The mod 2 homology of infinite loopspaces
aa r X i v : . [ m a t h . A T ] J un THE MOD 2 HOMOLOGY OF INFINITE LOOPSPACES
NICHOLAS J. KUHN AND JASON B. MCCARTY
Abstract.
Applying mod 2 homology to the Goodwillie tower of thefunctor sending a spectrum X to the suspension spectrum of its 0thspace, leads to a spectral sequence for computing H ∗ (Ω ∞ X ; Z / X is 0–connected. The E term is the homol-ogy of the extended powers of X , and thus is a well known functorof H ∗ ( X ; Z / A , and a left module overthe Dyer–Lashof operations. This paper is an investigation of how thisstructure is transformed through the spectral sequence.Hopf algebra considerations show that all pages of the spectral se-quence are primitively generated, with primitives equal to a subquotientof the primitives in E .We use an operad action on the tower, and the Tate construction, todetermine how Dyer–Lashof operations act on the spectral sequence. Inparticular, E ∞ has Dyer–Lashof operations induced from those on E .We use our spectral sequence Dyer–Lashof operations to determinedifferentials that hold for any spectrum X . The formulae for these uni-versal differentials then lead us to construct an algebraic spectral se-quence depending functorially on an A –module M . The topologicalspectral sequence for X agrees with the algebraic spectral sequence for H ∗ ( X ; Z /
2) for many spectra X , including suspension spectra and al-most all Eilenberg–MacLane spectra. The E ∞ term of the algebraicspectral sequence has form and structure similar to E , but now theright A –module structure is unstable. Our explicit formula involves thederived functors of destabilization as studied in the 1980’s by W. Singer,J. Lannes and S. Zarati, and P. Goerss. Introduction and main results
An infinite loopspace is a space of the form Ω ∞ X , the 0th space of afibrant spectrum X . Thus X consists of a sequence of spaces X , X , X , . . . ,together with homotopy equivalences X n ∼ −→ Ω X n +1 , and Ω ∞ X = X .The homology of X is defined by letting H ∗ ( X ) = colim n H ∗ + n ( X n ). Welet all homology be with mod 2 coefficients, and consider the following basicproblem. Problem 1.1.
How can one compute H ∗ (Ω ∞ X ) from knowledge of H ∗ ( X )? Date : September 30, 2012.2000
Mathematics Subject Classification.
Primary 55P47; Secondary 55S10, 55S12,55T99.This research was partially supported by National Science Foundation grant 0967649.
The graded vector space H ∗ ( X ) has a minimum of extra structure: it isan object in M , the category of locally finite right modules over the mod2 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 algebrasin the abelian category of left modules over the Dyer–Lashof algebra withcompatible unstable right A –module structure.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 surprisethat such a functor doesn’t exist, and we will see examples illustrating this.However, one punchline of this paper is that one can come surprisingly close.Our method is to carefully study the left half plane spectral sequence { E r ∗ , ∗ ( X ) } associated to Goodwillie tower of the functor X Σ ∞ Ω ∞ X .This converges strongly to H ∗ (Ω ∞ X ) when X is 0–connected, and has an E term that is a well known functor of H ∗ ( X ), including structure asa primitively generated bigraded Hopf algebra, with Steenrod operationsacting vertically on the right, and Dyer–Lashof operations acting on the leftand doubling horizontal grading.It is formal that pages of the spectral sequence will again be primitivelygenerated bigraded Hopf algebras equipped with Steenrod operations, butwe prove a more subtle phenomenon: for all r , the bigraded module of E r primitives will be a subquotient of the module of the E primitives. Itfollows that d r can be nonzero only when r = 2 t − s for some 0 ≤ s < t .We then determine universal differentials. After identifying d , we deducehow this propagates to give information about higher differentials. We areable to do this by using the Z / E ∞ operad. Another consequence is that the module of E ∞ primitives has a left action by Dyer–Lashof operations induced from theaction on E , though, curiously, this is not true for the intervening pages.Guided by our formula for universal differentials, we then construct, for M ∈ M , an algebraic spectral sequence depending functorially on M . Thetopological spectral sequence for X agrees with the algebraic spectral se-quence for H ∗ ( X ; Z /
2) for many spectra X , including suspension spectraand almost all Eilenberg–MacLane spectra, and may be a subquotient ingeneral. Our algebraic functor E alg, ∞∗ , ∗ ( M ) takes values in the category HQU ,and is built out of the derived functors of ‘destabilization’ which were thesubject of much research in the 1980’s by W. Singer [Si], J. Lannes andS. Zarati [LZ1], and P. Goerss [Goe].We now introduce our cast of characters, then describe our results in moredetail.1.1.
The tower for Σ ∞ + Ω ∞ X . We let T denote the category of based topo-logical spaces and S the category of S –modules as in [EKMM]. The suspen-sion 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 OD 2 HOMOLOGY OF INFINITE LOOPSPACES 3 for the suspension spectrum of Z + , the union of the space Z with a disjointbasepoint. Σ ∞ + Z comes with a natural augmentation map Σ ∞ + Z → Σ ∞ + ∗ = S to the sphere spectrum.T. Goodwillie’s general theory of the calculus of functors [Goo], appliedto the endofunctor of S sending X to Σ ∞ + Ω ∞ X , yields a natural tower P ( X )of fibrations augmented over S : ... (cid:15) (cid:15) P ( X ) (cid:15) (cid:15) P ( X ) (cid:15) (cid:15) Σ ∞ + Ω ∞ X e / / e ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ e ttttttttttttttttttttttttt P ( X ) . Basic properties include the following. • P ( X ) identifies with the product of spectra X × S , and the mapinduced by e on fibers of the augmentation corresponds to the eval-uation Σ ∞ Ω ∞ X → X . • The fiber of the map P d ( X ) → P d − ( X ) is naturally equivalent tothe spectrum D d X = ( X ∧ d ) h Σ d , the d th extended power of X . • If X is 0–connected, then e j is j –connected.We occasionally make use of the reduced tower ˜ P ( X ) defined by letting˜ P d ( X ) be the fiber of the augmentation P d ( X ) → S . There is a naturalequivalence P d ( X ) ≃ ˜ P d ( X ) × S for all d .1.2. The spectral sequence.
Applying mod 2 homology to the tower P ( X ) yields a left half plane spectral sequence { E r ∗ , ∗ ( X ) } . Proposition 1.2.
The spectral sequence satisfies the following properties. (a)
It converges strongly to H ∗ (Ω ∞ X ) when X is –connected. (b) E − d,d + ∗ ( X ) = H ∗ ( D d X ) . (c) The Steenrod algebra A acts on the columns of the spectral sequence. (d) Dyer–Lashof operations Q i , for all i ∈ Z , act on E ∗ , ∗ ( X ) , where theytake the form Q i : E − d,d + j = H j ( D d X ) → H j + i ( D d X ) = E − d, d + i + j . (e) The spectral sequence is a spectral sequence of Hopf algebras. The prod-uct and coproduct on E ∞ are induced by the H-space product and diagonal KUHN AND MCCARTY on Ω ∞ X , while the product and coproduct on E are induced by the mul-tiplication maps D b X ∧ D c X → D b + c X and the transfer maps D b + c X → D b X ∧ D c X associated to the subgroup inclusions Σ b × Σ c ֒ → Σ b + c . The first three listed properties are evident from the properties of thetower. Using G. Arone’s explicit model for this tower [Ar], further propertieswere explored in [AK], including property (e). There it was also shown thatthe action of the little cubes operad C ∞ on the infinite loop space Ω ∞ X induces a corresponding action on the tower. This leads to the Dyer–Lashofoperations of property (d). How these operations correspond to the Dyer-Lashof operations on H ∗ (Ω ∞ X ) at the level of E ∞∗ , ∗ ( X ), and how they acton the rest of the spectral sequence is part of the content of Theorem 1.6below.1.3. Lots of categories and a global description of E . We need to de-scribe our spectral sequence E term in a more global and functorial manner.We assume some familiarity with Dyer–Lashof operations and Steenrod op-erations, but see § § • M is the category of locally finite right A –modules. The Steenrodsquares go down in degree: given x ∈ M ∈ M , | xSq i | = | x | − i . Aright A –module M is locally finite if, for all x ∈ M , x · A is finitedimensional over Z / • U is the full subcategory of M consisting of modules satisfying theunstable condition: xSq i = 0 whenever 2 i > | x | . • Q is the category of graded vector spaces M acted on by Dyer–Lashofoperations Q i : M d → M d + i , for i ∈ Z , satisfying the Adem relationsand the unstable condition: Q i x = 0 whenever i < | x | . • QM is the full subcategory of M ∩ Q consisting of objects whoseDyer–Lashof structure is intertwined with the Steenrod structure viathe Nishida relations. • QU = QM ∩ U .All these categories are abelian, and admit tensor products, via the Cartanformula for both Steenrod and Dyer–Lashof operations. Then we definevarious categories of Hopf algebras. • HM is the category of bicommutative Hopf algebras in M . • HQM is the category of bicommutative Hopf algebras in QM sat-isfying the Dyer–Lashof restriction axiom: Q | x | x = x . • HQU = HQM ∩ U .We also need two ‘free’ functors. • R ∗ : M → QM is left adjoint to the forgetful functor. Explicitly, R ∗ M = L ∞ s =0 R s M where R s : M → M is given by R s M = h Q I x | l ( I ) = s, x ∈ M i / (unstable and Adem relations) . OD 2 HOMOLOGY OF INFINITE LOOPSPACES 5
Here, if I = ( i , . . . , i s ), Q I x = Q i · · · Q i s x , and l ( I ) = s . • U Q : QM → HQM is left adjoint to the functor taking an object H ∈ HQM to its module P H of primitives. Explicitly, U Q ( M ) = S ∗ ( M ) / ( Q | x | x − x ) , where S ∗ ( M ) is the free commutative algebra generated by M , andthe coalgebra structure is determined by making M primitive.The classic calculation of the mod 2 homology of extended powers can beinterpreted as saying the following. Proposition 1.3.
The natural maps R s ( H ∗ ( X )) → H ∗ ( D s ( X )) are inclu-sions, and induce an isomorphism U Q ( R ∗ ( H ∗ ( X ))) = ∞ M d =0 H ∗ ( D d X ) as objects in HQM . We begin our study of the spectral sequence with the following.
Theorem 1.4. (a) H ∗ (Ω ∞ X ) is an object in HQU . (b) E ∗ , ∗ ( X ) = U Q ( R ∗ ( H ∗ ( X ))) as an object in HQM , and under this iden-tification, the module of primitives R s ( H ∗ ( X )) is contained in E − s , s + ∗ ( X ) . (c) Steenrod operations act vertically, while Dyer–Lashof operations doublethe horizontal degree. (d)
Each E r ∗ , ∗ ( X ) is an object in HM , and each d r is A –linear and both aderivation and coderivation. (e) Each E r ∗ , ∗ ( X ) is primitively generated, and its bigraded module of prim-itives V r ∗ , ∗ ( X ) will be an A –module subquotient of R ∗ ( H ∗ ( X )) . (f ) The only possible nonzero differentials will be d r when r = 2 t − s , with t > s ≥ , and d t − s will be determined by its restriction to the primitivesin E t − s − s , ∗ ( X ) . The first four properties listed here are just restatements of parts of thelast two propositions. We will see that property (e) follows from these: itis standard that each E r will be primitively generated since this is true for E , but in our case we can also control where the E r primitives occur. SeeProposition 5.5 and the subsequent discussion. Property (f) follows from(e), as d r will send primitives to primitives. Remark . The careful reader may note that H ∗ (Ω ∞ X ) satisfies one morecondition than has been described: the dual of the classic restriction axiom KUHN AND MCCARTY for unstable A –algebras, Sq | x | x = x . This property is not preserved by thefiltration on H ∗ (Ω ∞ X ). The good news is then that this extra structure isavailable to be used to help determine extension problems.1.4. Universal differentials.
We now identify universal structure on thespectral sequence.
Theorem 1.6.
For all spectra X , the following hold in { E r ∗ , ∗ ( X ) } . (a) For all x ∈ H ∗ ( X ) , d ( x ) = X i ≥ Q i − ( xSq i ) . (b) If y ∈ H ∗ ( D d X ) lives to E r , and d r ( y ) is represented by z ∈ H ∗ ( D d + r X ) ,then Q i y ∈ H ∗ ( D d X ) lives to E r , and d r ( Q i y ) is represented by Q i ( z ) ∈ H ∗ ( D d +2 r X ) . (c) If y ∈ H ∗ ( D d X ) represents z ∈ H ∗ (Ω ∞ X ) in E ∞− d, ∗ ( X ) , then Q i y ∈ H ∗ ( D d X ) represents Q i z ∈ H ∗ (Ω ∞ X ) in E ∞− d, ∗ ( X ) . A consequence of the first two parts of the theorem is the following iden-tification of universal differentials.
Corollary 1.7.
For all spectra X , x ∈ H ∗ ( X ) , and I of length s , Q I x livesto E s − s , ∗ ( X ) and d s ( Q I x ) = X i ≥ Q I Q i − ( xSq i ) ∈ E s − s +1 , ∗ ( X ) . To further give context to what Theorem 1.6 says about how Dyer–Lashofoperations work in the spectral sequence, let = B ⊆ B ⊆ · · · ⊆ B r ⊆ · · · ⊆ Z r ⊆ · · · ⊆ Z ⊆ Z = E be cycles and boundaries as usual, so that E r = Z r /B r . Then Theo-rem 1.6(b) implies that for all r , Dyer–Lashof operations on E restrictto maps Q i : Z r → Z r and Q i : B r → B r − . As Z r /B r − both includes into E r − and projects onto E r , one getsDyer–Lashof operations of two flavors: Q i : E r → E r − and Q i : E r → E r . This discussion holds when r = ∞ , leading to the next corollary. Corollary 1.8.
For all spectra X , E ∞∗ , ∗ ( X ) ∈ HQM , with structure inducedfrom E : E ∞∗ , ∗ ( X ) = U Q ( V ∞∗ , ∗ ( X )) , with the bigraded module of primitives V ∞∗ , ∗ ( X ) ∈ QM equal to a subquotient of R ∗ ( H ∗ ( X )) . To the extent that thespectral sequence converges, this structure is also induced from H ∗ (Ω ∞ X ) .Remark . Though both E ∗ , ∗ ( X ) and E ∞∗ , ∗ ( X ) admit Dyer-Lashof oper-ations, Q i is not generally defined on the intervening pages, E r ∗ , ∗ ( X ) with OD 2 HOMOLOGY OF INFINITE LOOPSPACES 7 < r < ∞ , with the exception of the squaring operation x Q | x | ( x ), whichplays a special role in the Hopf algebra theory. See § Remark . Since H ∗ (Ω ∞ X ) is always an unstable A –module, it followsthat if X is 0–connected, then E ∞∗ , ∗ ( X ) ∈ HQU . We wonder if this is thecase for all spectra X .Theorem 1.6 will be proved in §
3, supported by the results in the precedingbackground section. We briefly comment on the proof.Statement (a) amounts to a calculation of δ ∗ , where δ : X → Σ D X isthe connecting map of the cofibration sequence D X → ˜ P X → ˜ P X ≃ X .When | x | >
0, this was calculated (in dual formulation) by the first authorin [K3] by means of universal example, and it is not too hard to extend thisto all x .We give proofs of statements (b) and (c) that show that versions of thesestatements will hold in the spectral sequence associated to any tower ofspectra admitting an action of the operad C ∞ . The key idea is to use the(once desuspended) Z / P d ( X ) ∧ ) h Z / = D ( P d ( X )) → P d ( X )compatible with the standard maps D ( D d ( X )) → D d ( X ). These don’texist, but we will show that one does have maps t Z / ( P d ( X ) ∧ ) → P d ( X )which do the job, where t Z / ( Y ∧ ) ≃ holim n Σ n D (Σ − n Y ), the colineariza-tion of D ( Y ) in McCarthy’s sense [McC]. A technical point is that, at anappropriate moment, we need to pass from towers of S –modules to towersof H Z / An algebraic spectral sequence.
We now build an algebraic spectralsequence using only the differentials given by the formula in Corollary 1.7.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, Ω M is the largest unstable submodule of Σ − M . • The functor Ω ∞ is left exact, and we let Ω ∞ s : M → U denote theassociated right derived functors.It is convenient to let L s M = ΩΩ ∞ s Σ − s M . (Note that L M = Ω ∞ M .)We observe that these functors to the category U have extra structure. KUHN AND MCCARTY
Proposition 1.11.
There are natural operations Q i : L s M → L s +1 M giving L ∗ M the structure of an object in QU ; indeed, L ∗ M is a natural subquotientof R ∗ M , viewed as an object in QM . Theorem 1.12.
For all M ∈ M , there is a left half plane spectral sequence { E alg,r ∗ , ∗ ( M ) } described by the following properties. (a) The spectral sequence is a functor of M taking values in HM , withSteenrod operations acting vertically, and with each d r both a derivation andcoderivation. (b) E alg, ∗ , ∗ ( M ) = U Q ( R ∗ M ) as an object in HQM , with the module of prim-itives R s M contained in E alg, − s , s + ∗ ( M ) . (c) d r is not zero only when r = 2 s , and d s is determined by the formulaein Corollary 1.7: for x ∈ M and I of length s , Q I x lives to E alg, s − s , ∗ ( M ) , and d s ( Q I x ) = X i ≥ Q I Q i − ( xSq i ) . (d) For all r , E alg,r ∗ , ∗ ( M ) is primitively generated with nonzero primitivesconcentrated in the − s lines. For all r > s , the module of primitives in E alg,r − s , s + ∗ ( M ) is naturally isomorphic to L s M . (e) E alg, ∞∗ , ∗ ( M ) ≃ U Q ( L ∗ M ) as an object in HQU . For a spectrum X , the spectral sequences { E r ∗ , ∗ ( X ) } and { E alg,r ∗ , ∗ ( H ∗ ( X )) } will agree exactly when all topological spectral sequence differentials d t − s with s < t − d t − s with s < t − rogue differential. Corollary 1.13. If d t − s is a rogue differential with t − s smallest, thenrestricted to the primitives on the − s line, it will be a nonzero map L s H ∗ ( X ) → E t − s − t , t + ∗− ( X ) . Recalling that L H ∗ ( X ) = Ω ∞ H ∗ ( X ), it follows that rogue differentialsoff of the − H ∗ (Ω ∞ X ) → Ω ∞ H ∗ ( X ) to be onto. Theorem 1.6(b) and the last corollary then tell usthat in some circumstances this can be the only source of rogue differentials. Corollary 1.14.
Suppose the following conditions hold for a spectrum X . (a) Ω ∞ H ∗ ( X ) = L H ∗ ( X ) generates L ∗ H ∗ ( X ) as a module over the Dyer–Lashof algebra. OD 2 HOMOLOGY OF INFINITE LOOPSPACES 9 (b)
The evaluation map H ∗ (Ω ∞ X ) → Ω ∞ H ∗ ( X ) is onto.Then { E r ∗ , ∗ ( X ) } = { E alg,r ∗ , ∗ ( H ∗ ( X )) } , and thus E ∞∗ , ∗ ( X ) ≃ U Q ( L ∗ H ∗ ( X ))) asan object in HQU .Remark . One might wonder if E ∞∗ , ∗ ( X ) is always a subquotient of E alg, ∞∗ , ∗ ( H ∗ ( X )) = U Q ( L ∗ H ∗ ( X ))). Our results say that the algebraic bound-aries, B alg, ∞ , are contained in the topological boundaries B ∞ . To con-clude that E ∞ is a subquotient of E alg, ∞ , it suffices to show that Z ∞ ⊆ B ∞ + Z alg, ∞ , when restricted to primitives.The development of our algebraic spectral sequence, and the proof of itsproperties as in Theorem 1.12, is given in §
5. Besides using the Hopf algebratheory needed in the proof of Theorem 1.4(e), this relies heavily on §
4, whichis focused on the connection between R s and Ω ∞ s . We say a bit about thisconnection here.We relabel: let R s = Σ R s Σ s − : M → M . Let d s : R s M → R s +1 M begiven by the formula d s ( Q I x ) = X i ≥ Q I Q i − ( xSq i ) , where we have suppressed some suspensions. The next theorem is a variantof theorems in [Goe] and [P]. All such results go back to work of Singer [Si]for inspiration. Theorem 1.16.
For all M ∈ M , R M d −→ R M d −→ R M d −→ · · · is a chain complex with H s ( R ∗ M ; d ) naturally isomorphic to Ω ∞ s M .Remark . Recall that L s M = ΩΩ ∞ s (Σ − s M ). Theorem 1.16 says that R s − (Σ − M ) d s − −−−→ R s ( M ) d s −→ R s +1 (Σ M )has homology Σ − Ω ∞ s (Σ − s M ) at R s ( M ), and we will see that L s M is theimage of a natural homomorphismΩ ∞ s (Σ − s M ) → Σ − Ω ∞ s (Σ − s M ) . This may make it plausible (though by no means obvious) that there mightbe an algebraic spectral sequence with E = U Q ( R ∗ M ) and E ∞ = U Q ( L ∗ M ).In §
4, we will give a complete presentation of Theorem 1.16, which ismuch more topologically based and less explicitly computational than sim-ilar results in the literature. Also included in this section is a proof ofProposition 1.11.
Examples. In §
6, we give a variety of examples illustrating the strengthof our main results and their limitations. Here we summarize some of ourfindings.Corollary 1.14 holds for the following families of spectra X , so that thealgebraic and topological spectral sequences agree, and thus E ∞∗ , ∗ ( X ) ≃ U Q ( L ∗ H ∗ ( X )) . • X = Σ n HA , the Eilenberg–MacLane spectrum of type ( A, n ), unless n = 0 or −
1, and A has 2-torsion of order at least 4. • X = Σ ∞ Z , a suspension spectrum. • X = S h i , the 1–connected cover of Σ ∞ S . • X = (Σ ∞ Z )( s ), the s th stage of an Adams resolution of Σ ∞ Z , with Z a connected space.Verifying the geometric hypothesis of Corollary 1.14 for this last familyrelies on unpublished work of Lannes and Zarati [LZ2] from the 1980’s.By contrast, we have examples of spectra for which the spectral sequencehas rogue differentials. • When X = hocofib { Σ ∞ R P −→ Σ ∞ R P } , d is nonzero. • When X = H Z / r , with r ≥ d r − is the only rogue differential. • When X = Σ − H Z / r , with r ≥
2, the rogue differentials are thefamily d s (2 r − , s ≥ X as in the first of these examples, X and the suspensionspectrum of R P ∨ Σ R P have isomorphic homology, but differing spectralsequences.In all of these examples, L H ∗ ( X ) turns out to generate L ∗ H ∗ ( X ) as amodule over the Dyer–Lashof algebra. For a simple example where this does not hold, one can let X = S ∪ η D : see § Remark . From our calculations, we learn that, if X is any Eilenberg–MacLane spectrum whose homotopy is a finite abelian 2-group, the topo-logical spectral sequence correctly computes H ∗ (Ω ∞ X ), even when X is not0–connected. In ongoing work, the second author has found a couple moreexamples of nonconnective spectra for which the spectral sequence correctlycomputes H ∗ (Ω ∞ X ). As of yet the authors have no good sense of when suchexotic convergence should be expected.1.7. Acknowledgments.
This research was partially supported by Na-tional Science Foundation grant 0967649. Some of these results were pre-sented by the first author at the September, 2011 Workshop on HomotopyTheory at Oberwolfach, with a report published as [K4], and by the secondauthor in the Special Session on the Calculus of Functors at the January2012 AMS Annual Meeting in Boston. The authors would like to thank thereferee for many constructive comments and, in particular, for nudging ustowards a much improved § OD 2 HOMOLOGY OF INFINITE LOOPSPACES 11 Preliminaries
Prerequisites on spectra. T will be the category of pointed topolog-ical 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 extrastructure, and we let X n denote its n th space. Thus Ω ∞ X = X .By a weak natural transformation F → G between two functors withvalues 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 aweak equivalence (on any object). We say that a diagram of such weaknatural transformations commutes if it induces a commutative diagram inthe 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 sum-marizes the formal properties of S –modules that are needed to make thearguments in [AK] work. Proposition 2.1.
The category S of S –modules has the following structure. • S is a category enriched over T . • S is tensored and cotensored over T : given K ∈ T and X ∈ S ,there are spectra K ∧ X and Map S ( K, X ) , natural in both variables,satisfying standard adjunction properties. • There are natural maps η : Map S ( K, X ) → Map S ( L ∧ K, L ∧ X ) . • There are natural maps
Map S ( K, X ) ∧ Map S ( L, Y ) → Map S ( 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 smashproduct. • There are natural maps e : Σ ∞ Map T ( K, Z ) → Map S ( K, Σ ∞ Z ) . • There is a weak natural equivalence hocolim n Σ − n Σ ∞ X n → X . Here and elsewhere we write Σ − n X for Map S ( S n , X ).It is only the last item that really needs comment. See Appendix A formore discussion of this point.We end this subsection by describing the setting for the ‘evaluation/diagonal’natural transformations ǫ : Σ F ( X ) → F (Σ X )which play a significant role in our work.As S is a category enriched over T , Mor S ( X, Y ) has the structure of abased topological space. A functor F : S → S is said to be continuous if F : Mor S ( X, Y ) → Mor S ( F ( X ) , F ( Y ))is a continuous function. If F is also reduced , i.e. F ( ∗ ) = ∗ , then thiscontinuous 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 K → Mor S ( X, K ∧ X ) F −→ Mor S ( F ( X ) , F ( K ∧ X )) , where the first map is the unit of the adjunctionMor S ( K ∧ X, Y ) ≃ Mor T ( K, Mor S ( X, Y )) . The Tate construction. If G is a finite group, we let G – S denote thecategory 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 thecategory of R –modules with G –action. (For us, R will eventually be H Z / Y ∈ G – R –mod, we let Y hG and Y hG respectively denote associatedhomotopy 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 anatural isomorphism of R –modules, R ∧ Y hG = ( R ∧ Y ) hG . There are various constructions in the literature, e.g. [ACD, AK, GM], ofa natural norm map N G ( Y ) : Y hG → Y hG . The Tate spectrum of Y is defined as the homotopy cofiber of N G ( Y ). Itwill be more convenient for us to desuspend this once and define t G ( Y ) to bethe homotopy fiber of N G ( Y ). Thus t G ( Y ) comes equipped with a naturaltransformation t G ( Y ) → Y hG .The next lemma lists the properties we need about this. Lemma 2.4. (a) t G 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, t G ( 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 t Z / ( Y ). Let ρ be the onedimensional real sign representation of Z /
2, and let S nρ be the one pointcompactification of nρ . Lemma 2.5. (Compare with [GM, Thm.16.1] .) For Y ∈ Z / – R –mod, thereis a natural weak equivalence t Z / ( Y ) ≃ holim n Map R -mod ( S nρ , Y ) h Z / . 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 D R ( X ) = ( X ∧ R X ) h Z / and D R ( X ) = t Z / ( X ∧ R X ). OD 2 HOMOLOGY OF INFINITE LOOPSPACES 13
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, D R ( R ∧ X ) = R ∧ D ( X ) . (b) For all R –modules X , there is a natural weak equivalence D R ( X ) ≃ holim n Σ n D R (Σ − n X ) . Thus D R is identified as the colinearization of D R in the sense of [McC]. Corollary 2.8. D R preserves cofibration sequences of R –modules. The homology of extended powers.
When X is a spectrum, aconstruction of the Dyer–Lashof operations Q i : H j ( D d X ) → H j + i ( D d X ) , for all i ∈ Z , is given by M. Steinberger in [BMMS, Thm.III.1.1]. Analternative 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 ∗ ( P X )as an algebra with both Dyer–Lashof and Steenrod operations, where P X = W ∞ d =0 D d X .The coproduct structure on H ∗ ( P X ) seems to be less well documentedin the literature. Recall that the coproduct ∆ is induced by the transfermaps t b,c : D b + c X → D b X ∧ D c X . The following lemma is presumably wellknown, and is analogous to [CLM, Thm.I.1.1(6)]. Lemma 2.9.
For all y ∈ H ∗ ( P X ) , if ∆( y ) = P y ′ ⊗ y ′′ , then ∆( Q k y ) = X i + j = k X Q i y ′ ⊗ Q j y ′′ . Sketch proof.
Let p : X → X ∨ X be the pinch map. If b + c = d , then t b,c is the ( b, c )th component of the composite D d X D d ( p ) −−−→ D d ( X ∨ X ) = _ b + c = d D b X ∧ D c X. The diagram D D d X (cid:15) (cid:15) D D d ( p ) / / D D d ( X ∨ X ) (cid:15) (cid:15) D d X D d ( p ) / / D d ( X ∨ X )commutes, and the lemma follows from this, using the Cartan formula Q k ( y ′ ⊗ y ′′ ) = P i + j = k Q i y ′ ⊗ Q j y ′′ . (cid:3) Crucial to us is the behavior of ǫ : Σ D d X → D d Σ X on homology. Lemma 2.10. (a) ǫ ∗ : H ∗ ( P X ) → H ∗ +1 ( P Σ X ) sends the algebra decom-posables to zero, and has image in the coalgebra primitives. (b) ǫ ∗ ( Q i y ) = Q i ( 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 ∗ (Σ D s (Σ − X )) → H ∗ ( D s X ) is pre-cisely the subspace of primitives: the span of the elements Q I x with l ( I ) = s and x ∈ H ∗ ( X ) . Dyer–Lashof operations for D .Lemma 2.12. (a) The sequence . . . ǫ ∗ −→ H ∗− ( D (Σ − X )) ǫ ∗ −→ H ∗− ( D (Σ − X )) ǫ ∗ −→ H ∗ ( D ( X )) is Mittag–Leffler. (b) π ∗ ( D H Z / ( H Z / ∧ X )) = lim n H ∗ (Σ n D (Σ − n X )) . Statement (a) follows from Lemma 2.10, and then (b) follows from (a),noting that π ∗ ( D H Z / ( H Z / ∧ X )) = H ∗ ( D ( X )). Corollary 2.13.
The natural transformation Q i : H ∗ ( X ) → H ∗ + i ( D ( X )) lifts to a natural transformation Q i : H ∗ ( X ) → π ∗ + i ( D H Z / ( H Z / ∧ X )) . This corollary will play a critical role in our proof of parts (b) and (c) ofTheorem 1.6: see § The cohomology of D X . In the proof of Theorem 1.6(a), it will beuseful to work with mod 2 cohomology. As in [K3], letˆ Q : H ∗ ( X ) → H ∗ ( D X )be the squaring operation, and then, for i >
0, letˆ Q i : H ∗ ( X ) → H ∗ + i ( D X )be defined to be the composite H ∗ ( X ) = H ∗ + i (Σ i X ) ˆ Q −−→ H ∗ +2 i ( D (Σ i X )) ǫ ∗ −→ H ∗ + i ( D X ) . One also has a product ∗ : H ∗ ( X ) ⊗ H ∗ ( X ) → H ∗ ( D X ) induced by t , : D X → X ∧ X . One has ˆ Q ( x + y ) = ˆ Q x + ˆ Q y + x ∗ y , while, for i > Q i is linear. Lemma 2.14. H ∗ ( D X ) is spanned by the elements ˆ Q i x and x ∗ y . OD 2 HOMOLOGY OF INFINITE LOOPSPACES 15
These operations are appropriately dual to the homology Dyer–Lashofoperations. In the next proposition, Q i x = Q i + | x | x , as is standard. Proposition 2.15.
Let h x, y i denote the cohomology/homology pairing. Given w, x ∈ H ∗ ( X ) and y, z ∈ H ∗ ( X ) , the following formulae hold. (a) h ˆ Q i x, Q j y i = ( h x, y i if i = j otherwise. (b) h ˆ Q i x, y ∗ z i = ( h x, y ih x, z i if i = 00 otherwise. (c) h w ∗ x, Q i y i = ( h w, y ih x, y i if i = 00 otherwise. (d) h w ∗ x, y ∗ z i = h w, y ih x, z i + h w, z ih x, y i . See [K3, Prop.A.1]. 3.
Proof of Theorem 1.6
Proof of Theorem 1.6(a).
It suffices to prove this formula assuming X is a spectrum whose homology is bounded below and of finite type. In thiscase, it is easiest to first prove the cohomology version of Theorem 1.6(a).Recall from § H ∗ ( D X ) is spanned by elements ˆ Q i x and x ∗ y ,with x, y ∈ H ∗ ( X ) and i ≥ δ : X → Σ D X be the connecting map of thecofibration sequence D X → ˜ P X → X . Proposition 3.1.
For x ∈ H n ( X ) , we have δ ∗ ( σ ˆ Q r x ) = Sq r + n +1 x .Proof. The proof uses ideas from [K3, Prop.4.3] and [K2, Appendix A].Let P ( r, n ) be the statement δ ∗ ( σ ˆ Q r x ) = Sq r + n +1 x for all x ∈ H n ( X ) . We need to prove that P ( r, n ) is true for all r ≥ n ∈ Z .We first observe that, for r > P ( r − , n + 1) implies P ( r, n ). To seethis, we use that the diagram X δ / / Σ D X ǫ (cid:15) (cid:15) X Σ − δ / / D Σ X commutes by the naturality of δ . So, if x ∈ H n ( X ) and P ( r − , n + 1) holds,then δ ∗ ( σ ˆ Q r x ) = δ ∗ ( ǫ ∗ ( ˆ Q r − σx ))= (Σ − δ ) ∗ ( ˆ Q r − σx )= σ − Sq ( r − 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 toshow that δ ∗ ( σ ˆ Q ι n ) = Sq n +1 ι n , where ι n ∈ H n (Σ n H Z /
2) is the fundamental class.We break this into cases.When n >
0, this was proven in [K3, Prop.4.3] as follows. As Σ n H Z / n H Z / H ∗ ( K ( Z / , n )). Thus the element Sq n +1 ι n must be an eventualboundary, as Sq n +1 ι n = 0 in H ∗ ( K ( Z / , n )). For degree reasons, the onlyway this could happen is if δ ∗ ( σ ˆ Q ι n ) = Sq n +1 ι n .When n < −
1, the degree of σ ˆ Q ι n is 2 n + 1 < n , so δ ∗ takes this elementto zero. As desired, Sq n +1 ι n is also zero since n + 1 < Sq is injective on A indegrees 0 and 1.We have Sq δ ∗ ( σ ˆ Q ι n ) = δ ∗ ( σSq ˆ Q ι n ). The Nishida relations for theoperation ˆ Q [K3, Prop.3.15] tell us that Sq ˆ Q ι n = (cid:18) n (cid:19) ˆ Q ι n + (cid:18) n − (cid:19) ˆ Q Sq ι n + ι n ∗ Sq ι n . Since δ ∗ takes nontrivial products to zero, we deduce that Sq δ ∗ ( σ ˆ Q ι n ) = (cid:18) n (cid:19) δ ∗ ( σ ˆ Q ι n ) + δ ∗ ( σ ˆ Q Sq ι n ) . When n = 0, this equation and the established fact P (0 ,
1) imply that Sq δ ∗ ( σ ˆ Q ι ) = Sq Sq ι . We deduce that δ ∗ ( σ ˆ Q ι ) = Sq ι .When n = −
1, the above equation and the established facts P (2 , − P (0 , P (0 ,
0) imply that Sq δ ∗ ( σ ˆ Q ι − ) = Sq ι − + Sq Sq ι − = Sq ι − . We deduce that δ ∗ ( σ ˆ Q ι − ) = ι − . (cid:3) If we define ˆ Q i x = ˆ Q i −| x | x , the proposition says that, for all x ∈ H ∗ ( X ), δ ∗ ( σ ˆ Q i x ) = Sq i +1 x. OD 2 HOMOLOGY OF INFINITE LOOPSPACES 17
By duality, we get the formula stated in Theorem 1.6(a): for all x ∈ H ∗ ( X ), δ ∗ ( x ) = X i ≥ σQ i − ( xSq i ) , Remark . Variants of the formula in Theorem 1.6(a) go back at least asfar as the 1966 paper [BCKQRS].3.2.
The strategy for the proof of statements (b) and (c).
We outlinethe strategy of the proof of Theorem 1.6(b) and (c).First of all, what do we have to show?In (c), the statement‘ y ∈ H ∗ ( D d X ) represents z ∈ H ∗ (Ω ∞ X )’means that, under the mapsΣ ∞ + Ω ∞ X e d −→ P d ( X ) i d ←− D d ( X ) , we have e d ∗ ( z ) = i d ∗ ( y ). So to prove (c), we just need to show that then,under the maps Σ ∞ + Ω ∞ X e d −−→ P d ( X ) i d ←−− D d ( X ) , we have e d ∗ ( Q i z ) = i d ∗ ( Q i y ).In (b), the statement‘ y ∈ H ∗ ( D d X ) lives to E r , and d r ( y ) is represented by z ∈ H ∗ ( D d + r X )’means that there is an element w ∈ H ∗ ( P d + r − ( X )), such that under themaps D d ( X ) i d −→ P d ( X ) p d + r − ,d ←−−−−− P d + r − ( X ) δ d + r − −−−−→ Σ D d + r ( X ) , we have i d ∗ ( y ) = p d + r − ,d ∗ ( w ) and δ d + r − ∗ ( w ) = σz . So to prove (b), wejust need to show that then, there is an element w i ∈ H ∗ ( P d +2 r − ( X )) suchthat under the maps D d ( X ) i d −−→ P d ( X ) p d +2 r − , d ←−−−−−−− P d +2 r − ( X ) δ d +2 r − −−−−−→ Σ D d +2 r ( X ) , we have i d ∗ ( Q i y ) = p d +2 r − ,d ∗ ( w i ) and δ d +2 r − ∗ ( w i ) = σQ i z .We can also pass to H Z / Sup-pressing this from our notation, we will assume this in what we do below.
For example, H ∗ ( D ( P d ( X ))) will ‘really’ mean π ∗ ( D H Z / ( H Z / ∧ P d ( X ))).Theorem 1.6(c) follows from the following. Proposition 3.3.
There is a commutative diagram of weak natural trans-formations D (Σ ∞ + Ω ∞ X ) D e d / / (cid:15) (cid:15) D ( P d ( X )) (cid:15) (cid:15) D ( D d ( X )) D i d o o (cid:15) (cid:15) Σ ∞ + Ω ∞ X e d / / P d ( X ) D d ( X ) i d o o in which left and right vertical maps are the composites D (Σ ∞ + Ω ∞ X ) → D (Σ ∞ + Ω ∞ X ) µ −→ Σ ∞ + Ω ∞ X and D ( D d ( X )) → D ( D d ( X )) µ −→ D d ( X ) , where µ is the standard operad action. To deduce Theorem 1.6(c) from this, suppose that e d ∗ ( z ) = i d ∗ ( y ) as inthe discussion above. Then the diagram shows that e d ∗ ( Q i z ) = i d ∗ ( Q i y ),where Q i z and Q i y can be viewed as being in the homology of the appro-priate Tate construction, courtesy of Corollary 2.13.Theorem 1.6(b) follows from the following. Proposition 3.4.
There is a commutative diagram of weak natural trans-formations D D d ( X ) D i / / (cid:15) (cid:15) D P d ( X ) (cid:15) (cid:15) D P d + r − ( X ) D δ / / D p o o (cid:15) (cid:15) Σ D D d + r ( X ) (cid:15) (cid:15) D d ( X ) i / / P d ( X ) P d +2 r − ( X ) δ / / p o o Σ D d +2 r ( 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 thatΣ D D d + r ( X ) ≃ D Σ D d + r ( X ), thanks to Corollary 2.8.To deduce Theorem 1.6(b) from this, the needed element w i ∈ H ∗ ( P d +2 r − ( X ))will then be the image of Q i w ∈ H ∗ ( D P d + r − ( X )) under the vertical mapsecond from the right.It remains to prove these two propositions. We do this at the end of thenext 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 / S with( P ∧ P ) d = holim b + c ≤ d P b ∧ P c . Suggestively, we will let D d denote the fiber of P d → P d − , and then let F d denote the fiber of ( P ∧ P ) d → ( P ∧ P ) d − . From [AK, Cor.5.3] we learn OD 2 HOMOLOGY OF INFINITE LOOPSPACES 19
Lemma 3.6.
There is a weak natural equivalence in Z / – S F d ≃ Y b + c = d D b ∧ D c . Note that there are Z / P ∧ P ) d +1 → ( P ∧ P ) d → P d ∧ P d and F d → D d ∧ D d . Lemma 3.7.
These maps induce equivalences of Tate spectra: t Z / (( P ∧ P ) d +1 ) ∼ −→ t Z / (( P ∧ P ) d ) ∼ −→ D ( P d ) and t Z / ( F d ) ∼ −→ D ( D d ) . Proof.
With ǫ either 0 or 1, filtered in the usual way, ( P ∧ P ) d + ǫ has com-position factors of two types: • D i ∧ D i with i ≤ d . • Z / + ∧ D i ∧ D j with i < j and i + j ≤ d + ǫ .Meanwhile P d ∧ P d has composition factors: • D i ∧ D i with i ≤ d . • Z / + ∧ D i ∧ D j with i < j ≤ d .The first type of factors match up, and after applying t Z / , the second typebecome null.The proof for F d is similar and easier. (cid:3) 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, thereis a map µ : C ∞ (2) × Z / (Ω ∞ X ) → Ω ∞ X. The next theorem is our key geometric input. It is quite easily deduced from[AK, Thm.1.10], and hopefully seems plausible. See Appendix B for a bitmore detail.
Theorem 3.8.
There is a weak natural transformation of towers µ : ( P ∧ P ) h Z / → P with the following properties. (a) There is a commutative diagram of weak natural transformations (Σ ∞ + (Ω ∞ X ) ) h Z / µ (cid:15) (cid:15) ( e ∧ e ) h Z / / / ( P ∧ P ) h Z / µ (cid:15) (cid:15) Σ ∞ + Ω ∞ X e / / P. (b) On fibers, µ corresponds to the maps D b ∧ D c → D b + c and D D d → D d .Proof of Proposition 3.3. We have a commutative diagram of weak naturaltransformations D (Σ ∞ + Ω ∞ X ) D e / / D ( P d ) D ( D d ) D i o o D (Σ ∞ + Ω ∞ X ) / / (cid:15) (cid:15) t Z / (( P ∧ P ) d ) ≀ O O (cid:15) (cid:15) t Z / ( F d ) o o ≀ O O (cid:15) (cid:15) D (Σ ∞ + Ω ∞ X ) / / (cid:15) (cid:15) (( P ∧ P ) d ) h Z / (cid:15) (cid:15) ( F d ) h Z / o o (cid:15) (cid:15) Σ ∞ + Ω ∞ X e / / P d D d . i o o Here the bottom squares commute by the last theorem, and the right twotop vertical maps are weak equivalences by Lemma 3.7. (cid:3)
Proof of Proposition 3.4.
This time we have a commutative diagram of weaknatural transformations D D d D i / / D P d D P d + r − D δ / / D p o o Σ D D d + r t Z / ( F d ) / / (cid:15) (cid:15) ≀ O O t Z / (( P ∧ P ) d ) (cid:15) (cid:15) ≀ O O t Z / (( P ∧ P ) d +2 r − ) / / o o (cid:15) (cid:15) ≀ O O Σ t Z / ( F d +2 r ) (cid:15) (cid:15) ≀ O O ( F d ) h Z / / / (cid:15) (cid:15) (( P ∧ P ) d ) h Z / (cid:15) (cid:15) (( P ∧ P ) d +2 r − ) h Z / / / o o (cid:15) (cid:15) Σ( F d +2 r ) h Z / (cid:15) (cid:15) D d i / / P d P d +2 r − δ / / p o o Σ D d +2 r . Again the top vertical maps are weak equivalences by Lemma 3.7. (cid:3) Derived functors of destabilization
In this section, we will carefully define the Singer complex R M d −→ R M d −→ R M d −→ R M d −→ . . . of the introduction and prove Theorem 1.16, which says that the homologyof this complex computes the derived functors Ω ∞ s M for all M ∈ M .As a free standing theorem, Theorem 1.16 is similar (and maybe identical)to [Goe, Thm.3.17]. Goerss works totally algebraically, and at key momentshis proof appeals to computations and ad hoc arguments by others includingSinger [Si], Brown and Gitler [BG], and Bousfield et. al. [BCKQRS]. Bycontrast, we give a geometrically based construction of this chain complex, OD 2 HOMOLOGY OF INFINITE LOOPSPACES 21 with explicit calculations bypassed by appealing to our knowledge of thehomology of the extended powers.Our proof of Theorem 1.16 makes use of the doubling functor Φ :
M →M , dual to Powell’s use of it in the cohomological setting [P]. Also includedin this section is a presentation of properties of Φ and Ω :
U → U neededin our construction of the algebraic spectral sequence in §
5. Much of whatwe 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 thecategory M .Since modules in M are locally finite, they are certainly also locally Noe-therian. The abelian category M is thus a locally Noetherian abelian cate-gory satisfying good exactness properties, and so one knows a priori [Gab,IV.2] that arbitrary direct sums of injectives in M are again injective, andinjectives can be written essentially uniquely as the direct sum of indecom-posable injectives. It is useful for us to show this explicitly.Let A ∗ ∈ M be the dual of A , so A ∗ = H ∗ ( H Z / V denote thecategory of Z –graded vector spaces. Given V ∈ V , we let IV = V ⊗ A ∗ .Note that ǫ : A ∗ → Z / ǫ V : IV → V . Lemma 4.1.
For all M ∈ M , the natural map ǫ M,V : Hom M ( M, IV ) ≃ Hom V ( M, V ) sending f to ǫ V ◦ f is an isomorphism.Sketch proof. If M is finite, ǫ M, Σ n Z / : Hom M ( M, Σ n A ∗ ) ≃ ( M n ) is readilychecked to be an isomorphism, and thus the same is true for ǫ M,V when M is finite and V is finite dimensional.For finite M and arbitrary V , one then sees that ǫ M,V is an isomorphismby filtering V by its finite dimensional subspaces.For arbitrary M and V , one then sees that ǫ M,V is an isomorphism byfiltering M by its finite submodules. (cid:3) Corollary 4.2.
The modules IV are injective objects of M , and every M ∈M admits an injective resolution of the form → M → IV (0) → IV (1) → IV (2) → . . . , for some graded vector spaces V ( s ) ∈ V .Proof. As the functor sending M to Hom V ( M, V ) is exact, we conclude that IV is injective in M .Given M ∈ M , the A –module map M → IM corresponding to 1 M ∈ Hom V ( M, M ) is clearly monic. It follows that injective resolutions of theasserted sort exist. (cid:3)
It follows that every injective in M is a direct sum of modules of the formΣ n A ∗ , and thus is isomorphic to IV for some V ∈ V . Exact functors from the category M via topology. Let H ∗ ( S ) ⊂M be the subcategory obtained as the image of H ∗ : S → M . Thus theobjects are the locally finite A –modules of the form H ∗ ( X ), with morphismsall A –module maps of the form f ∗ : H ∗ ( X ) → H ∗ ( Y ) for some f : X → Y .Let A b be an abelian category, for example M . Call a functor F : H ∗ ( S ) → A b homological if whenever X → Y → Z induces a sequence H ∗ ( X ) → H ∗ ( Y ) → H ∗ ( Z ) that is exact at H ∗ ( Y ), then F ( H ∗ ( X )) → F ( H ∗ ( Y )) → F ( H ∗ ( Z )) is exact at F ( H ∗ ( Y )). The following is a useful wayto construct exact functors from M , and natural transformations betweensuch. Proposition 4.3. (a)
Any homological functor F : H ∗ ( S ) → A b extendsuniquely to an exact functor F : M → A b . (b) Let
F, G : H ∗ ( S ) → A b be homological. Any natural transformation φ : F → G extends uniquely to a natural transformation between the extendedfunctors. To prove this, we first note that injectives in M can be topologicallyrealized: given V ∈ V , there is an associated generalized Eilenberg–MacLanespectrum HV , satisfying π ∗ ( HV ) = V and H ∗ ( HV ) = IV . The next lemmais clear. Lemma 4.4.
For all spectra X , the Hurewitz map induces an isomorphism [ X, HV ] ≃ Hom M ( H ∗ ( X ) , IV ) . Proof of Proposition 4.3.
Given a homological functor F : H ∗ ( S ) → A b , wedefine its extension F : M → A b as follows. Given M ∈ M , choose an exactsequence 0 → M → H ∗ ( HV (0)) f ∗ −→ H ∗ ( HV (1)), and then define F ( M ) tobe the kernel of F ( f ∗ ). Given a morphism α : M → N in M , one canconstruct a diagram0 / / M α (cid:15) (cid:15) / / H ∗ ( HV (0)) β ∗ (cid:15) (cid:15) f ∗ / / H ∗ ( HV (1)) γ ∗ (cid:15) (cid:15) / / N / / H ∗ ( HW (0)) g ∗ / / H ∗ ( HW (1))with exact rows. Applying F to the right square and taking kernels, definesa map F ( α ) : F ( M ) → F ( N ). It is routine to check that this gives awell defined exact functor which extends the original functor up to naturalisomorphism. This proves (a). The proof of (b) is similar. (cid:3) A topological definition of R s M . We construct various functorsand natural transformations using the method of Proposition 4.3.
Definition 4.5.
Define R s : H ∗ ( S ) → M by the formula R s H ∗ ( X ) = im { ǫ ∗ : H ∗ (Σ D s (Σ − X ) → H ∗ ( D s X ) } . OD 2 HOMOLOGY OF INFINITE LOOPSPACES 23
Thanks to our knowledge of ǫ ∗ as summarized in Lemma 2.10, we see that R s H ∗ ( X ) = h Q I x | l ( I ) = s, x ∈ H ∗ ( X ) i / (unstable and Adem relations) . We remind readers that the Dyer–Lashof Adem relations are Q r Q s = X i (cid:18) i − s − i − r (cid:19) Q r + s − i Q i , and that the Steenrod algebra acts via the Nishida relations( Q s x ) Sq r = X i (cid:18) s − rr − i (cid:19) Q s − r + i ( xSq i ) . Lemma 4.6. R s is homological.Proof. This follows immediately from the observation that the natural mapof graded vector spaces M n ∈ Z R s (Σ n Z / ⊗ H n ( X ) → R s ( H ∗ ( X ))sending Q I ι n ⊗ x to Q I x is an isomorphism. (In this formula, H n ( X ) shouldbe regarded as just a degree 0 vector space.) (cid:3) Definitions 4.7. (a)
Let R s : M → M be the exact extension of R s : H ∗ ( S ) → M . (b) Let ǫ : Σ R s M → R s Σ M be the natural A –module map induced by thenatural transformation ǫ : Σ D s X → D s Σ X . (c) Let µ : R s R t M → R s + t M be the natural A –module map induced bythe natural transformation µ : D s D t X → D s + t X . (d) Let Q i : ( R s M ) n → ( R s +1 M ) n + i be the natural Z / Q i : H n ( D s X ) → H n + i ( D s +1 X ).We can immediately deduce lots of properties of these natural transfor-mations. We note, in particular, a couple. Lemma 4.8. (a)
The operations Q i satisfy the Adem relations, the Nishidarelations, and the Dyer–Lashof unstable relation. (b) The diagram Σ R s R t M µ (cid:15) (cid:15) ǫ / / R s Σ R t M ǫ / / R s R t Σ M µ (cid:15) (cid:15) Σ R s + t M ǫ / / R s + t Σ M commutes. Remark . Observe that R ( M ) = M , and ǫ : Σ R M → R Σ M is justthe identity map on Σ M .Another elementary property we will need involves connectivity. Lemma 4.10. If M is ( n − connected, then R s M is s n − connected.Proof. This follows from the observation that if X is ( n −
1) connected, then D d X is dn − (cid:3) Now we introduce algebraic differentials. As before, δ : X → Σ D X isthe connecting map of the cofibration sequence D X → ˜ P X → X . Definition 4.11.
Define d s : R s ( M ) → R s +1 (Σ M ) to be the natural trans-formation induced by the composite δ s : D s X D s δ −−−→ D s Σ D X D s ǫ −−−→ D s D Σ X µ −→ D s +1 Σ X. Explicitly, the computation of δ ∗ given in Theorem 1.6(a) tells us that d s ( Q I x ) = X i ≥ Q I Q i − ( σxSq i ) . Proposition 4.12.
The composite R s − (Σ − M ) d s − −−−→ R s ( M ) d s −→ R s +1 (Σ M ) is zero. This is an immediate consequence of the following topological version,and since homology is compactly supported, we really just need this resultwhen X is a finite CW spectrum. Proposition 4.13.
The composite D s − ( X ) δ s − −−−→ D s (Σ X ) δ s −→ D s +1 (Σ X ) is null.Proof. It is easy to see that this composite factors through D s − applied tothe composite X δ −→ D (Σ X ) δ −→ D (Σ X ) . Thus we just need to show that this last composite is null.The trick now is to colinearize these functors and maps. Generalizing ourprevious notation D , for any d , let D d ( X ) = holim n Σ n D d (Σ − n X ). OD 2 HOMOLOGY OF INFINITE LOOPSPACES 25
Colinearization then yields a commutative diagram of weak natural trans-formations D ( X ) (cid:15) (cid:15) / / X δ (cid:15) (cid:15) D (Σ X ) (cid:15) (cid:15) / / D (Σ X ) δ (cid:15) (cid:15) D (Σ X ) / / D (Σ X ) . As the top horizontal map is clearly an equivalence, the proposition willfollow 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 dependson consequences of the Segal Conjecture for elementary abelian 2–groups.Namely, the first author showed [K1, Cor.5.3] that D ( X ) ≃ ∗ if X is finite.(In this case, it is also true that the top left vertical map is an equivalenceafter completing at 2.)A second, more elementary argument goes roughly as follows. The colin-earized functors D d preserves cofibration sequences, and are null unless d isa power of 2. Then it is not too hard to show that the left vertical sequenceis equivalent to the composite D ( X ) → Σ D ( X ) → Σ D ( X )of the two connecting maps associated to the colinearization of the tower˜ P ( X ) → ˜ P ( X ) → ˜ P ( X ) , so that their composite is null. (cid:3) Remark . A direct algebraic proof of Proposition 4.12 is possible. Usingboth the Dyer–Lashof Adem relations and the Adem relations in A , oneneeds to show that X i ≥ X j ≥ Q i − Q j − ( xSq i Sq j ) = 0 . Goerss [Goe, Lem.3.13] points to Brown and Gitler’s assertion that a calcu-lation like this is straightforward [BG, Lem.2.3], and one can check that itis.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 moduleconcentrated in even degrees, with Φ( M ) n = M n and with φ ( x ) Sq i = φ ( xSq i ). (Here, given x ∈ M n , we have written φ ( x ) for the correspondingelement in Φ( M ) n .)Basic properties are listed in the next lemma. Lemma 4.16. (a) Φ is an exact functor preserving unstable modules. (b) Φ( N ⊗ M ) = Φ( N ) ⊗ Φ( M ) . In particular, Φ(Σ M ) = Σ Φ( M ) . (c) Let Γ ( M ) = ( M ⊗ M ) Z / and S ( M ) = ( M ⊗ M ) Z / . The composite Γ ( M ) ֒ → M ⊗ M ։ S ( M ) naturally factors as a composite Γ ( M ) ։ Φ( M ) ֒ → S ( M ) , where the second map sends φ ( x ) to x . (d) Let sq : M → Φ( M ) be the linear map defined by letting sq ( x ) = φ ( xSq n ) if x ∈ M n . If M is unstable, then sq is A –linear. For a proof of (d) in the cohomological setting, see [S, p.26].
Definition 4.17.
Let q : Φ( M ) → R M be defined by the formula q ( φ ( x )) = Q | x | x . More generally, define q : Φ( R s M ) → R s +1 M to be the compositeΦ( R s M ) q −→ R R s M µ −→ R s +1 M . Lemma 4.18. q is A –linear.Proof. As usual, one need just check this when M = H ∗ ( X ). The identity Q | x | x = x ∈ H ∗ ( D X ) for all x ∈ H ∗ ( X ) implies that the compositeΦ( H ∗ ( X )) q −→ R ( H ∗ ( X )) ⊆ H ∗ ( D X )equals the compositeΦ( H ∗ ( X )) ֒ → S ( H ∗ ( X )) → H ∗ ( D X ) , and so is A –linear. (cid:3) The following lemma is crucial.
Lemma 4.19.
For all M ∈ M and s > , the sequence → Φ( R s − ( M )) q −→ R s ( M ) ǫ −→ Σ − R s (Σ M ) → is short exact.Proof. It is convenient to use lower indices for Dyer–Lashof operations: Q i x = Q | x | + i x . Suppose M has a homogeneous basis { x α } . Then the Ademrelations show that R s ( M ) then has a basis given by { Q i Q i . . . Q i s x α | ≤ i ≤ i ≤ · · · ≤ i s } . Since ǫ ( Q i Q i . . . Q i s x α ) = σ − Q i − Q i − . . . Q i s − σx α , and Q i x = 0 if i <
0, the lemma follows. (cid:3)
The next lemma is clear from the definitions.
OD 2 HOMOLOGY OF INFINITE LOOPSPACES 27
Lemma 4.20. (a)
The diagram R s R t ( M ) R s ( d t ) (cid:15) (cid:15) µ / / R s + t ( M ) d s + t (cid:15) (cid:15) R s R t +1 (Σ M ) µ / / R s + t +1 (Σ M ) commutes. (b) The diagram Φ( R s − ( M )) Φ( d s − ) (cid:15) (cid:15) q / / R s ( M ) d s (cid:15) (cid:15) ǫ / / Σ − R s (Σ M ) Σ − d s (cid:15) (cid:15) Φ( R s (Σ M )) q / / R s +1 (Σ M ) ǫ / / Σ − R s +1 (Σ M ) commutes. The derived functors of destabilization.
We now relabel as in theintroduction.
Definition 4.21.
Let R s = Σ R s Σ s − : M → M .With this notation, the chain complexΣ R (Σ − M ) d −→ Σ R ( M ) d −→ Σ R (Σ M ) d −→ Σ R (Σ M ) → . . . rewrites as R ( M ) d −→ R ( M ) d −→ R ( M ) d −→ R ( M ) → . . . . The following is a restatement of Theorem 1.16.
Theorem 4.22.
For all M ∈ M , there is a natural isomorphism H s ( R ∗ ( M ); d ∗ ) ≃ Ω ∞ s M. In the usual way, this theorem is a consequence of the next three lemmas.
Lemma 4.23. R s is exact for all s . Lemma 4.24. H ( R ∗ ( M ); d ∗ ) = Ω ∞ M . Lemma 4.25.
For all n ∈ Z and s > , H s ( R ∗ (Σ n 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 d : M → Σ R M ,and we recall that d ( x ) = X i ≥ σQ i − ( xSq i ) . Then x ∈ ker( d ) ⇔ Q i − ( xSq i ) = 0 for all i ≥ ⇔ xSq i = 0 whenever i − ≥ | x | − i ⇔ xSq i = 0 whenever 2 i > | x | . ⇔ x ∈ Ω ∞ M. (cid:3) The proof of Lemma 4.25 will take a bit of preparation. Firstly, Lemma 4.19and Lemma 4.20(b) combine to tell us the following.
Proposition 4.26. → Φ( R ∗− (Σ M )) q −→ Σ R ∗ ( M ) ǫ −→ R ∗ (Σ M ) → is ashort exact sequence of chain complexes. Temporarily, let H s ( M ) = H s ( R ∗ ( M ); d ∗ ). The short exact sequence ofProposition 4.26 induces a long exact sequence0 → Σ H ( M ) ǫ ∗ −→ H (Σ M ) ∂ −→ Φ( H (Σ M )) q −→ Σ H ( M ) → . . .. . . → H s − (Σ M ) ∂ −→ Φ( H s − (Σ M )) q −→ Σ H s ( M ) ǫ ∗ −→ H s (Σ M ) → . . . We need to identify the first boundary map.
Lemma 4.27. H (Σ M ) ∂ −→ Φ( H (Σ M )) identifies with the map Ω ∞ (Σ M ) sq −−→ Φ(Ω ∞ (Σ M )) . Proof. If σx ∈ Ω ∞ (Σ M ) has | σx | = 2 n , then we have the correspondence,under the maps Σ M Σ d −−→ Σ R ( M ) q ←− Φ(Σ M ),(Σ d )( σx ) = σd ( x ) = σQ n − ( xSq n ) = q ( φ ( σxSq n )) = q ( sq ( σx )) . Thus ∂ ( σx ) = sq ( σx ). (cid:3) In dual form, the following lemma corresponds to the familiar fact thatthe map Sq : F ( n ) → F ( n ), sending x to Sq | x | x , is monic. Here F ( n ) isthe free unstable A –module on an n –dimensional class. Lemma 4.28.
For all n ∈ Z , Ω ∞ (Σ n A ∗ ) sq −−→ Φ(Ω ∞ (Σ n A ∗ )) is onto. We are finally ready to prove Lemma 4.25. The proof is dual to the proofof [P, Prop.9.4.1].
Proof of Lemma 4.25.
By induction on s ≥
1, we prove that H s (Σ n A ∗ ) = 0.In all cases, we consider the exact sequence H s − (Σ n +1 A ∗ ) ∂ −→ Φ( H s − (Σ n +1 A ∗ )) q −→ Σ H s (Σ n A ∗ ) ǫ ∗ −→ H s (Σ n +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( H s − (Σ n +1 A ∗ )) = 0. Thus,in all cases, we can conclude that Σ H s (Σ n A ∗ ) ǫ ∗ −→ H s (Σ n +1 A ∗ ) is monic forall n . But, by Lemma 4.10, the connectivity of Σ − m H s (Σ m + n A ∗ ) is at least(2 s − m + 2 s ( n + s − m goes to infinity. (cid:3) OD 2 HOMOLOGY OF INFINITE LOOPSPACES 29
First consequences.
Theorem 4.22, when combined with Proposi-tion 4.26 and Lemma 4.27, implies the following.
Corollary 4.29.
For all M ∈ M , there is a natural long exact sequence → ΣΩ ∞ ( M ) ǫ ∗ −→ Ω ∞ (Σ M ) sq −−→ Φ(Ω ∞ (Σ M )) q −→ ΣΩ ∞ ( M ) → . . .. . . → Ω ∞ s − (Σ M ) sq −−→ Φ(Ω ∞ s − (Σ M )) q −→ ΣΩ ∞ s ( M ) ǫ ∗ −→ Ω ∞ s (Σ M ) → . . . Remark . This long exact sequence already appears (in dual form) in[LZ1, § M → I ∗ ( M ) is an injective resolution in M , then so is Σ M → Σ I ∗ ( M ), and0 → ΣΩ ∞ ( I ∗ ( M )) ǫ −→ Ω ∞ (Σ I ∗ ( M )) sq −−→ Φ(Ω ∞ (Σ I ∗ ( M ))) → sq identifies with the boundary map in our derivation,while q identifies with the boundary map in the Lannes–Zarati approach.Next we note that Lemma 4.10 implies the following general connectivityestimate. Corollary 4.31. If M ∈ M is n –connected, then Ω ∞ s ( M ) is at least s ( n + s ) –connected. For all M ∈ M , colim n Σ − n Ω ∞ s (Σ n M ) = 0 for all s ≥ . The reasoning we gave in the proof of Lemma 4.25 then proves the fol-lowing useful criterion for the vanishing of the higher derived functors.
Proposition 4.32. If sq : Ω ∞ (Σ n M ) → Φ(Ω ∞ (Σ n M )) is onto for all n ≥ , then Ω ∞ s ( M ) = 0 for all s ≥ . Following [LZ1] and [Goe], we now deduce some properties of Ω ∞ s (Σ − t M )when M is unstable. Lemma 4.33.
Suppose M is unstable. Then, for all s ≥ , R s ( M ) is alsounstable, and d s : R s (Σ − M ) → R s +1 ( M ) is zero.Proof. Though this admits an algebraic proof, to show how the algebrafollows the topology, we offer a topologically based proof.If M ⊂ H ∗ ( HV ) and is unstable, then M ֒ → Ω ∞ H ∗ ( HV ) և H ∗ (Ω ∞ HV ).Using the exactness of the functors R s , one easily sees that the conclusionsof the lemma for the module H ∗ (Ω ∞ HV ) imply the same for M . Thus itsuffices to prove the lemma when M = H ∗ ( Z ), where Z is a space.In this case, R s ( H ∗ ( Z )) ⊂ H ∗ ( D s Z ), which is unstable, as D s Z is aspace.To see that d s : R s (Σ − H ∗ ( Z )) → R s +1 ( H ∗ ( Z )) is zero, we recall thatit is induced by a geometric stable map δ s : D s (Σ − Z ) → D s +1 ( Z ). (Weidentify Z with Σ ∞ Z .) We observe that this map is null: δ s factors through D s ( δ ), and δ : Σ − Z → D ( Z ) is null as Σ δ is the first boundary mapin the tower associated to Σ ∞ Ω ∞ Σ ∞ Z , which splits into the product of itsfibers. (cid:3) As Ω ∞ s (Σ − s M ) is the homology at the middle term of the complexΣ R s − (Σ − M ) d s − −−−→ Σ R s ( M ) d s −→ Σ R s +1 (Σ M ) , the lemma leads to the next result. Theorem 4.34.
Suppose M is unstable. (a) Ω ∞ s (Σ − s M ) ≃ Σ R s ( M ) , so that ΩΩ ∞ s (Σ − s M ) ≃ R s ( M ) . (b) More generally, if s > t , then Ω ∞ s (Σ − t M ) ≃ Σ R s (Σ s − t − M ) , which isa quotient of Σ s − t R s ( M ) . Thus Ω ∞ s (Σ − t M ) is an ( s − t ) –fold suspension ofan unstable module, and so Ω s − t Ω ∞ s (Σ − t M ) = Σ − s + t +1 R s (Σ s − t − M ) . (c) Ω ∞ s (Σ − s M ) ≃ coker { d s − : Σ R s − (Σ − M ) → Σ R s (Σ − M ) } . The first statement here is the main algebraic theorem of [LZ1], and thelast was observed in [Goe, Cor.5.4].4.7.
Dyer–Lashof operations on derived functors.
We need to explainProposition 1.11, which said that the sum of the looped derived functorsΩΩ ∞∗ Σ −∗ M is an object in QM . Otherwise said, we need to explain whythere exist natural transformations µ : R s ΩΩ ∞ t Σ − t M → ΩΩ ∞ s + t Σ − s − t M 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 : R s R t M → R s + t M induce maps µ : R s Σ − Ω ∞ t Σ − t M → Σ − Ω ∞ s + t Σ − s − t M 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 Ω .For all M ∈ U , there is an exact sequence → ΣΩ M → M sq −−→ Φ( M ) → ΣΩ M → . From the long exact sequence of Corollary 4.29, we thus deduce the fol-lowing.
OD 2 HOMOLOGY OF INFINITE LOOPSPACES 31
Corollary 4.36.
For M ∈ M , the following diagram commutes, and thebottom row is short exact: Σ − ΦΩ ∞ s (Σ M ) q ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ (cid:15) (cid:15) (cid:15) (cid:15) Σ − Ω ∞ s +1 (Σ M )Ω Ω ∞ s (Σ M ) (cid:31) (cid:127) / / Ω ∞ s +1 ( M ) / / / / ǫ ∗ ♦♦♦♦♦♦♦♦♦♦♦♦ ΩΩ ∞ s +1 (Σ M ) . ?(cid:31) O O Proof of Proposition 1.11.
From the commutative diagram of Lemma 4.8,Σ R s R t M µ (cid:15) (cid:15) ǫ / / R s Σ R t M R s ǫ / / R s R t Σ M µ (cid:15) (cid:15) Σ R s + t M ǫ / / R s + t Σ M, we deduce that the following diagram commutes:Σ R s Σ − Ω ∞ t Σ − t M µ (cid:15) (cid:15) ǫ / / / / R s Ω ∞ t Σ − t M R s ǫ ∗ / / R s Σ − Ω ∞ t Σ − t M µ (cid:15) (cid:15) Ω ∞ s + t Σ − s − t M ǫ ∗ / / Σ − Ω ∞ s + t Σ − s − t M. For M ∈ M , we then define µ : R s ΩΩ ∞ t Σ − t M → ΩΩ ∞ s + t Σ − s − t M to be the natural transformation induced by taking the image of the top andbottom horizontal maps in this last diagram.These natural transformations for all s and t are equivalent to definingnatural Dyer–Lashof operations Q i : ΩΩ ∞ s Σ − s M → ΩΩ ∞ s +1 Σ − s M, for all i ∈ Z , which raise degree by i , and satisfy the usual properties. (cid:3) Definition 4.37.
Define q : Φ(ΩΩ ∞ s (Σ M )) → ΩΩ ∞ s +1 ( M ) by the formula q ( φ ( x )) = Q | x | x. Hopf algebras and the algebraic spectral sequence
In this section, we first discuss part (e) of Theorem 1.4, which said thateach E r ∗ , ∗ ( X ) is primitively generated, with its bigraded module of primitivesan A –module subquotient of R ∗ H ∗ ( X ). This is really a reflection of anaspect of the general theory of differential Hopf algebras.Using some of the same theory, we then go on to develop the algebraicspectral sequence, as described in Theorem 1.12.The Hopf algebras of this paper, E r ∗ , ∗ ( X ), can be viewed as connected bicommutative Hopf algebras, by viewing E r − i, ∗ ( X ) as having grading i . Asthese are our concern, in this section by the term Hopf algebra, we will meana connected bicommutative Hopf algebra over Z / Primitively generated Hopf algebras and barcode modules.
Werecall some classic observations about primitively generated Hopf algebras.One has as examples Z / x d ] and Z / x d ] / ( x s d ), with x d primitive andhomogeneous of some positive grading d . The work of Milnor and Moore[MM] then tells us that any primitively generated Hopf algebra will be atensor product of Hopf algebras of these types.A more basis–free way to discuss primitively generated Hopf algebras isvia modules of primitives.If A is a Hopf algebra, its module of primitives has the structure of apositively graded vector space V equipped with a linear map which doublesgrading q : V ∗ → V ∗ . We will call such a V a barcode module, as it is thesort of Z / q ]–module appearing in the persistent homology literature [ZC],where the classification is given by a barcode.There is then an equivalence of categoriesbarcode modules ≃ primitively generated Hopf algebras . In one direction, the correspondence takes a barcode module V to U q ( V ) = S ∗ ( V ) / ( x − q ( x ) : x ∈ V ) . In the other direction, the primitives of a Hopf algebra, together with thesquaring map, form a barcode module.
Examples 5.1. Z / x d ] = U q ( W ), where W = h x d , x d , x d , . . . i , and q ( x t d ) = x t +1 d . Similarly Z / x d ] / ( x s d ) = U q ( V ), where V = h x d , x d , . . . , x s − d i , and q ( x t d ) = ( x t +1 d if t < s −
10 if t = s − . Remark . In our spectral sequences, V will be a graded object in M ,equipped with an A –module map q : Φ( V ∗ ) → V ∗ . In this situation, U q ( V )becomes an object in HM , bigraded by giving V i,j bigrading ( − i, i + j ). Example 5.3. E ∗ , ∗ ( X ) = U q ( V ), where V i = ( R s H ∗ ( X ) if i = 2 s q ( x ) = Q | x | x .5.2. The homology of certain differential Hopf algebras. A differen-tial Hopf algebra is a Hopf algebra equipped with a homogeneous differen-tial of some degree r (not necessarily ±
1) which is both a derivation and acoderivation. It is easy to see that such differentials on U q ( V ) correspondto homogeneous linear maps d : V ∗ → V ∗ + r such that d = 0 = dq . We willcall such pairs ( V, d ) differential barcode modules.
OD 2 HOMOLOGY OF INFINITE LOOPSPACES 33
Results in [MM] show that a Hopf algebra (bicommutative over Z /
2) isprimitively generated if and only if squares are zero in the dual. Browder[B] notes that, when such a Hopf algebra has a differential, this propertywill be preserved by taking homology.It follows that given a differential barcode module (
V, d ), the homologyHopf algebra H ( U q ( V ); d ) will have the form U q ( W ) for some new barcodemodule W . The natural problem now is to try to determine W from ( V, d ).It seems difficult to say something useful about the problem in this gen-erality. In particular, the following simple example shows that W need notbe a subquotient of V . Example 5.4.
Let V = h x, y i , with q = 0, and dx = y . Then U q ( V )is the exterior algebra Λ( x, y ) and H ( U q ( V ); d ) = Λ( xy ) = U q ( W ), where W = h xy i .More generally, the identity d ( xdx ) = qdx holds in U q ( V ), for any pair( V, d ), so that elements in ker q ∩ im d lead to cycles in H ∗ ( U q ( V ); d ) similarto the cycle xy in this last example.Our discovery is that the differential Hopf algebras U q ( V ) arising in ourspectral sequences satisfy an extra condition avoiding this problem, and en-suring that the module of primitives in H ( U q ( V ); d ) is a natural subquotientof V . Proposition 5.5.
Suppose a differential barcode module ( V, d ) is nonzeroonly in degrees of the form k , and d has nonzero component d : V s → V t for some s < t . If also ( ♦ t ) q : V k → V k +1 is monic for k ≥ t, then H ( U q ( V ); d ) ≃ U q ( ¯ V ) , where ¯ V is the following explicit barcode modulesubquotient of V : ¯ V k = V k if k < t and k = s ker( d ) if k = sV k /q k − t im( d ) if k ≥ t. In particular, ¯ V still satisfies condition ( ♦ t ) . We will prove this in the next subsection.
Corollary 5.6.
Suppose that one has a left half plane homological spectralsequence { E r } of differential objects in HM with E = U q ( V (1 , , where V (1 , is as in Remark 5.2. If V (1 , is nonzero only in degrees of the form k and satisfies condition ( ♦ ) , then (a) d r can only be nonzero when r = 2 t − s with s < t , and (b) E t − s = U q ( V ( t, s )) where V ( t, s ) is a subquotient of V (1 , satisfyingcondition ( ♦ t ) . Proof.
Using the proposition, the corollary is proved by induction on theset of pairs { ( t, s ) | ≤ s < t } totally ordered by the value 2 t − s . (Thus(1 , < (2 , < (2 , < (3 , < . . . .) Note that, in passing from the pair( t,
0) to ( t + 1 , t ), one uses that ( ♦ t ) ⇒ ( ♦ t +1 ). (cid:3) Proof of Theorem 1.4(e) and (f ).
The corollary applies to { E r ∗ , ∗ ( X ) } , with V (1 , s = R s H ∗ ( X ), as this V (1 ,
0) satisfies condition ( ♦ ). (cid:3) A useful lemma.
Given two positively graded vector spaces U and W and a degree doubling map q : U → U ⊕ W , we letΓ( q ) = S ∗ ( U ⊕ W ) / ( q ( x ) − x ) , a primitively generated Hopf algebra.The map q has components q U : U → U and q W : U → W . If a homoge-neous map d : U → W satisfies dq U = 0, d will induce a differential on Γ( q ).It is useful to then let U ′ = ker( d ), W ′ = coker( d ), and q ′ : U ′ → U ′ ⊕ W ′ be the map induced by q .There is an evident inclusion into the cycles U ′ ⊕ W ֒ → Z ∗ (Γ( q ); d ) , and this induces a natural map of Hopf algebras α ( q, d ) : Γ( q ′ ) → H ∗ (Γ( q ); d ) . Lemma 5.7.
In this situation, α ( q, d ) is an isomorphism.Proof. We first consider the case when q is identically zero. As a Hopfalgebra, Γ(0) = Λ ∗ ( U ) ⊗ S ∗ ( W ) . We can also assume that our map d : U → W , say of degree r, has the form U ′ ⊕ B ։ B ∼ −→ Σ r B ֒ → Σ r B ⊕ W ′ . Thus, as differential Hopf algebras,(Γ(0) , d ) = Λ ∗ ( U ′ ) ⊗ (Γ B , d B ) ⊗ S ∗ ( W ′ ) , where (Γ B , d B ) = Λ ∗ ( B ) ⊗ S ∗ (Σ r B ) with the Koszul differential: d B ( b ⊗
1) =1 ⊗ σ r b . The complex (Γ B , d B ) 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 formΛ ∗ ( x ) ⊗ Z / dx ] whose homology is Z /
2. Thus we see that H ∗ (Γ(0); d ) = Λ ∗ ( U ′ ) ⊗ S ∗ ( W ′ ) = Γ(0 ′ ) , and thus the lemma is true for the case q = 0.Now we consider the case when just the component q W is assumed to bezero, so that q = q U for an arbitrary map q U : U → U satisfying dq U = 0.We reduce to the previous case with a little spectral sequence argument.We define an increasing filtration F ⊂ F ⊂ F ⊂ . . . of the chaincomplex (Γ( q U ) , d ) by letting F k = im { S ∗≤ k ( U ) ⊗ S ∗ ( W ) → Γ( q ) } . OD 2 HOMOLOGY OF INFINITE LOOPSPACES 35
Then d : F k → F k − , and in the induced spectral sequence E = E =(Γ(0) , d ), so that E = Γ(0 ′ ). Noting that E (Γ( q ′ U )) = Γ(0 ′ ), one simulta-neously sees that E = E ∞ and α ( q U , d ) is an isomorphism.Finally we consider the case of a general q = ( q U , q W ). Once again wereduce to the previous case with a spectral sequence argument.This time, we define a decreasing filtration F ⊃ F ⊃ F ⊃ . . . of thechain complex (Γ( q ) , d ) by letting F k = im { S ∗ ( U ) ⊗ S ∗≥ k ( W ) → Γ( q ) } . Then d : F k → F k +1 , and in the induced spectral sequence E = E =(Γ( q U ) , d ), so that E = Γ( q ′ U ). Noting that E (Γ( q ′ )) = Γ( q ′ U ), one deducesthat E = E ∞ and α ( q, d ) is an isomorphism. (cid:3) Example 5.8.
Here is an example that illustrates the various reductionsmade in the last proof. Let U = h x , x , x i , W = h y i , q ( x ) = x , q ( x ) = x + y , q ( x ) = 0, and d ( x ) = y . ThenΓ( q ) = Z / x , x , x , y ] / ( x − x , x − x − y , x ) ,E (Γ( q )) = Γ( q U ) = Z / x ] / ( x ) ⊗ Z / y ] , and E E (Γ( q )) = Γ(0) = Λ( x , x , x ) ⊗ Z / y ] . The associated homology Hopf algebras are H (Γ( q ); d ) = H (Γ( q U ); d ) = Z / x ] / ( x ) , H (Γ(0); d ) = Λ( x , x ) . Proof of Proposition 5.5.
We are given V q −→ V q −→ V q −→ . . . , and V s d −→ V t such that dq : V s − → V t is zero. We also know that( ♦ t ) q : V k → V k +1 is monic for k ≥ t, For k > t choose a subspace W k ⊂ V k such that W k ⊕ q ( V k − ) = V k ,and let U = L k
In this section we explain why,given M ∈ M , there is a well defined spectral sequence of Hopf algebras asin Theorem 1.12: • E alg, ∗ , ∗ ( M ) = U Q ( R ∗ ( M )). • Nonzero differentials are only the d s , and, for x ∈ M and I of length s , Q I x lives to E alg, s ∗ , ∗ ( M ), and d s ( Q I x ) = P i ≥ Q I Q i − ( xSq i ). • For all r , E alg,r ∗ , ∗ ( M ) is primitively generated with primitives concen-trated in the − s lines. For all r > s , the module of primitives in E alg,r − s , s + ∗ ( M ) is naturally isomorphic to L s M . • E alg, ∞∗ , ∗ ( M ) = U Q ( L ∗ M ).(We recall that L s M = ΩΩ ∞ s Σ − s M .)We do this by explicitly describing all the intermediate pages E alg, s ∗ , ∗ ( M ),so that Proposition 5.5 applies.In the rest of this subsection we use the following notation. Notation 5.9.
We let H s M be the homology at R s M in the sequence R s − Σ − M d s − −−−→ R s M d s −→ R s +1 Σ M . We generically use ‘ q ’ to denote mapsinduced by bottom Dyer–Lashof operations (a.k.a. squaring). For k ≥ s , welet ¯ R k,s M = R k M/ im { q k − s d s − : R s − Σ − M → R k M } .We note a few of the relationships between these: • → H s M → ¯ R s,s M d s −→ R s +1 Σ M is exact. • L s M = im { Σ H s Σ − M ǫ ∗ −→ H s M } , by Corollary 4.36. • For k ≥ s , the square R k M (cid:15) (cid:15) q / / R k +1 M (cid:15) (cid:15) ¯ R k,s M q / / ¯ R k +1 ,s M is a pushout square with epic vertical map, thus defining the lowerhorizontal map. Lemma 5.10. d s : R s Σ − M → R s +1 M induces a map d s : ¯ R s,s M → ¯ R s +1 ,s M .Proof. The module ¯ R s,s M is the quotient of R s Σ − M by the subspace gen-erated by the images of the maps d s − : R s − Σ − M → R s Σ − M and q : R s − Σ − M → R s Σ − M . Meanwhile ¯ R s +1 ,s M is the quotient of R s +1 M by the subspace given by the image of d s q : R s − Σ − M → R s +1 M . Clearly d s : R s Σ − M → R s +1 M carries the former subspace to the latter. (cid:3) Armed with these various constructions, we can define our spectral se-quence.
OD 2 HOMOLOGY OF INFINITE LOOPSPACES 37
Definition 5.11.
Let ( E alg, s ∗ , ∗ ( M ) , d s ) = ( U q ( V ( s )) , d s ), where the pair( V ( s ) , d s ) is defined by letting V ( s ) k = ( L k M for k < s ¯ R k,s M for k ≥ s, and by letting the nonzero component of d s be the map d s : ¯ R s,s M → ¯ R s +1 ,s M of the last lemma.Proposition 5.5 applies to prove H ( E alg, s ∗ , ∗ ( M ); d s ) = E alg, s +1 ∗ , ∗ ( M ) oncewe check the next proposition. Proposition 5.12.
There are natural isomorphisms ker { d s : ¯ R s,s M → ¯ R s +1 ,s M } ≃ L s M and coker { d s : ¯ R s,s M → ¯ R s +1 ,s M } ≃ ¯ R s +1 ,s +1 M. Proving this is slightly subtle. We apply the next lemma, letting thediagram R s − Σ − M q (cid:15) (cid:15) d s − / / R s M q (cid:15) (cid:15) R s − Σ − M ǫ (cid:15) (cid:15) d s − / / R s Σ − M ǫ (cid:15) (cid:15) d s / / R s +1 M ǫ (cid:15) (cid:15) R s − Σ − M (cid:15) (cid:15) d s − / / R s M (cid:15) (cid:15) d s / / R s +1 Σ M (cid:15) (cid:15) ♥ ).So suppose given a diagram of vector spaces( ♥ ) U q U (cid:15) (cid:15) d U / / V q V (cid:15) (cid:15) U ′ e U (cid:15) (cid:15) d ′ U / / V ′ e V (cid:15) (cid:15) d ′ V / / W ′ e W (cid:15) (cid:15) U (cid:15) (cid:15) d U / / V (cid:15) (cid:15) d V / / W (cid:15) (cid:15) Let ¯ V = V / im d U , ¯ W = W , ¯ V ′ = V ′ / (im d ′ U + im q U ), and ¯ W ′ = W ′ / im d ′ V ◦ q U . The diagram ( ♥ ) maps in an evident way to the diagram( ♠ ) 0 (cid:15) (cid:15) / / ¯ V q (cid:15) (cid:15) / / ¯ V ′ ¯ e V (cid:15) (cid:15) d ′ / / ¯ W ′ ¯ e W (cid:15) (cid:15) / / ¯ V (cid:15) (cid:15) d / / ¯ W (cid:15) (cid:15) . Lemma 5.13.
The following properties hold. (a) ¯ e V : ¯ V ′ → ¯ V is an isomorphism. (b) ker( d ′ ) = im { e V ∗ : H ( V ′ ) → H ( V ) } . (c) coker( d ′ ) ≃ coker( d ′ V ) . (d) The third column of ( ♠ ) is exact.Proof. Diagram chasing with the left two columns of ( ♥ ) shows that thereis an exact sequence U q U −→ V ′ / im d ′ U e V −→ V / im d U → , and statement (a) follows.It is standard that given maps A f −→ B g −→ C in an abelian category, thereis an exact sequence coker f → coker gf → coker g →
0. Apply this to U d U −−→ V q V −→ W ′ to deduce statement (d). Apply this to V ′ d ′ V −−→ W ′ ։ ¯ W ′ to deduce statement (c), noting thatcoker { V ′ → ¯ W ′ } = coker { ¯ V ′ d ′ −→ ¯ W ′ } . To deduce (b), let ˜ V ′ = V ′ / im d ′ U . One has a commutative diagram˜ V ′ d ′ V (cid:15) (cid:15) / / ¯ V ′ d ′ (cid:15) (cid:15) ∼ / / ¯ V d (cid:15) (cid:15) W ′ / / ¯ W ′ / / ¯ W , where the indicated isomorphism is the isomorphism of (a). Taking kernels,one gets H ( V ′ ) → ker d ′ ֒ → H ( V ) , OD 2 HOMOLOGY OF INFINITE LOOPSPACES 39 and we need to check that the first map here is onto. But this follows becausethe left square fits into a commutative diagram U / / ˜ V ′ d ′ V (cid:15) (cid:15) / / ¯ V ′ d ′ (cid:15) (cid:15) / / U / / W ′ / / ¯ W ′ / / (cid:3) Examples
Eilenberg–MacLane spectra.
We begin our discussion of how thespectral sequence behaves when X = Σ n HA by noting that all of our con-structions behave well with respect to filtered colimits, localization at 2, anddirect sums in the variable A . It follows that the key cases to understandare when A = Z , Z /
2, and Z / r with r ≥ L ∗ ( H ∗ (Σ n HA )) in these cases.Recall that H ∗ ( H Z /
2) = A , H ∗ ( H Z ) = A / A Sq , and H ∗ ( H Z / r ) = H ∗ ( H Z ) ⊕ Σ H ∗ ( H Z ) for r ≥
2. For convenience, let ¯ A ∗ = H ∗ ( H Z ). Lemma 6.1.
For all s > , Ω ∞ s Σ − s + n ¯ A ∗ = Σ Z / if n = 0 Z / if n = − otherwise . Proof.
We work with the equivalent dual left A –module situation. Let F ( n ) = Ω ∞ Σ n A , the free unstable A –module on an n –dimensional class.(This is 0, if n < A / A Sq has a projective resolution · · · → Σ A · Sq −−→ Σ A · Sq −−→ A → A / A Sq → . Applying Ω ∞ Σ − s + n yields the complex · · · → F (1 − s + n + 2) · Sq −−→ F (1 − s + n + 1) · Sq −−→ F (1 − s + n ) . The module Ω ∞ s Σ − s + n ¯ A ∗ is thus dual to the homology of F ( n + 2) · Sq −−→ F ( n + 1) · Sq −−→ F ( n ) . By inspection, one sees that this is exact except when n = − (cid:3) Corollary 6.2. (a) L s H ∗ (Σ n H Z /
2) = 0 for all s > and all n . (b) For all s > , L s H ∗ (Σ n H Z ) = ( Z / if n = 00 otherwise . (c) For all s > and r ≥ , L s H ∗ (Σ n H Z / r ) = L s H ∗ (Σ n H Z ) ⊕ L s H ∗ (Σ n +1 H Z ) = ( Z / if n = − , otherwise . Now we need to know how the Dyer–Lashof operation Q acts. Lemma 6.3. Q : L s H ∗ ( H Z ) → L s +1 H ∗ ( H Z ) is an isomorphism for all s ≥ .Proof. The key point is that the exact sequenceΩ ∞ s Σ − s ¯ A ∗ sq −−→ Φ(Ω ∞ s Σ − s ¯ A ∗ ) q −→ ΣΩ ∞ s +1 Σ − s ¯ A ∗ identifies with the exact sequenceΣ Z / sq −−→ Σ Z / q −→ Σ Z / . As the first map here is clearly zero, the second is an isomorphism. (cid:3)
This lemma and the previous corollary combine to give us the next cal-culations.
Proposition 6.4. (a) U Q ( L ∗ H ∗ ( H Z )) = Z / x ] where x is the nonzero 0dimensional class in Ω ∞ H ∗ ( H Z ) . (b) For r ≥ , U Q ( L ∗ H ∗ ( H Z / r )) = Z / x ] ⊗ Λ ∗ ( y ) where x and y are thenonzero 0 and 1 dimensional classes in Ω ∞ H ∗ ( H Z / r ) . (c) For r ≥ , U Q ( L ∗ H ∗ (Σ − H Z / r )) = Z / y ] where y is the nonzero 0dimensional class in Ω ∞ Σ − H ∗ ( H Z / r ) . (d) If A is either Z or Z / r , then U Q ( L ∗ H ∗ (Σ n HA )) = Λ ∗ (Ω ∞ H ∗ (Σ n HA )) ,except in the cases covered by (a), (b), and (c). By inspection we see that the algebraic condition of Corollary 1.14 holdsfor all Eilenberg–MacLane spectra.
Corollary 6.5.
For all abelian groups A and all n ∈ Z , L ∗ H ∗ (Σ n HA ) isgenerated as a module over the Dyer–Lashof algebra by L H ∗ (Σ n HA ) =Ω ∞ H ∗ (Σ n HA ) . The classic calculations of H ∗ ( K ( A, n )) when A is either Z or Z / r allowus to determine when the geometric condition of Corollary 1.14 holds. Lemma 6.6. If A is either Z or Z / r , the evaluation map H ∗ ( K ( A, n )) → Ω ∞ H ∗ (Σ n HA ) is onto except when A = Z / r with r ≥ and n = 0 or − . We conclude that the algebraic spectral sequence equals the topologicalspectral sequence for all Eilenberg–MacLane spectra Σ n HA , unless A has2–torsion of order at least 4, and n = 0 or − H ∗ ( K ( Z / , n )) equals Λ ∗ ( F ( n ) ∗ ) with no nonzeroDyer–Lashof operations. OD 2 HOMOLOGY OF INFINITE LOOPSPACES 41
Now we use our calculations to say more about how the topological spec-tral sequence behaves in the cases covered by Proposition 6.4(a), (b), and(c).6.2.
Convergence of the spectral sequence for H Z . We have shownthat E ∞∗ , ∗ ( H Z ) = Z / x ]. The spectral sequence converges to the cor-rect answer as well as possible: H ∗ (Ω ∞ H Z ) = H ∗ ( Z ) = Z / t, t − ] whilelim d H ∗ ( P d ( H Z )) = Z / x ]], and the former embeds densely in the latter viathe homomorphism sending t to x + 1.6.3. The spectral sequence for H Z / r with r ≥ . When r ≥
2, wehave shown that E alg, ∞∗ , ∗ ( H ∗ ( H Z / r )) = Z / x ] ⊗ Λ ∗ ( y ). This time only x isin the image of the evaluation, so there should be a rogue differential off of y . The elements x s are the only nonzero 0 dimensional primitive classes in E , so the first rogue differential must hit one of these.We claim that d r − ( y ) = x r , this is the only rogue differential, and E ∞∗ , ∗ ( H Z / r ) = Z / x ] / ( x r ). Furthermore, the spectral sequence convergesto the correct answer: H ∗ (Ω ∞ H Z / r ) = H ∗ ( Z / r ) = Z / t ] / ( t r −
1) = Z/ x ] / ( x r ), when t = x + 1.To prove the claim, we first make some observations about the beginningof the spectral sequence in low degrees. In total degree 0, E is spannedby the classes x n , and in total degree 1, E is spanned by the classes x n y ,and x n Q x . If z ∈ H ∗ ( H Z / r ) is the two dimensional class with zSq = x ,then d ( x n z ) = x n Q x . It follows that the only classes in E in degrees 0and 1 will be x n and x n y , none of which can possibly be in the image of analgebraic differential.We now show that x r = 0 in E ∞∗ , ∗ ( H Z / r ). To see this, we consider thediagram Z/ t, t − ] (cid:15) (cid:15) / / Z / x ]] (cid:15) (cid:15) Z/ t ] / ( t r − / / lim d H ∗ ( P d ( H Z / r ))in which both horizontal maps send t − x . As ( t − r = t r − Z/ t ] / ( t r − x r = 0 in lim d H ∗ ( P d ( H Z / r )), and thus in E ∞∗ , ∗ ( H Z / r ).Finally we show that x s = 0 for all s < r , or equivalently, that y livesto E r − . This we show by induction on r . The r = 2 case is true because d ( y ) = 0. For the inductive step, let E alg, ∞∗ , ∗ ( Z / r − ) = Z / x ′ ] ⊗ Λ ∗ ( y ′ ).The inclusion Z / r − → Z / r induces a map of both the topological andalgebraic spectral sequences sending x ′ to 0, and y ′ to y . Then the inductivehypothesis — that y ′ lives to E r − − ∗ , ∗ ( H Z / r − ) and d r − − ( y ′ ) = ( x ′ ) r − — implies that y lives to E r − − ∗ , ∗ ( H Z / r ) and d r − − ( y ) = 0, i.e. y lives to E r − ∗ , ∗ ( H Z / r ), and thus to E r − ∗ , ∗ ( H Z / r ). The spectral sequence for Σ − H Z / r with r ≥ . Our most com-plicated example is the spectral sequence for Σ − H Z / r , with r ≥ x and y be the nonzero classes in H ∗ (Σ − H Z / r ) of dimensions − E alg, ∞∗ , ∗ ( H ∗ (Σ − H Z / r )) = Z / y ], and obviously y is not in theimage of the evaluation. The only primitive elements in E of total degree − Q ) s x ∈ E − s , s − , so a first rogue differential musthit one of these.We claim that y lives to E r , and d r − ( y ) = ( Q ) r x . To see this, we com-pare this example to our previous one, using the map of spectral sequencesinduced by Σ P (Σ − H Z / r ) → P ( H Z / r ) . This sends the elements x and y to the elements with the same name inthe last example. It also induces an isomorphism from the primitives oftotal degree − E (Σ − H Z / r ) to the primitives of total degree 0 in E ( H Z / r ). The calculation that d r − ( y ) = x r = ( Q ) r x in the spectralsequence for H Z / r then implies that d r − ( y ) = ( Q ) r x in the spectralsequence for Σ − H Z / r .The formula d r − ( y ) = ( Q ) r x then implies that, for any s ≥ d s (2 r − ( y s ) = d s (2 r − (( Q ) s y ) = ( Q ) s d r − ( y ) = ( Q ) s + r x. We also note that d ( x ) = Q − x = x , and it follows that, for any s ≥ d s (( Q ) s x ) = ( Q ) s Q − x = Q − ( Q ) s x = (( Q ) s x ) . We now explain how these calculations completely determine how thealgebraic and topological spectral sequences differ. Let x s = ( Q ) s x . Usingthe standard primitive generators, the E term of both spectral sequencesdecomposes: E = Z / y, x , x , x , . . . ] ⊗ E ⊥ , . This, in fact, represents a decomposition of both spectral sequences, wherethe algebraic and topological spectral sequences agree on E ⊥ , ∗ , and thedifferentials on Z / y, x , x , x , . . . ] go as follows: • The algebraic spectral sequence has d s ( x s ) = x s . • The topological spectral sequence also has d s (2 r − ( y s ) = x s + r .It is then easy to compute that, for all s ≥ E alg, s = Z / y, x s , x s +1 , x s +2 , . . . ] ⊗ E ⊥ , s , while, for all s ≥ r , E top, s = Z / y s +1 − r , x s +1 , x s +2 , . . . ] ⊗ E ⊥ , s . The spectral sequence for suspension spectra.
Let X = Σ ∞ Z ,a suspension spectrum, so that H ∗ ( X ) is unstable. The tower is known tosplit; for example, when Z is connected,Σ ∞ Ω ∞ Σ ∞ Z ≃ _ d Σ ∞ D d Z. OD 2 HOMOLOGY OF INFINITE LOOPSPACES 43
Thus the spectral sequence collapses at E , and so E ∞∗ , ∗ ( X ) ≃ U Q ( R ∗ H ∗ ( X )) . As we clearly have no rogue differentials, our works says that E ∞∗ , ∗ ( X ) ≃ U Q ( L ∗ H ∗ ( X )) . This is in agreement with Theorem 4.34(a), which says that, since H ∗ ( X )is unstable, L ∗ H ∗ ( X ) = R ∗ H ∗ ( X ).6.6. The spectral sequence for S h i . The partially published work ofLannes and Zarati [LZ2] suggests that one might be able to ‘mix and match’the suspension spectra and Eilenberg–MacLane spectra examples. Here isthe simplest such example.Let S h i be the cofiber of S → H Z .By dimension shifting, one can easily compute that, for all s ≥ L s H ∗ ( S h i ) ≃ L s +1 H ∗ ( S ) = R s +1 H ∗ ( S ) , and this is compatible with Dyer–Lashof operations.Meanwhile, there is a map Σ ∞ D S t −→ S h i such that t ∗ realizes theisomorphism H ∗ ( D S ) = R H ∗ ( S ) ≃ Ω ∞ H ∗ ( S h i ): t can be taken tobe the bottom horizontal map in the commutative square of symmetricproducts SP ( S ) (cid:15) (cid:15) / / SP ∞ ( S ) (cid:15) (cid:15) SP ( S ) /S / / SP ∞ ( S ) /S. One formally concludes that H ∗ (Ω ∞ S h i ) → Ω ∞ H ∗ ( S h i ) is onto.Corollary 1.14 now applies to say that E ∞∗ , ∗ ( S h i ) = U Q ( R ∗≥ H ∗ ( S )).This is in agreement with known calculation: Ω ∞ S h i is the fiber of thesplit fibration Ω ∞ Σ ∞ S → S , and it is not hard to see that, localized at 2,Σ ∞ Ω ∞ S h i ≃ _ d D d S . Adams resolutions of suspension spectra.
Here is a much moresophisticated version of the last example. Let Z be a connected space, and,for s ≥
0, recursively define spectra Z ( s ) and K ( s ) by letting Z (0) = Σ ∞ Z , K ( s ) = Z ( s ) ∧ H Z /
2, and Z ( s ) = hofib { Z ( s − i −→ K ( s − } .In [LZ2], an only partially finished manuscript from the 1980’s but sup-ported by [LZ1], Lannes and Zarati study H ∗ (Ω ∞ Z ( s )). Enroute, theyshow (in dual form) [LZ2, Prop.2.5.2(iv)] that H ∗ (Ω ∞ Z ( s )) → Ω ∞ H ∗ ( Z ( s ))is onto. (This is proved by a rather elaborate induction on s using theEilenberg–Moore spectral sequence.)Meanwhile, dimension shifting immediately shows that, for t >
0, thereare isomorphisms L t H ∗ ( Z ( s )) ≃ R s + t H ∗ ( Z ) . We claim that there is also an epimorphism L H ∗ ( Z ( s )) → R s H ∗ ( Z ),so that L H ∗ ( Z ( s )) generates L ∗ H ∗ ( Z ( s )), and once again Corollary 1.14applies.To see this, we begin with the exact sequence0 → Ω ∞ H ∗ ( Z ( s − i ∗ −→ Ω ∞ H ∗ ( K ( s − → Ω ∞ Σ H ∗ ( Z ( s )) → Ω ∞ H ∗ ( Z ( s − → . Let M denote the cokernel of i ∗ , so there is a short exact sequence0 → M → Ω ∞ Σ H ∗ ( Z ( s )) → Ω ∞ H ∗ ( Z ( s − → . Applying Ω to this yields an exact sequence0 → Ω M → L H ∗ ( Z ( s )) → R s H ∗ ( Z ) → Ω M → . . . . Now we note that Ω M = 0. Lemma 4.35 tells us that ΣΩ M is the cokernelof sq : M → Φ( M ). But this map is onto, as it fits into a squareΩ ∞ H ∗ ( K ( s − sq (cid:15) (cid:15) / / M sq (cid:15) (cid:15) Φ(Ω ∞ H ∗ ( K ( s − / / Φ( M )in which the horizontal maps are onto by construction, and the left verticalarrow is onto by Lemma 4.28.To conclude, E ∞∗ , ∗ ( Z ( s )) = U Q ( L ∗ H ∗ ( Z ( s ))), and this fits into a shortexact sequence in HQU : Z / → Λ ∗ (Ω M ) → U Q ( L ∗ H ∗ ( Z ( s ))) → U Q ( R ∗≥ s H ∗ ( Z )) → Z / . A rogue differential for a 0–connected finite complex.
Here isperhaps the simplest example of a rogue differential occurring in the spectralsequence of a 0–connected spectrum X .Let the spectrum X be the cofiber of 4 : R P → R P , so that X fits intoa cofibration sequence R P → X → Σ R P . As 4 has Adams filtration 2, we are guaranteed that H ∗ ( X ) ≃ H ∗ ( R P ∨ Σ R P ) ≃ H ∗ ( R P ) ⊕ Σ H ∗ ( R P ) , as right A –modules. For i = 1 , , ,
4, let a i ∈ H i ( X ) be the image ofthe nonzero element under the inclusion R P ֒ → X , and let b i ∈ H i +1 ( X )project to a nonzero element under the projection X → Σ R P .As H ∗ ( X ) ∈ U , if there were no rogue differentials, then E ∞∗ , ∗ ( X ) = E ∗ , ∗ ( X ). We show this is impossible. Proposition 6.7.
In the spectral sequence, d ( b ) = a . Before proving this, we note some properties that X must (not) have. Lemma 6.8. X is not homotopy equivalent to R P ∨ Σ R P . OD 2 HOMOLOGY OF INFINITE LOOPSPACES 45 a a b a , b a a , b a , b a
211 0 − − − − s \ t Figure 1. E s,t ( X ) Proof.
This follows easily from the fact that the identity on R P has stableorder 8, not 4 [T]. (cid:3) Corollary 6.9.
The evaluation H ∗ (Ω ∞ X ) → H ∗ ( X ) is not onto.Proof. R P ∨ Σ R P is the wedge of two (dual) Brown–Gitler spectra, andthus is homotopy equivalent to any other 2–complete connective spectrum Y with isomorphic mod 2 homology such that H ∗ (Ω ∞ Y ) → H ∗ ( Y ) is onto[HK]. (cid:3) Proof of Proposition 6.7.
Figure 1 shows the − E ∗ , ∗ ( X ) for the spectral sequence converg-ing to H ∗ (Ω ∞ X ).Recalling that d ≡
0, and that differentials take primitives to primitives,the only possible nonzero differential off of the − d ( b ) = a .Thus if d ( b ) = a did not hold, then we could conclude that E ∞− , ∗ ( X ) = E − , ∗ ( X ), so that H ∗ (Ω ∞ X ) → H ∗ ( X ) would be onto, contradicting thecorollary. (cid:3) The spectral sequence for S ∪ η D . Let X = S ∪ η D = Σ − Σ ∞ C P ,so the spectral sequence is converging to H ∗ (ΩΩ ∞ Σ ∞ C P ). Then H ∗ ( X ) = h x, y i with | x | = 1, | y | = 3, and ySq = x . Thus d ( y ) = Q x , and so d ( Q y ) = Q Q x , which is zero by the Dyer–Lashof Adem relations. Itfollows that Q y is an element in L H ∗ ( X ) that is not in the Dyer–Lashofalgebra module generated by L H ∗ ( X ) = h x i . Thus this is an examplewhere the algebraic condition of Corollary 1.14 fails to hold, even while thegeometric condition clearly does.Another aspect of our theory easily seen here is that, though E and E ∞ support Dyer–Lashof operations, the in between pages needn’t. Forexample, Q x ∈ B , but Q Q x = Q Q x B . Appendix A. Proof of Proposition 2.1
We need to explain the last property of S –modules listed in Proposi-tion 2.1. This said that, given an S –module X , there is a weak naturalequivalence hocolim n Σ − n Σ ∞ X n → X. We thank Mike Mandell for helping us be accurate in the following dis-cussion.Let Σ ∞ n : T →
Spectra be left adjoint to X X n . Recall that an S –module is a special sort of L –module. The functor sending a spectrum X tothe S –module S ∧ L L X is left adjoint to the functor sending an S –module X to F L ( S, X ), just regarded as a spectrum (and not as an L –module).There is a weak equivalence of S –modules S ∧ L L (Σ ∞ n X n ) → Σ − n Σ ∞ X n given as the adjoint to the composite of maps of spectraΣ ∞ n X n → Σ − n Σ ∞ X n → Σ − n F L ( S, Σ ∞ X n ) = F L ( S, Σ − n Σ ∞ X n ) . There is a map of S –modules S ∧ L L (Σ ∞ n X n ) → X given as the adjoint to the composite of maps of spectraΣ ∞ n X n → Σ ∞ n F L ( S, X ) n → F L ( S, X ) . The desired weak natural equivalence is obtained by taking the hocolimitover n of the zig-zagΣ − n Σ ∞ X n ∼ ←− S ∧ L L (Σ ∞ n X n ) → X We note that the n = 0 case of the zig-zag here has the formΣ ∞ 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 suitablyon the tower P ( X ) as described in Theorem 3.8.The paper [AK] explored the explicit model from [Ar] for the tower asso-ciated to the functor sending a space Z to the spectrum Σ ∞ + Map T ( K, Z ),where K is a fixed CW complex. Call this tower P ( K, Z ), indicating its func-toriality in both variables. (The more awkward notation P K ( X ) was usedin [AK].) It comes with a natural transformation e : Σ ∞ + Map T ( K, Z ) → P ( K, Z ) which is an equivalence if the dimension of K is less than the con-nectivity of Z .We note that the properties of our category of spectra needed to form ourconstructions correspond to the first five properties of S listed in Proposi-tion 2.1. OD 2 HOMOLOGY OF INFINITE LOOPSPACES 47
The product theorem, [AK, Thm.1.4], says that there is a weak naturalequivalence of towers P ( K ∨ L, Z ) ∼ −→ P ( K ) ∧ P ( L ) . This generalizes to more than two factors in a straightforward way. Inparticular, if W d K denotes the wedge of d copies of K , there is a Σ d –equivariant map of towers of spectra P ( _ d K, X ) → P ( K ) ∧ d which is a nonequivariant equivalence.Specialized to K = S n , one gets a tower P ( S n , Z ) approximating Σ ∞ + Ω n Z with d th fiber naturally weakly equivalent to C n ( d ) + ∧ Σ d (Σ − n Z ) ∧ d , as ex-pected. Here C n 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 towersΘ( d ) : C n ( d ) + ∧ Σ d P ( _ d S n , Z ) → P ( S n , Z )compatible with the usual C n operad action on Ω n Z [Ma]. In particular,from [AK, Thm.1.10], we learn that the square in the diagramΣ ∞ + C n ( d ) × Σ d (Ω n Z ) d C n ( d ) + ∧ Σ d e ∧ d (cid:15) (cid:15) Θ( d ) / / Σ ∞ + Ω n Z e (cid:15) (cid:15) C n ( d ) + ∧ Σ d P ( S n , Z ) ∧ d C n ( d ) + ∧ Σ d P ( W d S n , Z ) Θ( d ) / / ∼ o o P ( S n , Z )commutes. Furthermore, the map on fibers induced by the map of towerscorresponds to the maps induced by the operad structure in the expectedway.Given a spectrum X , our tower is then defined to be P ( X ) = hocolim n P ( S n , X n ) , where the homotopy colimit is over natural transformations P ( S n , X n ) ∧ −→ P ( S n +1 , Σ X n ) → P ( S n +1 , X n +1 ) . Here the first map is the smashing map from [AK, Thm.1.1].The d th fiber of the tower P ( X ) then naturally identifies withhocolim n C n ( d ) + ∧ Σ d (Σ − n Σ ∞ X n ) ∧ d ≃ C ∞ ( d ) + ∧ Σ d X ∧ d = D d X. Finally the weak natural transformation e : Σ ∞ + Ω ∞ X → P ( X ) is definedas the compositeΣ ∞ + Ω ∞ X ∼ ←− hocolim n Σ ∞ + Ω n X n hocolim n e −−−−−−→ hocolim n P ( S n , X 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,
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