homotopy orbit and homotopy fixed point R–modules. The homotopy orbit construction satisfies a change-of-rings lemma. Lemma 2.3. Given Y ∈ G–S and a commutative S–algebra R, there is a natural isomorphism of R–modules, R ∧ YhG = (R ∧ Y )hG . There are various constructions in the literature, e.g. [ACD, AK, GM], of a natural norm map NG (Y ) : YhG → Y hG . The Tate spectrum of Y is defined as the homotopy cofiber of NG (Y ). It will be more convenient for us to desuspend this once and define tG (Y ) to be the homotopy fiber of NG (Y ). Thus tG (Y ) comes equipped with a natural transformation tG (Y ) → YhG . The next lemma lists the properties we need about this.

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KUHN AND MCCARTY

Lemma 2.4. (a) tG takes weak equivalences and cofibration sequences in G–R–mod to weak equivalences and cofibration sequences in R–mod. (b) If X is a nonequivariant R–module, tG (G+ ∧ X) ≃ ∗. See [GM, Part I] for these sorts of facts. Statement (b) also follows from [AK, Prop.2.10]. When G = Z/2, there is a well known model for tZ/2 (Y ). Let ρ be the one dimensional real sign representation of Z/2, and let S nρ be the one point compactification of nρ. Lemma 2.5. (Compare with [GM, Thm.16.1].) For Y ∈ Z/2–R–mod, there is a natural weak equivalence tZ/2 (Y ) ≃ holim MapR-mod (S nρ , Y )hZ/2 . n

We now specialize to the special case Y = X ∧R X, with X an R–module. Notation 2.6. Let X be an R–module. We let D2R (X) = (X ∧R X)hZ/2 and D2R (X) = tZ/2 (X ∧R X). One easily checks the following. Lemma 2.7. (a) For all S–modules X and commutative S–algebras R, there is an isomorphism of R–modules, D2R (R ∧ X) = R ∧ D2 (X). (b) For all R–modules X, there is a natural weak equivalence D2R (X) ≃ holim Σn D2R (Σ−n X). n

Thus

D2R

is identified as the colinearization of D2R in the sense of [McC].

Corollary 2.8. D2R preserves cofibration sequences of R–modules. 2.3. The homology of extended powers. When X is a spectrum, a construction of the Dyer–Lashof operations Qi : Hj (Dd X) → Hj+i (D2d X), for all i ∈ Z, is given by M. Steinberger in [BMMS, Thm.III.1.1]. An alternative construction is given later in the same book by J. McClure [BMMS, Prop.VIII.3.3]. He [BMMS, Thm.IX.2.1] also computes H∗ (PX) as W∞an algebra with both Dyer–Lashof and Steenrod operations, where PX = d=0 Dd X. The coproduct structure on H∗ (PX) seems to be less well documented in the literature. Recall that the coproduct ∆ is induced by the transfer maps tb,c : Db+c X → Db X ∧ Dc X. The following lemma is presumably well known, and is analogous to [CLM, Thm.I.1.1(6)]. P ′ Lemma 2.9. For all y ∈ H∗ (PX), if ∆(y) = y ⊗ y ′′ , then X X ∆(Qk y) = Qi y ′ ⊗ Qj y ′′ . i+j=k


13

Sketch proof. Let p : X → X ∨ X be the pinch map. If b + c = d, then tb,c is the (b, c)th component of the composite _ Dd (p) Dd X −−−→ Dd (X ∨ X) = Db X ∧ Dc X. b+c=d

The diagram D2 Dd X

D2d X

D2 Dd (p)

/ D2 Dd (X ∨ X) / D2d (X ∨ X)

D2d (p)

commutes, andPthe lemma follows from this, using the Cartan formula Qk (y ′ ⊗ y ′′ ) = i+j=k Qi y ′ ⊗ Qj y ′′ . Crucial to us is the behavior of ǫ : ΣDd X → Dd ΣX on homology.

Lemma 2.10. (a) ǫ∗ : H∗ (PX) → H∗+1 (PΣX) sends ∗–decomposables to zero, and has image in the primitives. (b) ǫ∗ (Qi y) = Qi (ǫ∗ (y)). One reference for (a) is [AK, Ex.6.7]. For statement (b), see [BMMS, Lem.II.5.6] (or alternatively, deduce it from [BMMS, Prop.VIII.3.2]). Corollary 2.11. The image of ǫ∗ : H∗ (ΣD2s (Σ−1 X)) → H∗ (D2s X) is precisely the subspace of primitives: the span of the elements QI x with l(I) = s and x ∈ H∗ (X). 2.4. Dyer–Lashof operations for D2 . Lemma 2.12. (a) The sequence ǫ

ǫ

ǫ

∗ ∗ ∗ . . . −→ H∗−2 (D2 (Σ−2 X)) −→ H∗−1 (D2 (Σ−1 X)) −→ H∗ (D2 (X))

is Mittag–Leffler. HZ/2

(b) π∗ (D2

(HZ/2 ∧ X)) = lim H∗ (Σn D2 (Σ−n X)). n

Statement (a) follows from Lemma 2.10, and then (b) follows from (a), HZ/2 noting that π∗ (D2 (HZ/2 ∧ X)) = H∗ (D2 (X)). Corollary 2.13. The natural transformation Qi : H∗ (X) → H∗+i (D2 (X)) lifts to a natural transformation HZ/2

Qi : H∗ (X) → π∗+i (D2

(HZ/2 ∧ X)).

14

KUHN AND MCCARTY

2.5. The cohomology of D2 X. In the proof of Theorem 1.3(a), it will be useful to work with mod 2 cohomology. As in [K3], let ˆ 0 : H ∗ (X) → H 2∗ (D2 X) Q be the squaring operation, and then, for i > 0, let ˆ i : H ∗ (X) → H 2∗+i (D2 X) Q be defined to be the composite ˆ0 Q

ǫ∗

H ∗ (X) = H ∗+i (Σi X) −−→ H 2∗+2i (D2 (Σi X)) −→ H 2∗+i (D2 X). One also has a product ∗ : H ∗ (X) ⊗ H ∗ (X) → H ∗ (D2 X) induced by t1,1 : ˆ 0 (x + y) = Q0 x + Q0 y + x ∗ y, while, for i > 0, D2 X → X ∧ X. One has Q ˆ i is linear. Q ˆ i x and x ∗ y. Lemma 2.14. H ∗ (D2 X) is spanned by the elements Q These operations are appropriately dual to the homology Dyer–Lashof operations. In the next proposition, Qi x = Qi+|x| x, as is standard. Proposition 2.15. Let hx, yi denote the cohomology/homology pairing. Given w, x ∈ H ∗ (X) and y, ( z ∈ H∗ (X), the following formulae hold. ˆ i x, Qj yi = hx, yi if i = j (a) hQ 0 otherwise. ( ˆ i x, y ∗ zi = hx, yihx, zi if i = 0 (b) hQ 0 otherwise. ( hw, yihx, yi if i = 0 (c) hw ∗ x, Qi yi = 0 otherwise. (d) hw ∗ x, y ∗ zi = hw, yihx, zi + hw, zihx, yi. See [K3, Prop.A.1]. 3. Proof of Theorem 1.3 3.1. Proof of Theorem 1.3(a). It suffices to prove this formula assuming X is a spectrum whose homology is bounded below and of finite type. In this case, it is easiest to first prove the cohomology version of Theorem 1.3(a). ˆ i x and x ∗ y, Recall from §2.5 that H ∗ (D2 X) is spanned by elements Q ∗ with x, y ∈ H (X) and i ≥ 0. As in the introduction, let δ : X → ΣD2 X be the connecting map of the cofibration sequence D2 X → P˜2 X → X. ˆ r x) = Sq r+n+1 x. Proposition 3.1. For x ∈ H n (X), we have δ∗ (σ Q Proof. The proof uses ideas from [K3, Proposition 4.3] and [K2, Appendix A]. Let P (r, n) be the statement ˆ r x) = Sq r+n+1 x for all x ∈ H n (X). δ∗ (σ Q


15

We need to prove that P (r, n) is true for all r ≥ 0 and n ∈ Z. We first observe that, for r > 0, P (r − 1, n + 1) implies P (r, n). To see this, we use that the diagram X

δ

/ ΣD2 X ǫ

X

Σ−1 δ

/ D2 ΣX

commutes by the naturality of δ. So, if x ∈ H n (X) and P (r − 1, n + 1) holds, then ˆ r x) = δ∗ (ǫ∗ (Q ˆ r−1 σx)) δ∗ (σ Q ˆ r−1 σx) = (Σ−1 δ)∗ (Q = σ −1 Sq (r−1)+(n+1)+1 σx = Sq r+n+1 x. Thus it suffices to show P (0, n) for all n. By naturality, it is enough to show that ˆ 0 ιn ) = Sq n+1 ιn , δ∗ (σ Q where ιn ∈ H n (Σn HZ/2) is the fundamental class. We break this into cases. When n > 0, this was proven in [K3, Proposition 4.3] as follows. As Σn HZ/2 is 0–connected, the cohomology tower spectral sequence for Σn HZ/2 strongly converges to H ∗ (K(Z/2, n)). Thus the element Sq n+1 ιn must be an eventual boundary, as Sq n+1 ιn = 0 in H ∗ (K(Z/2, n)). For degree reasons, ˆ 0 ιn ) = Sq n+1 ιn . the only way this could happen is if δ∗ (σ Q ˆ 0 ιn is 2n + 1 < n, so δ∗ takes this element When n < −1, the degree of σ Q to zero. As desired, Sq n+1 ιn is also zero since n + 1 < 0. For the remaining cases, we use the fact that Sq 2 is injective on A in degrees 0 and 1. ˆ 0 ιn ) = δ∗ (σSq 2 Q ˆ 0 ιn ). The Nishida relations for the We have Sq 2 δ∗ (σ Q ˆ operation Q0 [K3, Proposition 3.15] tell us that ˆ 0 Sq 1 ιn + ιn ∗ Sq 2 ιn . ˆ 2 ιn + n − 1 Q ˆ 0 ιn = n Q Sq 2 Q 0 2 Since δ∗ takes nontrivial products to zero, we deduce that n ∗ ˆ 2 ∗ ˆ ˆ 0 Sq 1 ιn ). Sq δ (σ Q0 ιn ) = δ (σ Q2 ιn ) + δ∗ (σ Q 2

When n = 0, this equation and the established fact P (0, 1) imply that ˆ 0 ι0 ) = Sq 2 Sq 1 ι0 . Sq 2 δ∗ (σ Q ˆ 0 ι0 ) = Sq 1 ι0 . We deduce that δ∗ (σ Q

16

KUHN AND MCCARTY

When n = −1, the above equation and the established facts P (2, −1) (implied by P (0, 1)) and P (0, 0) imply that ˆ 0 ι−1 ) = Sq 2 ι−1 + Sq 1 Sq 1 ι−1 = Sq 2 ι−1 . Sq 2 δ∗ (σ Q ˆ 0 ι−1 ) = ι−1 . We deduce that δ∗ (σ Q

ˆ ix = Q ˆ i−|x| x, the proposition says that, for all x ∈ H ∗ (X), If we define Q ˆ i x) = Sq i+1 x. δ∗ (σ Q By duality, we get the formula stated in Theorem 1.3(a): for all x ∈ H∗ (X), X δ∗ (x) = σQi−1 (xSq i ), i≥0

Remark 3.2. Variants of the formula in Theorem 1.3(a) go back at least as far as the 1966 paper [BCKQRS]. 3.2. The strategy for the proof of statements (b) and (c). We outline the strategy of the proof of Theorem 1.3(b) and (c). First of all, what do we have to show? In (c), the statement ‘y ∈ H∗ (Dd X) represents z ∈ H∗ (Ω∞ X)’ means that, under the maps p

i

d d ∞ − Dd (X), Σ∞ → Pd (X) ← +Ω X −

we have pd∗ (z) = id∗ (y). So to prove (c), we just need to show that then, under the maps p

i

2d 2d ∞ − → P2d (X) ←− − D2d (X), Σ∞ +Ω X −

we have p2d∗ (Qi z) = i2d∗ (Qi y). In (b), the statement ‘y ∈ H∗ (Dd X) lives to E r , and dr (y) is represented by z ∈ H∗ (Dd+r X)’ means that there is an element w ∈ H∗ (Pd+r−1 (X)), such that under the maps i

pd+r−1,d

δd+r−1

d Dd (X) − → Pd (X) ←−−−−− Pd+r−1 (X) −−−−→ ΣDd+r (X),

we have id∗ (y) = pd+r−1,d ∗ (w) and δd+r−1 ∗ (w) = σz. So to prove (b), we just need to show that then, there is an element wi ∈ H∗ (P2d+2r−1 (X)) such that under the maps i

p2d+2r−1,2d

δ2d+2r−1

2d D2d (X) −− → P2d (X) ←−−−−−−− P2d+2r−1 (X) −−−−−→ ΣD2d+2r (X),

we have i2d∗ (Qi y) = p2d+2r−1,d ∗ (wi ) and δ2d+2r−1 ∗ (wi ) = σQi z. We can also pass to HZ/2–modules, and use elements in homotopy. Repressing this from our notation, we will assume this in what we do below.

MOD 2 HOMOLOGY OF INFINITE LOOPSPACES HZ/2

For example, H∗ (D2 (Pd (X))) will ‘really’ mean π∗ (D2

17

(HZ/2 ∧ Pd (X))).

Theorem 1.3(c) follows from the following. Proposition 3.3. There is a commutative diagram of weak natural transformations ∞ D2 (Σ∞ + Ω X)

D2 p

p

∞ Σ∞ +Ω X

/ D2 (Pd (X)) o

D2 i

/ P2d (X) o

D2 (Dd (X))

i

D2d (X)

in which left and right vertical maps are the composites µ

∞ ∞ ∞ ∞ D2 (Σ∞ → Σ∞ + Ω X) → D2 (Σ+ Ω X) − +Ω X

and

µ

D2 (Dd (X)) → D2 (Dd (X)) − → D2d (X), where µ is the standard operad action. Theorem 1.3(b) follows from the following. Proposition 3.4. There is a commutative diagram of weak natural transformations D2 Dd (X)

D2d (X)

D2 i

i

/ D2 Pd (X) o / P2d (X) o

D2 p

p

D2 Pd+r−1 (X)

P2d+2r−1 (X)

D2 δ

δ

/ ΣD2 Dd+r (X) / ΣD2d+2r (X)

in which left and right vertical maps are as in the previous proposition. In interpreting the right square in this diagram, one should recall that ΣD2 Dd+r (X) ≃ D2 ΣDd+r (X), thanks to Corollary 2.8. To deduce Theorem 1.3(b), the needed element wi ∈ H∗ (P2d+2r−1 (X)) will then be the image of Qi w ∈ H∗ (D2 Pd+r−1 (X)) under the vertical map second from the right. It remains to prove these two propositions. We do this at the end of the next subsection. 3.3. Operad actions on towers. The following definition is from [AK]. Definition 3.5. If P is a tower in S, then P ∧ P is the tower in Z/2–S with (P ∧ P )d = holim Pb ∧ Pc . b+c≤d

Suggestively, we will let Dd denote the fiber of Pd → Pd−1 , and then let Fd denote the fiber of (P ∧ P )d → (P ∧ P )d−1 . From [AK, Cor.5.3] we learn

18

KUHN AND MCCARTY

Lemma 3.6. There is a weak natural equivalence in Z/2–S Y Fd ≃ Db ∧ Dc . b+c=d

Note that there are Z/2–equivariant maps

(P ∧ P )2d+1 → (P ∧ P )2d → Pd ∧ Pd and F2d → Dd ∧ Dd . Lemma 3.7. These maps induce equivalences of Tate spectra: ∼

∼

tZ/2 ((P ∧ P )2d+1 ) − → tZ/2 ((P ∧ P )2d ) − → D2 (Pd ) and

∼

tZ/2 (F2d ) − → D2 (Dd ). Proof. With ǫ either 0 or 1, filtered in the usual way, (P ∧ P )2d+ǫ has composition factors of two types: • Di ∧ Di with i ≤ d. • Z/2+ ∧ Di ∧ Dj with i < j and i + j ≤ 2d + ǫ. Meanwhile Pd ∧ Pd has composition factors: • Di ∧ Di with i ≤ d. • Z/2+ ∧ Di ∧ Dj with i < j ≤ d. The first type of factors match up, and after applying tZ/2 , the second type become null. The proof for F2d is similar and easier. ∞ Now let P be the tower P (X), the Goodwillie tower for Σ∞ + Ω X. ∞ Recall that the C∞ operad acts on the space Ω X. In particular, there is a map µ : C∞ (2) ×Z/2 (Ω∞ X)2 → Ω∞ X. The next proposition is our key geometric input. It is quite easily deduced from [AK, Thm. 1.10], and hopefully seems plausible. See Appendix B for a bit more detail.

Theorem 3.8. There is a weak natural transformation of towers µ : (P ∧ P )hZ/2 → P with the following properties. (a) There is a commutative diagram of weak natural transformations ∞ 2 (Σ∞ + (Ω X) )hZ/2

(e∧e)hZ/2

µ

µ

∞ Σ∞ +Ω X

/ (P ∧ P )hZ/2

e

/ P.

(b) On fibers, µ corresponds to the maps Db ∧Dc → Db+c and D2 Dd → D2d .


19

Proof of Proposition 3.3. We have a commutative diagram of weak natural transformations ∞ D2 (Σ∞ + Ω X)

D2 p

/ D2 (Pd ) o O

D2 i

D2 (Dd ) O

≀

≀ ∞ D2 (Σ∞ + Ω X)

/ tZ/2 ((P ∧ P )2d ) o

tZ/2 (F2d )

/ ((P ∧ P )2d )hZ/2 o

(F2d )hZ/2

∞ D2 (Σ∞ + Ω X)

/ P2d o

p

∞ Σ∞ +Ω X

i

D2d .

Here the bottom squares commute by the last proposition, and the right two top vertical maps are weak equivalences by Lemma 3.7. Proof of Proposition 3.4. This time we have a commutative diagram of weak natural transformations D2 D O d

D2 i

/ D2 Pd o O

≀

D2 p

D2 Pd+r−1

D2 δ

/ ΣD2 Dd+r O

O

≀

≀

≀

tZ/2 (F2d )

/ tZ/2 ((P ∧ P )2d ) o

tZ/2 ((P ∧ P )2d+2r−1 )

/ ΣtZ/2 (F2d+2r )

/ ((P ∧ P )2d )hZ/2 o

((P ∧ P )2d+2r−1 )hZ/2

/ Σ(F2d+2r )hZ/2

(F2d )hZ/2

D2d

i

/ P2d o

p

P2d+2r−1

δ

Again the top vertical maps are weak equivalences by Lemma 3.7.

/ ΣD2d+2r .

4. Derived functors of destabilization In this section, we will carefully define the Singer complex d

d

d

d

0 1 2 3 R0 M −→ R1 M −→ R2 M −→ R3 M −→ ...

of the introduction and prove Theorem 1.12, which says that the homology of this complex computes the derived functors Ω∞ s M for all M ∈ M. As a free standing theorem, Theorem 1.12 is similar (and maybe identical) to [Goe, Thm.3.17]. Goerss works totally algebraically, and at key moments his proof appeals to computations and ad hoc arguments by others including Singer [Si], Brown and Gitler [BG], and Bousfield et. al. [BCKQRS]. By contrast, we give a geometrically based construction of this chain complex,

20

KUHN AND MCCARTY

with explicit calculations bypassed by appealing to our knowledge of the homology of the extended powers. Our proof of Theorem 1.12 makes use of the doubling functor Φ : M → M, dual to Powell’s use of it in the cohomological setting [P]. Also included in this section is a presentation of properties of Φ and Ω : U → U needed in our construction of the algebraic spectral sequence in §5. Much of what we say about these functors is dual to cohomological presentations in [LZ1] and [S]. 4.1. Injective resolutions in M. We say a bit about injectives in the category M. Since M is a locally Noetherian abelian category satisfying good exactness properties, one knows a priori [Gab, IV.2] that arbitrary direct sums of injectives in M are again injective, and injectives can be written essentially uniquely as the direct sum of indecomposable injectives. It is useful for us to show this explicitly. Let A∗ ∈ M be the dual of A, so A∗ = H∗ (HZ/2). Let V denote the category of Z–graded vector spaces. Given V ∈ V, we let IV = V ⊗ A∗ . Note that ǫ : A∗ → Z/2 induces a map of graded vector spaces ǫV : IV → V . Lemma 4.1. For all M ∈ M, the natural map ǫM,V : HomM (M, IV ) ≃ HomV (M, V ) sending f to ǫV ◦ f is an isomorphism. Sketch proof. If M is finite, ǫM,Σn Z/2 : HomM (M, Σn A∗ ) ≃ (Mn )# is readily checked to be an isomorphism, and thus the same is true for ǫM,V when M is finite and V is finite dimensional. For finite M and arbitrary V , one then sees that ǫM,V is an isomorphism by filtering V by its finite dimensional subspaces. For arbitrary M and V , one then sees that ǫM,V is an isomorphism by filtering M by its finite submodules. Corollary 4.2. The modules IV are injective objects of M, and every M ∈ M admits an injective resolution of the form 0 → M → IV (0) → IV (1) → IV (2) → . . . , for some graded vector spaces V (s) ∈ V. Proof. As the functor sending M to HomV (M, V ) is exact, we conclude that IV is injective in M. Given M ∈ M, the A–module map M → IM corresponding to 1M ∈ HomV (M, M ) is clearly monic. It follows that injective resolutions of the asserted sort exist. It follows that every injective in M is a direct sum of modules of the form Σn A∗ , and thus is isomorphic to IV for some V ∈ V.


21

4.2. Exact functors from the category M via topology. Let H∗ (S) ⊂ M be the subcategory obtained as the image of H∗ : S → M. Thus the objects are the locally finite A–modules of the form H∗ (X), with morphisms all A–module maps of the form f∗ : H∗ (X) → H∗ (Y ) for some f : X → Y . Let Ab be an abelian category, for example M. Call a functor F : H∗ (S) → Ab homological if whenever X → Y → Z is a cofibration sequence, F (H∗ (X)) → F (H∗ (Y )) → F (H∗ (Z)) is exact at F (H∗ (Y )). The following is a useful way to construct exact functors from M, and natural transformations between such. Proposition 4.3. (a) Any homological functor F : H∗ (S) → Ab extends uniquely to an exact functor F : M → Ab. (b) Let F, G : H∗ (S) → Ab be homological. Any natural transformation φ : F → G extends uniquely to a natural transformation between the extended functors. To prove this, we first note that injectives in M can be topologically realized: given V ∈ V, there is an associated generalized Eilenberg–MacLane spectrum HV , satisfying π∗ (HV ) = V and H∗ (HV ) = IV . The next lemma is clear. Lemma 4.4. For all spectra X, the Hurewitz map induces an isomorphism [X, HV ] ≃ HomM (H∗ (X), IV ). Proof of Proposition 4.3. Given a homological functor F : H∗ (S) → Ab, we define its extension F : M → Ab as follows. Given M ∈ M, choose an exact f∗

sequence 0 → M → H∗ (HV (0)) −→ H∗ (HV (1)), and then define F (M ) to be the kernel of F (f∗ ). Given a morphism α : M → N in M, one can construct a diagram 0

/M

/ H∗ (HV (0))

α

0

/N

f∗

/ H∗ (HV (1)) γ∗

β∗

/ H∗ (HW (0))

g∗

/ H∗ (HW (1))

with exact rows. Applying F to the right square and taking kernels, defines a map F (α) : F (M ) → F (N ). It is routine to check that this gives a well defined exact functor which extends the original functor up to natural isomorphism. This proves (a). The proof of (b) is similar. 4.3. A topological definition of Rs M . We construct various functors and natural transformations using the method of Proposition 4.3. Definition 4.5. Define Rs : H∗ (S) → M by the formula Rs H∗ (X) = im{ǫ∗ : H∗ (ΣD2s (Σ−1 X) → H∗ (D2s X)}.

22

KUHN AND MCCARTY

Thanks to our knowledge of ǫ∗ as summarized in Lemma 2.10, we see that Rs H∗ (X) = hQI x | l(I) = s, x ∈ H∗ (X)i/(unstable and Adem relations). We remind readers that the Dyer–Lashof Adem relations are X i − s − 1 r s Q Q = Qr+s−i Qi , 2i − r i

and that the Steenrod algebra acts via the Nishida relations Xs−r s r (Q x)Sq = Qs−r+i (xSq i ). r − 2i i

Lemma 4.6. Rs is homological. Proof. This follows immediately from the observation that the natural map of graded vector spaces M Rs (Σn Z/2) ⊗ Hn (X) → Rs (H∗ (X)) n∈Z

sending QI ιn ⊗ x to QI x is an isomorphism. (In this formula, Hn (X) should be regarded as just a degree 0 vector space.) Definitions 4.7. (a) Let Rs : M → M be the exact extension of Rs : H∗ (S) → M. (b) Let ǫ : ΣRs M → Rs ΣM be the natural A–module map induced by the natural transformation ǫ : ΣD2s X → D2s ΣX. (c) Let µ : Rs Rt M → Rs+t M be the natural A–module map induced by the natural transformation µ : D2s D2t X → D2s+t X. (d) Let Qi : (Rs M )n → (Rs+1 M )n+i be the natural Z/2–linear map induced by the Dyer–Lashof operation Qi : Hn (D2s X) → Hn+i (D2s+1 X). We can immediately deduce lots of properties of these natural transformations. We note, in particular, a couple. Lemma 4.8. (a) The operations Qi satisfy the Adem relations, the Nishida relations, and the Dyer–Lashof unstable relation. (b) The diagram ΣRs Rt M

ǫ

/ Rs ΣRt M

ǫ

/ Rs Rt ΣM µ

µ

ΣRs+tM commutes.

ǫ

/ Rs+t ΣM


23

Remark 4.9. Observe that R0 (M ) = M , and ǫ : ΣR0 M → R0 ΣM is just the identity map on ΣM . Another elementary property we will need involves connectivity. Lemma 4.10. If M is (n − 1) connected, then Rs M is 2s n − 1 connected. Proof. This follows from the observation that if X is (n − 1) connected, then Dd X is dn − 1 connected. Now we introduce algebraic differentials. As before, δ : X → ΣD2 X is the connecting map of the cofibration sequence D2 X → P˜2 X → X. Definition 4.11. Define ds : Rs (M ) → Rs+1 (ΣM ) to be the natural transformation induced by the composite D

sδ

D

sǫ

µ

δs : D2s X −−2−→ D2s ΣD2 X −−2−→ D2s D2 ΣX − → D2s+1 ΣX. Explicitly, the computation of δ∗ given in Theorem 1.3(a) tells us that X ds (QI x) = QI Qi−1 (σxSq i ). i≥0

Proposition 4.12. The composite ds−1

d

s Rs−1 (Σ−1 M ) −−−→ Rs (M ) −→ Rs+1 (ΣM )

is zero. This is an immediate consequence of the following topological version, and since homology is compactly supported, we really just need this result when X is a finite CW spectrum. Proposition 4.13. The composite δs−1

δ

s D2s−1 (X) −−−→ D2s (ΣX) −→ D2s+1 (Σ2 X)

is null. Proof. It is easy to see that this composite factors through D2s−1 applied to the composite δ

δ

0 1 X −→ D2 (ΣX) −→ D4 (Σ2 X).

Thus we just need to show that this last composite is null. The trick now is to colinearize these functors and maps. Generalizing our previous notation D2 , for any d, let Dd (X) = holimn Σn Dd (Σ−n X).

24

KUHN AND MCCARTY

Colinearization then yields a commutative diagram of weak natural transformations /X D1 (X) δ0

D2 (ΣX)

/ D2 (ΣX)

δ1

D4 (Σ2 X)

/ D4 (Σ2 X).

As the top horizontal map is clearly an equivalence, the proposition will follow if we can show the left composite is null. We offer two rather different reasons for this. The first argument only seems to hold when X is finite, and depends on consequences of the Segal Conjecture for elementary abelian 2–groups. Namely, the first author showed [K1, Cor.5.3] that D4 (X) ≃ ∗ if X is finite. (In this case, it is also true that the top left vertical map is an equivalence after completing at 2.) A second, more elementary argument goes roughly as follows. The colinearized functors Dd preserves cofibration sequences, and are null unless d is a power of 2. Then it is not too hard to show that the left vertical sequence is equivalent to the composite D1 (X) → ΣD2 (X) → Σ2 D4 (X) of the two connecting maps associated to the colinearization of the tower P˜4 (X) → P˜2 (X) → P˜1 (X), so that their composite is null.

Remark 4.14. A direct algebraic proof of Proposition 4.12 is possible. Using both the Dyer–Lashof Adem relations and the Adem relations in A, one needs to show that XX Qi−1 Qj−1 (xSq i Sq j ) = 0. i≥0 j≥0

Goerss [Goe, Lem.3.13] points to Brown and Gitler’s assertion that a calculation like this is straightforward [BG, Lem.2.3], and one can check that it is. 4.4. The doubling functor and R∗ (M ). In the cohomological setting, the following definition should be familiar to readers of [LZ1] and [S].

Definition 4.15. If M ∈ M, Φ(M ) ∈ M is defined to be the module concentrated in even degrees, with Φ(M )2n = Mn and with φ(x)Sq 2i = φ(xSq i ). (Here, given x ∈ Mn , we have written φ(x) for the corresponding element in Φ(M )2n .) Basic properties are listed in the next lemma.


25

Lemma 4.16. (a) Φ is an exact functor preserving unstable modules. (b) Φ(N ⊗ M ) = Φ(N ) ⊗ Φ(M ). In particular, Φ(ΣM ) = Σ2 Φ(M ). (c) Let Γ2 (M ) = (M ⊗ M )Z/2 and S 2 (M ) = (M ⊗ M )Z/2 . The composite Γ2 (M ) ֒→ M ⊗ M ։ S 2 (M ) naturally factors as a composite Γ2 (M ) ։ Φ(M ) ֒→ S 2 (M ), where the second map sends φ(x) to x2 . (d) Let sq0 : M → Φ(M ) be the linear map defined by letting sq0 (x) = φ(xSq n ) if x ∈ M2n . If M is unstable, then sq0 is A–linear. For a proof of (d), see [S, p.26]. Definition 4.17. Let q0 : Φ(M ) → R1 M be defined by the formula q0 (φ(x)) = Q|x| x. More generally, define q0 : Φ(Rs M ) → Rs+1 M to be the composite q0 µ Φ(Rs M ) −→ R1 Rs M − → Rs+1 M . Lemma 4.18. q0 is A–linear. Proof. As usual, one need just check this when M = H∗ (X). The identity Q|x| x = x2 ∈ H∗ (D2 X) for all x ∈ H∗ (X) implies that the composite q0

Φ(H∗ (X)) −→ R1 (H∗ (X)) ⊆ H∗ (D2 X) equals the composite Φ(H∗ (X)) ֒→ S 2 (H∗ (X)) → H∗ (D2 X), and so is A–linear.

The following lemma is crucial. Lemma 4.19. For all M ∈ M and s > 0, the sequence q0

ǫ

0 → Φ(Rs−1 (M )) −→ Rs (M ) − → Σ−1 Rs (ΣM ) → 0 is short exact. Proof. It is convenient to use lower indices for Dyer–Lashof operations: Qi x = Q|x|+i x. Suppose M has a homogeneous basis {xα }. Then the Adem relations show that Rs (M ) then has a basis given by {Qi0 Qi1 . . . Qis xα | 0 ≤ i0 ≤ i1 ≤ · · · ≤ is }. Since ǫ(Qi0 Qi1 . . . Qis xα ) = σ −1 Qi0 −1 Qi1 −1 . . . Qis −1 σxα , and Qi x = 0 if i < 0, the lemma follows. The next lemma is clear from the definitions.

26

KUHN AND MCCARTY

Lemma 4.20. (a) The diagram µ

Rs Rt (M )

/ Rs+t (M ) ds+t

Rs (dt )

µ

Rs Rt+1 (ΣM )

/ Rs+t+1 (ΣM )

commutes. (b) The diagram Φ(Rs−1 (M ))

q0

/ Rs (M )

Φ(ds−1 )

Φ(Rs (ΣM ))

ǫ

/ Σ−1 Rs (ΣM )

ds q0

/ Rs+1 (ΣM )

ds ǫ

/ Σ−1 Rs+1 (Σ2 M )

commutes. 4.5. The derived functors of destabilization. We now relabel as in the introduction. Definition 4.21. Let Rs = ΣRs Σs−1 : M → M. With this notation, the chain complex d

d

d

0 1 2 ΣR0 (Σ−1 M ) −→ ΣR1 (M ) −→ ΣR2 (ΣM ) −→ ΣR3 (Σ2 M ) → . . .

rewrites as d

d

d

0 1 2 R0 (M ) −→ R1 (M ) −→ R2 (M ) −→ R3 (M ) → . . . .

The following is a restatement of Theorem 1.12. Theorem 4.22. For all M ∈ M, there is a natural isomorphism Hs (R∗ (M ); d∗ ) ≃ Ω∞ s M. In the usual way, this theorem is a consequence of the next three lemmas. Lemma 4.23. Rs is exact for all s. Lemma 4.24. H0 (R∗ (M ); d∗ ) = Ω∞ M . Lemma 4.25. For all n ∈ Z and s > 0, Hs (R∗ (Σn A∗ ); d∗ ) = 0. The first of these lemmas is evident, and we quickly check the second. Proof of Lemma 4.24. We need to compute the kernel of d0 : M → ΣR1 M , and we recall that X d0 (x) = σQi−1 (xSq i ). i≥0


27

Then x ∈ ker(d0 ) ⇔ Qi−1 (xSq i ) = 0 for all i ≥ 0 ⇔ xSq i = 0 whenever i − 1 ≥ |x| − i ⇔ xSq i = 0 whenever 2i > |x|. ⇔ x ∈ Ω∞ M. The proof of Lemma 4.25 will take a bit of preparation. Firstly, Lemma 4.19 and Lemma 4.20(b) combine to tell us the following. q0

ǫ

Proposition 4.26. 0 → Φ(R∗−1 (ΣM )) −→ ΣR∗ (M ) − → R∗ (ΣM ) → 0 is a short exact sequence of chain complexes. Temporarily, let Hs (M ) = Hs (R∗ (M ); d∗ ). The short exact sequence of Proposition 4.26 induces a long exact sequence ǫ

q0

∂

∗ 0 → ΣH0 (M ) −→ H0 (ΣM ) − → Φ(H0 (ΣM )) −→ ΣH1 (M ) → . . .

q0

∂

ǫ

∗ . . . → Hs−1 (ΣM ) − → Φ(Hs−1 (ΣM )) −→ ΣHs (M ) −→ Hs (ΣM ) → . . .

We need to identify the first boundary map. ∂

Lemma 4.27. H0 (ΣM ) − → Φ(H0 (ΣM )) identifies with the map sq0

Ω∞ (ΣM ) −−→ Φ(Ω∞ (ΣM )). Proof. If σx ∈ Ω∞ (ΣM ) has |σx| = 2n, then we have the correspondence, Σd

q0

0 under the maps ΣM −−→ ΣR1 (M ) ←− Φ(ΣM ),

(Σd0 )(σx) = σd0 (x) = σQn−1 (xSq n ) = q0 (φ(σxSq n )) = q0 (sq0 (σx)). Thus ∂(σx) = sq0 (σx).

In dual form, the following lemma corresponds to the familiar fact that the map Sq0 : F (n) → F (n), sending x to Sq |x| x, is monic. Here F (n) is the free unstable A–module on an n–dimensional class. sq0

Lemma 4.28. For all n ∈ Z, Ω∞ (Σn A∗ ) −−→ Φ(Ω∞ (Σn A∗ )) is onto. We are finally ready to prove Lemma 4.25. The proof is dual to the proof of [P, Proposition 9.4.1]. Proof of Lemma 4.25. By induction on s ≥ 1, we prove that Hs (Σn A∗ ) = 0. In all cases, we consider the exact sequence ∂

q0

ǫ

∗ Hs−1 (Σn+1 A∗ ) − → Φ(Hs−1 (Σn+1 A∗ )) −→ ΣHs (Σn A∗ ) −→ Hs (Σn+1 A∗ ).

In the initial case when s = 1, the previous two lemmas show that ∂ is onto. If s > 1, then, under the inductive hypothesis, Φ(Hs−1 (Σn+1 A∗ )) = 0. Thus, ǫ∗ in all cases, we can conclude that ΣHs (Σn A∗ ) −→ Hs (Σn+1 A∗ ) is monic for −m all n. But, by Lemma 4.10, the connectivity of Σ Hs (Σm+n A∗ ) is at least (2s − 1)m + 2s (n + s − 1), and so goes to infinity as m goes to infinity.

28

KUHN AND MCCARTY

4.6. First consequences. Theorem 4.22, when combined with Proposition 4.26 and Lemma 4.27, implies the following. Corollary 4.29. For all M ∈ M, there is a natural long exact sequence sq0

ǫ

q0

∗ 0 → ΣΩ∞ → Ω∞ −→ Φ(Ω∞ → ΣΩ∞ 0 (M ) − 0 (ΣM ) − 0 (ΣM )) − 1 (M ) → . . .

sq0

q0

ǫ

∗ . . . → Ω∞ −→ Φ(Ω∞ → ΣΩ∞ → Ω∞ s−1 (ΣM ) − s−1 (ΣM )) − s (M ) − s (ΣM ) → . . .

Remark 4.30. This long exact sequence already appears (in dual form) in [LZ1, §4.1]. One observes that, if M → I∗ (M ) is an injective resolution in M, then so is ΣM → ΣI∗ (M ), and sq0

ǫ

0 → ΣΩ∞ (I∗ (M )) − → Ω∞ (ΣI∗ (M )) −−→ Φ(Ω∞ (ΣI∗ (M ))) → 0 is short exact. This short exact sequence of chain complexes then induces the long exact sequence of the corollary. It is amusing that sq0 identifies with the boundary map in our derivation, while q0 identifies with the boundary map in the Lannes–Zarati approach. Next we note that Lemma 4.10 implies the following general connectivity estimate. s Corollary 4.31. If M ∈ M is n–connected, then Ω∞ s (M ) is at least 2 (n + −n ∞ n s)–connected. For all M ∈ M, colim Σ Ωs (Σ M ) = 0 for all s ≥ 1. n

The reasoning we gave in the proof of Lemma 4.25 then proves the following useful criterion for the vanishing of the higher derived functors. Proposition 4.32. If sq0 : Ω∞ (Σn M ) → Φ(Ω∞ (Σn M )) is onto for all n ≥ 1, then Ω∞ s (M ) = 0 for all s ≥ 1. −t Following [LZ1] and [Goe], we now deduce some properties of Ω∞ s (Σ M ) when M is unstable.

Lemma 4.33. Suppose M is unstable. Then, for all s ≥ 0, Rs (M ) is also unstable, and ds : Rs (Σ−1 M ) → Rs+1 (M ) is zero. Proof. Though this admits an algebraic proof, to show how the algebra follows the topology, we offer a topologically based proof. We begin by observing that every unstable module embeds in the homology of a space: if M ⊂ IV and is unstable, then M ⊂ Ω∞ IV = H∗ (Ω∞ HV ). Thus it suffices to prove the lemma when M = H∗ (Z), where Z is a space. In this case, Rs (H∗ (Z)) ⊂ H∗ (D2s Z), which is unstable, as D2s Z is a space. To see that ds : Rs (Σ−1 H∗ (Z)) → Rs+1 (H∗ (Z)) is zero, we recall that it is induced by a geometric stable map δs : D2s (Σ−1 Z) → D2s+1 (Z). (We identify Z with Σ∞ Z.) We observe that this map is null: δs factors through D2s (δ0 ), and δ0 : Σ−1 Z → D2 (Z) is null as Σδ0 is the first boundary map in the tower associated to Σ∞ Ω∞ Σ∞ Z, which splits into the product of its fibers.


29

1−s M ) is the homology at the middle term of the complex As Ω∞ s (Σ ds−1

d

s ΣRs−1 (Σ−1 M ) −−−→ ΣRs (M ) −→ ΣRs+1 (ΣM ),

the lemma leads to the next result. Theorem 4.34. Suppose M is unstable. 1−s M ) ≃ ΣR (M ), so that ΩΩ∞ (Σ1−s M ) ≃ R (M ). (a) Ω∞ s s s (Σ s −t s−t−1 M ), which is (b) More generally, if s > t, then Ω∞ s (Σ M ) ≃ ΣRs (Σ s−t ∞ −t a quotient of Σ Rs (M ). Thus Ωs (Σ M ) is an (s − t)–fold suspension of −t −s+t+1 R (Σs−t−1 M ). an unstable module, and so Ωs−t Ω∞ s s (Σ M ) = Σ −s −2 −1 (c) Ω∞ s (Σ M ) ≃ coker{ds−1 : ΣRs−1 (Σ M ) → ΣRs (Σ M )}.

The first statement here is the main algebraic theorem of [LZ1], and the last was observed in [Goe, Cor.5.4]. 4.7. Dyer–Lashof operations on derived functors. We need to explain Proposition 1.7, which said that the sum of the looped derived functors 1−∗ M is an object in QM. Otherwise said, we need to explain why ΩΩ∞ ∗ Σ there exist natural transformations 1−t 1−s−t µ : Rs ΩΩ∞ M → ΩΩ∞ M t Σ s+t Σ

compatible in the usual way. Firstly we note that Lemma 4.20(a) and Theorem 4.22 together imply, when one is careful with suspensions, that the maps µ : Rs Rt M → Rs+t M induce maps 1−t 1−s−t µ : Rs Σ−1 Ω∞ M → Σ−1 Ω∞ M t Σ s+t Σ

We now need a better understanding of ΩΩ∞ s (M ) for general M ∈ M. The following lemma is dual to [S, Prop.1.7.5]. Lemma 4.35. Ω : U → U has only one nonzero right derived functor Ω1 . For all M ∈ U , there is an exact sequence sq0

0 → ΣΩM → M −−→ Φ(M ) → ΣΩ1 M → 0. From the long exact sequence of Corollary 4.29, we thus deduce the following.

30

KUHN AND MCCARTY

Corollary 4.36. For M ∈ M, the following diagram commutes, and the bottom row is short exact: Σ−1 Ω∞ s+1 (ΣM )

Σ−1 ΦΩ∞ s (ΣM )

o7

OOO OOOq0 OOO OOO ' /

ooo ooo

Ω∞ s+1 (M )

Ω1 Ω∞ s (ΣM )

O

ǫ∗ oooo

? / / ΩΩ∞ (ΣM ). s+1

Proof of Proposition 1.7. From the commutative diagram of Lemma 4.8, ǫ

ΣRs Rt M

/ Rs ΣRt M

Rs ǫ

/ Rs Rt ΣM

µ

µ

ǫ

ΣRs+tM

/ Rs+t ΣM,

we deduce that the following diagram commutes: −t ΣRs Σ−1 Ω∞ t Σ M

ǫ

/ / Rs Ω∞ Σ−t M t

Rs ǫ∗

/ Rs Σ−1 Ω∞ Σ1−t M t µ

µ

−s−t M Ω∞ s+t Σ

ǫ∗

/ Σ−1 Ω∞ Σ1−s−t M. s+t

For M ∈ M, we then define 1−t 1−s−t µ : Rs ΩΩ∞ M → ΩΩ∞ M t Σ s+t Σ

to be the natural transformation induced by taking the image of the top and bottom horizontal maps in this last diagram. These natural transformations for all s and t are equivalent to defining natural Dyer–Lashof operations 1−s −s Qi : ΩΩ∞ M → ΩΩ∞ s Σ s+1 Σ M,

for all i ∈ Z, which raise degree by i, and satisfy the usual properties.

∞ Definition 4.37. Define q0 : Φ(ΩΩ∞ s (ΣM )) → ΩΩs+1 (M ) by the formula q0 (φ(x)) = Q|x| x.

5. The algebraic spectral sequence In this section we explain why, given M ∈ M, there is a well defined spectral sequence of Hopf algebras as in Theorem 1.8: alg,1 • E∗,∗ (M ) = UQ (R∗ (M )). s • Nonzero differentials are only the d2 , and, for x ∈ M and I of length P s s alg,2 (M ), and d2 (QI x) = i≥0 QI Qi−1 (xSq i ). s, QI x lives to E∗,∗ alg,r • For all r, E∗,∗ (M ) is primitively generated with primitives concenalg,r trated in the −2s lines. For all r > 2s , P E−2 s ,2s +∗ (M ) ≃ Ls M . alg,∞ 1−∗ (M )). • E∗,∗ (M ) = UQ (ΩΩ∞ ∗ Σ


31

5.1. A complete description of the algebraic spectral sequence. We alg,1 alg,∞ will rewrite both E∗,∗ (M ) and E∗,∗ (M ) in a manner that will allow us alg,2s (M ). to also describe the intermediate pages E∗,∗ For clarity, in the rest of this section we use the following notation. alg,r 1−s M , C M = Notation 5.1. We let E r (M ) = E∗,∗ (M ), Ls M = ΩΩ∞ s s Σ ds−1

coker{Rs−1 Σ−1 M −−−→ Rs M }, and Hs M be the homology at Rs M in the ds−1

d

s sequence Rs−1 Σ−1 M −−−→ Rs M −→ Rs+1 ΣM .

We note a couple of relationships between these: d

s • 0 → Hs M → Cs M −→ Rs+1 ΣM is exact. ǫ∗ −1 • Ls M = im{ΣHs Σ M −→ Hs M }, by Corollary 4.36. We use the following module structures: • For any N ∈ M, the natural inclusion Φ(N ) ⊆ S 2 (N ) makes S ∗ (N ) into a S ∗ (Φ(N )) module. • The natural map q0 : Φ(Rs M ) → Rs+1 M makes S ∗ (Rs+1 M ) into an S ∗ (Φ(Rs M ))–module. • The natural map q0 : Φ(Ls M ) → Ls+1 M makes S ∗ (Ls+1 M ) into an S ∗ (Φ(Ls M ))–module.

We will have: E 1 (M ) = S ∗ (R0 M ) ⊗S ∗ (Φ(R0 M )) S ∗ (R1 M ) ⊗S ∗ (Φ(R1 M )) S ∗ (R2 M ) ⊗ . . . , E 2 (M ) = S ∗ (L0 M ) ⊗S ∗ (Φ(L0 M )) S ∗ (C1 M ) ⊗S ∗ (Φ(R1 M )) S ∗ (R2 M ) ⊗ . . . , E 4 (M ) = S ∗ (L0 M ) ⊗S ∗ (Φ(L0 M )) S ∗ (L1 M ) ⊗S ∗ (Φ(L1 M )) S ∗ (C2 M ) ⊗ . . . , etc., ending with E ∞ (M ) = S ∗ (L0 M ) ⊗S ∗ (Φ(L0 M )) S ∗ (L1 M ) ⊗S ∗ (Φ(L1 M )) S ∗ (L2 M ) ⊗ . . . . We say this more formally. Definition 5.2. Let Γs (M ) = S ∗ (Cs M ) ⊗S ∗ (Φ(Rs M )) S ∗ (Rs+1 M ). Theorem 5.3. (a) Γs (M ) is a differential graded algebra with a differential d of degree -1 induced by ds : Rs Σ−1 M → Rs+1 M as follows: d(y) = 0 for y ∈ Rs+1 M , and for x ∈ Cs M , d(x) = ds (¯ x) where x ¯ ∈ Rs (Σ−1 M ) is any element such that σ¯ x maps to x under the composite ǫ

∗ ΣRs Σ−1 M −→ Rs M ։ Cs (M ).

(b) H∗ (Γs (M ); d) ≃ S ∗ (Ls M ) ⊗S ∗ (Φ(Ls M )) S ∗ (Cs+1 M ).

32

KUHN AND MCCARTY

We assume this key result, and continue with our presentation. We have an addendum to this last theorem. Lemma 5.4. The differential d : Γs (M ) → Γs (M ) commutes with both the S ∗ (Ls (M ))–module structure and the S ∗ (Rs+1 M )–module structure. Thus the isomorphism of part (b) is as S ∗ (Ls M )–S ∗ (Rs+1 M )–bimodules. Proof. The only nontrivial point to check is that, if x ∈ Ls M , then d(x) = 0, where d(x) is calculated as in the theorem. For this, one has the following commutative diagram 0

/ H Σ−1 M s

/ C Σ−1 M s

ǫ∗

ds

/ Rs+1 M

ǫ∗

/ Cs M

Ls M

where the top row is exact, and the vertical maps have degree +1. One can thus lift any x ∈ Ls M to an element in Hs Σ−1 M , which will then map to zero in Rs+1 M , and thus also in Γs (M ). This lemma allows us to define the pages of our spectral sequence. s

s

Definition 5.5. Let (E 2 (M ); d2 ) be the differential graded algebra As (M ) ⊗S ∗ (Φ(Ls−1 M )) (Γs (M ); d) ⊗S ∗ (Φ(Rs+1 M )) Bs (M ), where As (M ) = S ∗ (L0 M ) ⊗S ∗ (Φ(L0 M )) S ∗ (L1 M ) ⊗ . . . ⊗S ∗ (Φ(Ls−2 M )) S ∗ (Ls−1 M ) and Bs (M ) = S ∗ (Rs+2 M ) ⊗S ∗ (Φ(Rs+2 M )) S ∗ (Rs+3 M ) ⊗S ∗ (Φ(Rs+3 M )) . . . . Lemma 5.6. (a) As (M ) is a free S ∗ (Φ(Ls−1 M ))–module. (b) Bs (M ) is a free S ∗ (Φ(Rs M ))–module. Proof. Statement (a) is a consequence of the familiar general fact that, for any N ∈ M, S ∗ (N ) is a free S ∗ (Φ(N ))–module. Statement (b) similarly follows from the fact that, for any t and M , q0 : Φ(Rt M ) → Rt+1 M is monic, so that S ∗ (Rt+1 M ) is a free S ∗ (Φ(Rt M ))–module. We now can show that we really have defined a spectral sequence. s

s

s+1

Proposition 5.7. H∗ (E 2 (M ); d2 ) = E 2 2s

(M ).

2s

Proof. H∗ (E (M ); d ) is the homology of As (M ) ⊗S ∗ (Φ(Ls−1 M )) (Γs (M ); d) ⊗S ∗ (Φ(Rs+1 M )) Bs (M ). By the last lemma, this equals As (M ) ⊗S ∗ (Φ(Ls−1 M )) H∗ (Γs (M ); d) ⊗S ∗ (Φ(Rs+1 M )) Bs (M ).


33

By Theorem 5.3, this is As (M )⊗S ∗ (Φ(Ls−1 M )) S ∗ (Ls M )⊗S ∗ (Φ(Ls M )) S ∗ (Cs+1 M )⊗S ∗ (Φ(Rs+1 M )) Bs (M ). This regroups as As+1 (M ) ⊗S ∗ (Φ(Ls M )) Γs+1 (M ) ⊗S ∗ (Φ(Rs+2 M )) Bs+1 (M ), s+1

which is just E 2

(M ).

Remark 5.8. We need to say a little bit about the coproduct structure on our alg,2s (M ) is generated as an algebra by Lt (M ) spectral sequence. Note that E∗,∗ with t < s, Cs (M ), and Rt (M ) with t > s. Assuming by induction that these s are all primitive, our formula for d2 sends these generating primitives to s alg,2s+1 (M ) primitives, and it follows that d2 will be a coderivation, and so E∗,∗ will again be a primitively generated coalgebra of the same form. With this Hopf algebra structure in place, we can explain why Corollary 1.9 is true. This corollary concerned when the algebraic and topological spectral sequences agree. Primitives get mapped to primitives under r (X) → E r (X), and the algebraic spectral sequence has primidr : E∗,∗ ∗,∗ tives concentrated on the −2s lines. It follows that a first differential in the topological spectral sequence for X differing from the differentials in the algebraic spectral sequence for H∗ (X) would, in lowest filtration degree, necessarily have the form s (2t −1)

d2

2s (2t −1)

: E−2s ,∗

2s (2t −1)

(X) → E−2s+t ,∗+2s (2t −1)−1 (X) 2s (2t −1)

for some t ≥ 2, and that the primitives in E−2s ,∗ Ls H∗ (X). This proves Corollary 1.9.

(X) would identify with

It remains to prove Theorem 5.3. 5.2. Reduction of Theorem 5.3 to a proposition. Theorem 5.3 will be a special case of a general proposition. We start with a diagram of graded vectors spaces (or locally finite right A–modules) (♣)

Φ(U )

Φ(dU )

/ Φ(V ) qV

qU

U′

d′U

/V′

eU

U

0

d′V

/ W′

eV dU

/V

0

eW dV

/W

0

where eU , eV , and eW have degree +1, other maps are of degree 0, the columns are exact, and the two bottom rows are chain complexes.

34

KUHN AND MCCARTY

We let p : V → V / im dU be the projection, and we let im e∗ denote the image of eV ∗ : H(V ′ ) → H(V ). Now let Γ = S ∗ (V / im dU ) ⊗S ∗ (Φ(V )) S ∗ (W ′ ). Proposition 5.9. In this situation, the following hold. (a) Γ admits the structure of a differential graded algebra with differential d′

p◦e

V W ′ : d(w) = 0 for d of degree -1 defined by the maps V / im dU ←−−V− V ′ −−→ ′ ′ w ∈ W , and for v¯ ∈ V / im dU , d(¯ v ) = dV (v) if p(e(v)) = v¯.

(b) H∗ (Γ; d) ≃ S ∗ (im e∗ ) ⊗S ∗ (Φ(im e∗ )) S ∗ (W ′ / im d′V ). Then Theorem 5.3 is the proposition applied to the diagram Φ(Rs−1 Σ−1 M )

Φ(ds−1 )

/ Φ(Rs M )

q0

Rs−1 Σ−2 M

ds−1

q0

ds

/ R Σ−1 M s

/ Rs+1 M

ǫ

ǫ

Rs−1 Σ−1 M

ǫ

ds−1

ds

/ Rs M

0

/ Rs+1 ΣM

0

0

5.3. Proof of Proposition 5.9. The idea of the proof of Proposition 5.9 is to reduce the situation of the proposition to simpler and simpler cases. Our first reduction is most dramatic: one can assume that U ′ , and thus also U , is 0. To see this, we construct a ‘quotient’ of diagram (♣). ¯ = W , V¯ ′ = V ′ /(im d′ + im qU ), and W ¯′ = Let V¯ = V / im dU , W U ′ ′ W / im dV ◦ qU . Our original diagram (♣) will map in an evident way to the diagram (♦)

/ Φ(V¯ )

0

q

0

/ V¯ ′

d′

/W ¯′

d

e¯V

0

/ V¯

0

0

e¯W

/W ¯

0

in which all maps are induced from the corresponding map in (♣).


35

¯ = S ∗ (V¯ ) ⊗ ∗ ¯ S ∗ (W ¯ ′ ). The next lemma shows that proving Let Γ S (Φ(V )) Proposition 5.9 for the situation of diagram (♣) reduces to proving it in the situation of diagram (♦). Lemma 5.10. (a) e¯V : V¯ ′ → V¯ is an isomorphism, of degree +1. (b) The third column of (♦) is exact. ¯ ′ / im d′ . (c) W ′ / im d′V ≃ W (d) im{eV∗ : H(V ′ ) → H(V )} ≃ im{¯ eV∗ : ker d′ → ker d} = ker d′ . ¯ is an isomorphism. (e) The natural algebra map Γ → Γ Proof. Diagram chasing with the left two columns of (♣) shows that there is an exact sequence qU

e

V Φ(U ) −→ V ′ / im d′U −→ V / im dU → 0,

and statement (a) follows. f

g

It is standard that given maps A − →B− → C in an abelian category, there is an exact sequence coker f → coker gf → coker g → 0. Apply this to Φ(dU )

d′

qV

V Φ(U ) −−−−→ Φ(V ) −→ W ′ to deduce statement (b). Apply this to V ′ −−→ ′ ′ ¯ to deduce statement (c), noting that W ։W ′

d ¯ ′ }. ¯ ′ } = coker{V¯ ′ − →W coker{V ′ → W To deduce (d), let V˜ ′ = V ′ / im d′U . One has a commutative diagram

V˜ ′

/ V¯ ′

d′V

W′

∼

/ V¯

d′

/W ¯′

d

/W ¯,

where the indicated isomorphism is the isomorphism of (a). Taking kernels, one gets H(V ′ ) → ker d′ ֒→ H(V ), and we need to check that the first map here is onto. But this follows because the left square fits into a commutative diagram Φ(U )

/ ˜′ V d′V

Φ(U )

/ W′

/ V¯ ′

/0

d′

/W ¯′

/0

with exact rows. Finally, we need to prove (e), which says that the evident quotient map ¯ ′) S ∗ (V¯ ) ⊗S ∗ (Φ(V )) S ∗ (W ′ ) → S ∗ (V¯ ) ⊗S ∗ (Φ(V¯ )) S ∗ (W

36

KUHN AND MCCARTY

is an isomorphism. On one hand, S ∗ (V¯ ) ⊗S ∗ (Φ(V )) S ∗ (W ′ ) = S ∗ (V¯ ⊕ W ′ )/I, where I is the ideal generated by the elements −p(v)2 + qV (φ(v)), v ∈ V . On the other hand, ¯ ′ ) = S ∗ (V¯ ⊕ W ′ )/I, ¯ S ∗ (V¯ ) ⊗S ∗ (Φ(V¯ )) S ∗ (W where I¯ is generated by the elements −p(v)2 + qV (φ(v)), v ∈ V , and also the elements qV (φ(dU (u))), u ∈ U . But this second family of elements is included among the first, as p(dU (u)) = 0. We thus just need to prove Proposition 5.9 for diagrams of the form (♦). Simplifying notation, this means we need to prove the following proposition. δ

q

Proposition 5.11. Suppose given V − →W ← − Φ(V ), with δ having degree -1, and let Γ(δ, q) = S ∗ (V ) ⊗S ∗ (Φ(V )) S ∗ (W ). Then Γ(δ, q) admits the structure of a differential graded algebra with differential d given on generators by d(v, w) = (0, δ(v)), and there is a natural isomorphism H∗ (Γ(δ, q); d) ≃ S ∗ (ker δ) ⊗S ∗ (Φ(ker δ)) S ∗ (coker δ). Proof. We first explain why Γ(δ, q) admits a derivation d as claimed. Very generally, if U is a graded Z/2–vector space, and M is an S ∗ (U )– module, one can compute Der(S ∗ (U ), M ), the vector space of derivations S ∗ (U ) → M , by the formula Der(S ∗ (U ), M ) = HomS ∗ (U ) (ΩS ∗ (U ) , M ) = HomZ/2 (U, M ), where ΩR is the R–module of K¨ ahler differentials of a commutative ring R [W, 8.8.1]. The first equality here is tautological, the second then follows from the calculation ΩS ∗ (U ) ≃ S ∗ (U ) ⊗ U , which can be deduced from [W, 9.2.4]. Specialized to our situation, we learn that there is a unique derivation d˜ : S ∗ (V ⊕ W ) → S ∗ (V ⊕ W ) which restricts to the linear map V ⊕ W → V ⊕ W , (v, w) 7→ (0, δ(v)). As ˜ d˜ = 0. this linear map gives zero when composed with itself, it follows that d◦ Checking that this passes to a differential on the quotient algebra, d : Γ(δ, q) → Γ(δ, q), amounts to the observation that, for any v ∈ V , ˜ 2 − q(φ(v))) = 2v d(v) ˜ − d(q(φ(v))) ˜ d(v = 0. We now turn to computing the homology of Γ(δ, q).


37

We first consider the special case when q is identically zero, and write Γ(δ) for Γ(δ, 0). Since S ∗ (V ) ⊗S ∗ (Φ(V )) Z/2 = Λ∗ (V ), we see that, as an algebra, Γ(δ) = Λ∗ (V ) ⊗ S ∗ (W ). We can also assume that our map δ : V → W has the form 1

U K ⊕ U ։ U −− → Σ−1 U ֒→ Σ−1 U ⊕ C,

where K = ker δ, C = coker δ, and the identity map 1U is viewed as a map of degree -1 from U to its desuspension. In this case, Γ(δ) = Λ∗ (K) ⊗ Γ(1U ) ⊗ S ∗ (C), as differential graded algebras. But Γ(1U ) = Λ∗ (U ) ⊗ S ∗ (Σ−1 U ) with the Koszul differential, which is well known to be acyclic [W, Cor.4.5.5], and easily checked to be: it is the tensor product of complexes of the form Λ∗ (x) ⊗ Z/2[dx] whose homology is Z/2. Thus we see that H∗ (Γ(δ)) = Λ∗ (K) ⊗ S ∗ (C), and the proposition is true for the case q = 0. Now we consider the case of a general q, and begin by constructing a natural map α∗ : S ∗ (ker δ) ⊗S ∗ (Φ(ker d)) S ∗ (coker δ) → H∗ (Γ(δ, q)). The evident inclusion into the cycles ker δ ⊕ W ֒→ Z∗ (Γ(δ, q)) extends to an algebra map S ∗ (ker δ) ⊗ S ∗ (W ) → Z∗ (Γ(δ, q)) which is easily seen to descend to an algebra map α ˜ ∗ : S ∗ (ker δ) ⊗S ∗ (Φ(ker δ)) S ∗ (W ) → Z∗ (Γ(δ, q)). As the ideal (im δ) ⊂ Z∗ (Γ(δ, q)) is contained in the boundaries B∗ (Γ(δ, q)), α ˜ ∗ induces the needed algebra map α∗ : S ∗ (ker δ) ⊗S ∗ (Φ(ker δ)) S ∗ (coker δ) → H∗ (Γ(δ, q)). We show that α∗ is an isomorphism with a little spectral sequence argument. Filter Γ(δ, q) by powers of the augmentation ideal, i.e. let Fp = (V ⊕ W )p ⊂ Γ(δ, q), and consider the associated spectral sequence converging to H∗ (Γ(δ, q)). The generating relations in Γ(δ, q), q(φ(v)) = v 2 , for v ∈ V , imply that q(φ(v)) ≡ 0 mod F2 , so E 1 is identified with Γ(δ), and thus E 2 = H∗ (Γ(δ)) = Λ∗ (ker δ) ⊗ S ∗ (coker δ), by the q = 0 case of the proposition already discussed. As the generators ker δ ⊕coker δ are clearly permanent cycles in the image of α∗ , we see that the spectral sequence collapses at E 2 and α∗ is onto. Filtering the domain of α∗ by powers of its augmentation ideal reveals that

38

KUHN AND MCCARTY

it too has Λ∗ (ker δ) ⊗ S ∗ (coker δ) as an associated graded algebra, and we conclude α∗ is an isomorphism. 6. Examples We fill in some detail with some of the examples given in the introduction. 6.1. Generalized Eilenberg–MacLane spectra. We discuss Example 1.14 which concerned our spectral sequences for HA, where A is a graded abelian group. We begin by noting that all of our constructions behave well with respect to filtered colimits and direct sums in the variable A, so that the key cases to understand are when A = Σn Z and A = Σn Z/2r . Recall that H ∗ (HZ/2) = A, H ∗ (HZ) = A/ASq 1 , and H ∗ (HZ/2r ) = H ∗ (HZ) ⊕ ΣH ∗ (HZ) for r ≥ 2. For convenience, let A¯∗ = H∗ (HZ).   ΣZ/2 if n = 0 1−s+n ∞ ¯ A∗ = Z/2 Lemma 6.1. For all s > 0, Ωs Σ if n = −1   0 otherwise.

Proof. We work with the equivalent dual left A–module situation. Let F (n) = Ω∞ Σn A, the free unstable A–module on an n–dimensional class. (This is 0, if n < 0.) The module A/ASq 1 has a projective resolution ·Sq 1

·Sq 1

· · · → Σ2 A −−→ ΣA −−→ A → A/ASq 1 → 0. Applying Ω∞ Σ1−s+n yields the complex ·Sq 1

·Sq 1

· · · → F (1 − s + n + 2) −−→ F (1 − s + n + 1) −−→ F (1 − s + n). 1−s+n A ¯∗ is thus dual to the homology of The module Ω∞ s Σ ·Sq 1

·Sq 1

F (n + 2) −−→ F (n + 1) −−→ F (n). By inspection, one sees that this is exact except when n = −1 or 0.

Corollary 6.2. (a) Ls H∗ (Σn HZ/2) = 0 for all s > 0 and all n. ( Z/2 if n = 0 (b) For all s > 0, Ls H∗ (Σn HZ) = 0 otherwise. (c) For all s > 0 and r ≥ 2, n

r

n

Ls H∗ (Σ HZ/2 ) = Ls H∗ (Σ HZ) ⊕ Ls H∗ (Σ

n+1

( Z/2 HZ) = 0

if n = −1, 0 otherwise.

Now we need to know how the Dyer–Lashof operation Q0 acts. Lemma 6.3. Q0 : Ls H∗ (HZ) → Ls+1 H∗ (HZ) is an isomorphism for all s ≥ 0.


39

Proof. The key point is that the exact sequence q0 1−s ¯ sq0 1−s ¯ −s ¯ Ω∞ A∗ −−→ Φ(Ω∞ A∗ ) −→ ΣΩ∞ s Σ s Σ s+1 Σ A∗

identifies with the exact sequence sq0

q0

ΣZ/2 −−→ Σ2 Z/2 −→ Σ2 Z/2. As the first map here is clearly zero, the second is an isomorphism.

This lemma and the previous corollary combine to give us the next calculations. Corollary 6.4. (a) UQ (L∗ H∗ (HZ)) = Z/2[x] where x is the nonzero 0 dimensional class in Ω∞ H∗ (HZ). (b) For r ≥ 2, UQ (L∗ H∗ (HZ/2r )) = Z/2[x] ⊗ Λ∗ (y) where x and y are the nonzero 0 and 1 dimensional classes in Ω∞ H∗ (HZ/2r ) . (c) For r ≥ 2, UQ (L∗ H∗ (Σ−1 HZ/2r )) = Z/2[y] where y is the nonzero 0 dimensional class in Ω∞ Σ−1 H∗ (HZ/2r ). Now we use our calculations to determine how the topological spectral sequence behaves for HZ, HZ/2r , and Σ−1 HZ/2r , for r ≥ 2. alg,∞ 6.2. The spectral sequence for HZ. For HZ, E∗,∗ (H∗ (HZ)) = Z/2[x]. ∞ ∞ As x is in the image of ǫ∗ : H∗ (Ω HZ) → Ω H∗ (HZ), all the xn are infinite cycles, and we conclude that there can be no rogue differentials. The spectral sequence converges to the correct answer as well as possible: H∗ (Ω∞ HZ) = H∗ (Z) = Z/2[t, t−1 ], lim H∗ (Pd (HZ)) = Z/2[[x]], and the d

former embeds densely in the latter via the homomorphism sending t to x + 1. 6.3. The spectral sequence for HZ/2r with r ≥ 2. For HZ/2r , with alg,∞ r ≥ 2, E∗,∗ (H∗ (HZ/2r )) = Z/2[x] ⊗ Λ∗ (y). This time only x is in the image of ǫ∗ , so there might be a rogue differential off of y. The elements s x2 are the only nonzero 0 dimensional primitive classes in E 1 , so the first rogue differential must hit one of these. r r We claim that d2 −1 (y) = x2 , this is the only rogue differential, and ∞ (HZ/2r ) = Z/2[x]/(x2r ). Furthermore, the spectral sequence converges E∗,∗ r to the correct answer: H∗ (Ω∞ HZ/2r ) = H∗ (Z/2r ) = Z/2[t]/(t2 − 1) = r Z/2[x]/(x2 ), when t = x + 1. To prove the claim, we first make some observations about the beginning of the spectral sequence in low degrees. In total degree 0, E 1 is spanned by the classes xn , and in total degree 1, E alg,1 is spanned by the classes xn y, and xn Q1 x. If z ∈ H∗ (HZ/2r ) is the two dimensional class with zSq 2 = x, then d1 (xn z) = xn Q1 x. It follows that the only classes in E 2 in degrees 0 and 1 will be xn and xn y, none of which can possibly be in the image of an algebraic differential.

40

KUHN AND MCCARTY r

∞ (HZ/2r ). To see this, we consider the We now show that x2 = 0 in E∗,∗ diagram

Z/2[t, t−1 ]

/ Z/2[[x]]

/ limd H∗ (Pd (HZ/2r ))

r

Z/2[t]/(t2 − 1)

r

r

in which both horizontal maps send t − 1 to x. As (t − 1)2 = t2 − 1 = 0 r r in Z/2[t]/(t2 − 1), we see that x2 = 0 in lim H∗ (Pd (HZ/2r )), and thus in d

∞ (HZ/2r ). E∗,∗ s Finally we show that x2 6= 0 for all s < r, or equivalently, that y lives r to E 2 −1 . This we show by induction on r. The r = 2 case is true because alg,∞ d1 (y) = 0. For the inductive step, let E∗,∗ (Z/2r−1 ) = Z/2[x′ ] ⊗ Λ∗ (y ′ ). The inclusion Z/2r−1 → Z/2r induces a map of both the topological and algebraic spectral sequences sending x′ to 0, and y ′ to y. Then the inductive 2r−1 −1 (HZ/2r−1 ) and d2r−1 −1 (y ′ ) = (x′ )2r−1 hypothesis — that y ′ lives to E∗,∗ 2r−1 −1 (HZ/2r ) and d2r−1 −1 (y) = 0, i.e. y lives — implies that y lives to E∗,∗ 2r−1 (HZ/2r ), and thus to E 2r −1 (HZ/2r ). to E∗,∗ ∗,∗

6.4. The spectral sequence for Σ−1 HZ/2r with r ≥ 2. Our most complicated example is the spectral sequence for Σ−1 HZ/2r , with r ≥ 2. Let x and y be the nonzero classes in H∗ (Σ−1 HZ/2r ) of dimensions −1 alg,∞ and 0. E∗,∗ (H∗ (HZ/2r )) = Z/2[y], and obviously y is not in the image of ǫ∗ . The only primitive elements in E 1 of total degree -1 are the elements 1 (Q0 )s x ∈ E−2 s ,2s −1 , so a first rogue differential must hit one of these. r r We claim that y lives to E 2 , and d2 −1 (y) = (Q0 )r x. To see this, we compare this example to our previous one, using the map of spectral sequences induced by ΣP (Σ−1 HZ/2r ) → P (HZ/2r ). This sends the elements x and y to the elements with the same name in the last example. It also induces an isomorphism from the primitives of total degree -1 in E 1 (Σ−1 HZ/2r ) to the primitives of total degree 0 in E 1 (HZ/2r ). r r The calculation that d2 −1 (y) = x2 = (Q0 )r x in the spectral sequence for r HZ/2r then implies that d2 −1 (y) = (Q0 )r x in the spectral sequence for Σ−1 HZ/2r . r The formula d2 −1 (y) = (Q0 )r x then implies that, for any s ≥ 0, s (2r −1)

d2

s

s (2r −1)

(y 2 ) = d2

r −1

((Q0 )s y) = (Q0 )s d2

(y) = (Q0 )s+r x.

We also note that d1 (x) = Q−1 x = x2 , and it follows that, for any s ≥ 0, s

d2 ((Q0 )s x) = (Q0 )s Q−1 x = Q−1 (Q0 )s x = ((Q0 )s x)2 . We now explain how these calculations completely determine how the algebraic and topological spectral sequences differ. Let xs = (Q0 )s x. Using


41

the standard primitive generators, the E 1 term of both spectral sequences decomposes: E 1 = Z/2[y, x0 , x1 , x2 , . . . ] ⊗ E ⊥,1 . This, in fact, represents a decomposition of both spectral sequences, where the algebraic and topological spectral sequences agree on E ⊥,∗ , and the differentials on Z/2[y, x0 , x1 , x2 , . . . ] go as follows: s • The algebraic spectral sequence has d2 (xs ) = x2s . s r s • The topological spectral sequence also has d2 (2 −1) (y 2 ) = xs+r . It is then easy to compute that, for all s ≥ 0, s

s

E alg,2 = Z/2[y, xs , xs+1 , xs+2 , . . . ] ⊗ E ⊥,2 , while, for all s ≥ r, s

s+1−r

E top,2 = Z/2[y 2

s

, xs+1 , xs+2 , . . . ] ⊗ E ⊥,2 .

6.5. A rogue differential for a 0-connected finite complex. We discuss Example 1.17. Let the spectrum X be the cofiber of 4 : RP 4 → RP 4 , so that X fits into a cofibration sequence RP 4 → X → ΣRP 4 . As 4 has Adams filtration 2, we are guaranteed that H∗ (X) ≃ H∗ (RP 4 ∨ ΣRP 4 ) ≃ H∗ (RP 4 ) ⊕ ΣH∗ (RP 4 ), as right A–modules. For i = 1, 2, 3, 4, let ai ∈ Hi (X) be the image of the nonzero element under the inclusion RP 4 ֒→ X, and let bi ∈ Hi+1 (X) project to a nonzero element under the projection X → ΣRP 4 . ∞ (X) = As H∗ (X) ∈ U , if there were no rogue differentials, then E∗,∗ 1 (X). We show this is impossible. E∗,∗ Proposition 6.5. In the spectral sequence, d3 (b4 ) = a41 . Before proving this, we note some properties that X must (not) have. Lemma 6.6. X is not homotopy equivalent to RP 4 ∨ ΣRP 4 . Proof. This follows easily from the fact that the identity on RP 4 has stable order 8, not 4 [T]. Corollary 6.7. ǫ∗ : H∗ (Σ∞ Ω∞ X) → H∗ (X) is not onto. Proof. RP 4 ∨ ΣRP 4 is the wedge of two (dual) Brown–Gitler spectra, and thus is homotopy equivalent to any other 2–complete connective spectrum Y with isomorphic mod 2 homology such that ǫ∗ : H∗ (Σ∞ Ω∞ Y ) → H∗ (Y ) is onto [HK]. Proof of Proposition 6.5. Figure 1 shows the −1 line, and the bottom nonzero 1 (X) for the spectral sequence convergelements in the next few lines, of E∗,∗ ∞ ing to H∗ (Ω X).

42

KUHN AND MCCARTY

a41 a31 a21

−4 −3 −2

b4 a4 , b3 a3 , b2 a2 , b1 a1

−1

8 7 6 5 4 3 2 1 1 0 0 s\t

1 (X) Figure 1. Es,t

Recalling that d1 ≡ 0, and that differentials take primitives to primitives, the only possible nonzero differential off of the −1 line would be d3 (b4 ) = a41 . ∞ (X) = Thus if d3 (b4 ) = a41 did not hold, then we could conclude that E−1,∗ ∞ ∞ 1 E−1,∗ (X), so that ǫ∗ : H∗ (Σ Ω X) → H∗ (X) would be onto, contradicting the corollary. Appendix A. Proof of Proposition 2.1 We need to explain the last property of S–modules listed in Proposition 2.1. This said that, given an S–module X, there is a weak natural equivalence hocolim Σ−n Σ∞ Xn → X. n

We thank Mike Mandell for helping us be accurate in the following discussion. Xn . Recall that an S– Let Σ∞ n : T → Spectra be left adjoint to X module is a special sort of L–module. The functor sending a spectrum X to the S–module S ∧L LX is left adjoint to the functor sending an S–module X to FL (S, X), just regarded as a spectrum (and not as an L–module). There is a weak equivalence of S–modules −n ∞ S ∧L L(Σ∞ Σ Xn n Xn ) → Σ

given as the adjoint to the composite of maps of spectra −n ∞ Σ∞ Σ Xn → Σ−n FL (S, Σ∞ Xn ) = FL (S, Σ−n Σ∞ Xn ). n Xn → Σ

There is a map of S–modules S ∧L L(Σ∞ n Xn ) → X given as the adjoint to the composite of maps of spectra ∞ Σ∞ n Xn → Σn FL (S, X)n → FL (S, X).


43

The desired weak natural equivalence is obtained by taking the hocolimit over n of the zig-zag ∼

Σ−n Σ∞ Xn ← − S ∧L L(Σ∞ n Xn ) → X We note that the n = 0 case of the zig-zag here has the form ∼

Σ∞ Ω∞ X ← − S ∧L L(Σ∞ Ω∞ X) → X, which induces the evaluation (counit) map in the homotopy category.

Appendix B. The tower P (X) with its operad action We explain how the results of [AK] show that the operad C∞ acts suitably on the tower P (X) as described in Theorem 3.8. The paper [AK] explored the explicit model from [Ar] for the tower associated to the functor sending a space Z to the spectrum Σ∞ + Map(K, Z), where K is a fixed CW complex. Call this tower P (K, Z), indicating its functoriality in both variables. (The more awkward notation P K (X) was used in [AK].) It comes with a natural transformation e : Σ∞ + Map(K, Z) → P (K, Z) which is an equivalence if the dimension of K is less than the connectivity of Z. We note that the properties of our category of spectra needed to form our constructions correspond to the first five properties of S listed in Proposition 2.1. The product theorem, [AK, Thm.1.4], says that there is a weak natural equivalence of towers ∼

P (K ∨ L, Z) − → P (K) ∧ P (L). This generalizes W to more than two factors in a straightforward way. In particular, if d K denotes the wedge of d copies of K, there is a Σd – equivariant map of towers of spectra _ P ( K, X) → P (K)∧d d

which is a nonequivariant equivalence. n Specialized to K = S n , one gets a tower P (S n , Z) approximating Σ∞ +Ω Z −n ∧d with dth fiber naturally weakly equivalent to Cn (d)+ ∧Σd (Σ Z) , as expected. Here Cn is the little n–cubes operad. The naturality and continuity of the P (K, Z) construction in the variable K make it quite easy to define maps of towers _ Θ(d) : Cn (d)+ ∧Σd P ( S n , Z) → P (S n , Z) d

44

KUHN AND MCCARTY

compatible with the usual Cn operad action on Ωn Z [Ma]. In particular, from [AK, Thm.1.10], we learn that the square in the diagram n d Σ∞ + Cn (d) ×Σd (Ω Z)

Θ(d)

n / Σ∞ +Ω Z

Θ(d)

Cn (d)+ ∧Σd e∧d

Cn (d)+ ∧Σd P (S n , Z)∧d o

∼

Cn (d)+ ∧Σd

W P ( d S n , Z)

e

/ P (S n , Z)

commutes. Furthermore, the map on fibers induced by the map of towers corresponds to the maps induced by the operad structure in the expected way. Given a spectrum X, our tower is then defined to be P (X) = hocolim P (S n , Xn ), n

where the homotopy colimit is over natural transformations ∧

P (S n , Xn ) − → P (S n+1 , ΣXn ) → P (S n+1 , Xn+1 ). Here the first map is the smashing map from [AK, Thm.1.1]. The dth fiber of the tower P (X) then naturally identifies with hocolim Cn (d)+ ∧Σd (Σ−n Σ∞ Xn )∧d ≃ C∞ (d)+ ∧Σd X ∧d = Dd X. n

∞ Finally the weak natural transformation e : Σ∞ + Ω X → P (X) is defined as the composite hocolim e

∼

n ∞ n n Σ∞ − hocolim Σ∞ + Ω Xn −−−−−−→ hocolim P (S , Xn ), +Ω X ←

n

n

and the diagram of Theorem 3.8 is obtained by taking the hocolimit over n of diagrams as above (with d specialized to 2). References [ACD] A. Adem, R. L. Cohen, and W. G. Dwyer, Generalized Tate homology, homotopy fixed points and the transfer, Algebraic topology (Evanston, 1988), A.M.S. Cont. Math. 96 (1989), 1–13. [AK] S. T. Ahearn and N. J. Kuhn, Product and other fine structure in polynomial resolutions of mapping spaces, Alg. Geom. Topol. 2 (2002), 591–647. [Ar] G. Arone, A generalization of Snaith–type filtration, Trans. A.M.S. 351 (1999), 1123– 1250. [BCKQRS] A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, and J. W. Schlesinger, The mod–p lower central series and the Adams spectral sequence, Topology 5 (1966), 331–342. [BG] E. H. Brown and S. Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283–295. [BMMS] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, H∞ ring spectra and their applications, Springer L. N. Math. 1176, 1986. [CLM] F. R. Cohen, T. J. Lada, and J. P. May, The homology of iterated loop spaces, Springer L. N. Math. 533, 1976. [EKMM] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, A.M.S. Math. Surveys and Monographs 47, 1997. [Gab] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math France 90 (1962), 323–448.


45

[Goe] P. G. Goerss, Unstable projectives and stable Ext: with applications, Proc. L.M.S. 53 (1986), 539–561. [Goo] T. G. Goodwillie, Calculus III: Taylor series, Geometry and Topology 7 (2003), 645–711. [GM] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Memoirs A.M.S. 113 (1995), no. 543. [HM] J. R. Harper and H. R. Miller, Looping Massey–Peterson towers, Advances in homotopy theory (Cortona, 1988), L.M.S. Lecture Note Series 139 (1989), 69–86. [HK] D. J. Hunter and N. J. Kuhn, Characterizations of spectra with U–injective cohomology which satisfy the Brown–Gitler property, Trans.A.M.S. 352 (1999), 1171–1190. [K1] N. J. Kuhn, Chevalley group theory and the transfer in the homology of symmetric groups, Topology 24 (1985), 247–264. [K2] N. J. Kuhn, Tate cohomology and periodic localization of polynomial functors, Invent. Math. 157 (2004), 345–370. [K3] N. J. Kuhn, Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology, Alg. Geo. Topol. 8 (2008), 2109–2129. (Correction: 10 (2010), 531–533.) [LZ1] J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Zeit. 194 (1987), 25–59. (Correction: 10 (2010).) [LZ2] J. Lannes and S. Zarati, Invariants de Hopf d’order superieur et la suite spectrale ´ d’Adams, preprint Ecole Polytechnique, mid 1980’s. [LMMS] L. G. Lewis, J. P. May, J. McClure, and M. Steinberger, Equivariant stable homotopy theory, Springer L. N. Math. 1213, 1986. [Ma] J. P. May, The geometry of iterated loop spaces, Springer L. N. Math. 271, 1972. [McC] R. McCarthy, Dual calculus for functors to spectra, Homotopy methods in algebraic topology (Boulder, 1999), A.M.S. Contemp. Math. Series 271 (2001), 183–215. [P] G. M. L. Powell, On the derived functors of destabilization at odd primes, arXiv:1101.0226. [S] L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Math., University of Chicago Press, 1994. [Si] W. M. Singer, Iterated loop functors and the homology of the Steenrod algebra II: a chain complex for Ωks M , J.P.A.A. 16 (1980), 85–97. [T] H. Toda, Order of the identity class of a suspension space, Ann. Math. 78 (1963), 300–325. [W] C. A. Weibel, An introduction to homological algebra, Camb. Studies Adv. Math. 38, 1994. Department of Mathematics, University of Virginia, Charlottesville, VA 22904 E-mail address: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22904 E-mail address: [email protected]

THE MOD 2 HOMOLOGY OF INFINITE LOOPSPACES

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