TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 7, July 1998, Pages 2629–2664 S 0002-9947(98)02263-6
THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE h i
JESUS´ GONZALEZ´
Abstract. We give a complete description of the quotient complex obtained C by dividing out the Fp Eilenberg-Mac Lane wedge summands in the first term of the BP 1 –Adams spectral sequence for the sphere spectrum S0.Wealso h i give a detailed computation of the cohomology groups Hs,t( ) and obtain as 2 1 C a consequence a vanishing line of slope (p p 1)− in their usual (t s, s) representation. These calculations are interpreted− − as giving general simple− conditions to lift homotopy classes through a BP 1 resolution of S0. h i
1. Introduction One of the main motivations in algebraic topology is the need to have com- putable methods to analyze homotopical properties of spaces. Indeed, it is often the case that the solution of a given geometrical problem can be reduced to the study of certain key elements in the homotopy groups of a suitable space. The generalized Adams spectral sequence provides a convenient theoretical framework to study these questions; however in practice, even determining whether or not a class is null-homotopic may be an extremely difficult task. This is reflected in the fact that the complexity of calculations with the spectral sequence gets rapidly out of hand. However the analysis of a particular homotopy class can be greatly simplified if the Adams spectral sequence is based at a suitably chosen spectrum, one detecting in a “natural” way the homotopy problem under consideration. Since topological K-theory has shown to be a powerful tool in the solution of geometric problems (Atiyah-Singer Index Theorem or Adams’ determination of tangent fields to spheres) it is desirable to have a manageable model to handle calculations in the K-theory Adams spectral sequence. A good deal of homotopical information has al- ready been obtained through calculations directly related to this spectral sequence; for instance this was the approach used to obtain a homotopy theoretical descrip- tion of the v1–periodic classes in the stable homotopy groups of spheres in [11], [20] and [21]. Subsequently the techniques have been applied to the study of the stable geometric dimension of vector bundles over projective spaces [9] and also in a geometric construction of the K–theory localization of odd primary Moore spaces [8]. A more recent application of the theory has been made to the classification of the stable p–local homotopy types of stunted spaces arising from the classifying
Received by the editors August 2, 1994. 1991 Mathematics Subject Classification. Primary 55T15; Secondary 55P42. Key words and phrases. Adams spectral sequence, Adams resolutions, BP 1 spectrum, Eilenberg-Mac Lane spectrum, cobar complex, weight spectral sequence, vanishingh line.i The author held a fellowship from the Conacyt while this research was performed.
c 1998 American Mathematical Society
2629
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space for the group of permutations on p letters (p an odd prime) [14]. The solution of each of the problems above depends on the triviality of certain key homotopy classes or “obstructions”, and the success of using the K–theory Adams spectral sequence to analyze them is based in the fact that each time such an obstruction is trivial it vanishes already at the E2 term of the spectral sequence. Thus while the complete classification of the stable homotopy types of stunted projective spaces [10] was accomplished by showing that many of the relevant obstructions were killed by high–order differentials in the classical Adams spectral sequence, the odd pri- mary problem (stunted lens spaces) was solved in [14] by replacing the analysis of Adams’ high–order differentials by the K–theory calculations in that paper. 0 The second term E2(S ) of the Adams spectral sequence based at (a stable sum- mand of connective p–local) K–theory is still far from being completely described, nevertheless the obstructions above can be easily managed within the context of this spectral sequence. The aim of the paper is to give full details on how this can be accomplished. Roughly speaking the analysis is done through a certain quotient complex of the first term E (S0) in the spectral sequence, whose co- C 1 homology H∗( ) detects the above obstructions. In more detail, letting V stand C 0 for the kernel of the projection E1(S ) ,the usual three-term long exact se- →C 0 quence associated to the extension 0 V E1(S ) 0 relates H∗( ) 0 → → →C→ C to E2(S ). The cohomology of is computed in full and, together with the long exact sequence just described, producesC enough information to show the triviality 0 0 of the obstructions in E (S ). In a sense H∗( ) is an approximation of E (S ), 2 C 2 so that a closer study of H∗(V ) would be in order. In this direction we should mention the work in [6] computing V 1, the homological degree s = 1 of the com- plex V, in the 2-primary case. An alternative approach uses a common subcomplex 0 (1) of both and E1(S ). The long exact sequence associated to the extension 0C (1) EC (S0) E (S0)/ (1) 0 yields a long exact sequence of the form →C → 1 → 1 C → s s 0 s s+1 H ( (1)) E (S ) Ext (Z/p) H ( (1)) , ···→ C → 2 → rel → C →··· s where Extrel(Z/p) is a graded group of extensions arising in a certain context of relative homological algebra. H∗( (1)) is computed in full in this paper, and the relative “Ext” groups are studied inC [15]. We now say a few words about the techniques used and the organization of the paper. Section 2 recollects the preliminary results we need and sets up some notation. This section is mainly descriptive, and the results are stated only with references to the literature. We only remark that Theorem 2.7 is a partial gener- alization of results in [18] and (as noted in that paper) can be proved with similar methods. Moreover the proof of Theorem 2.9 has appeared in print only for the two primary situation ([19]); however the techniques used in that case generalize directly to the odd primary version. The remainder of the paper can be read independently of [19]. Section 3 defines and studies the quotient complex (X); in particular a “filtered” homological interpretation of it is derived. Section 4C introduces our main technical tool: Lellmann and Mahowald’s weight spectral sequence computing the cohomology of (X). The section studies the E1 term of this spectral sequence, and sections 5 andC 6 settle all higher differentials as well as the E extensions for a number of spectra related to the sphere spectrum. A full description∞ of Hs,t( (X)) 2 1 C follows, as well as a vanishing line of slope (p p 1)− in the (t s, s) chart for these groups. The calculations are summarized− in− Corollary 6.11;− however, as
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2631 h i it stands, it is difficult to appreciate its full strength. Thus the final section de- rives a topological interpretation (Theorem 7.5) which gives a powerful criterion to lift homotopy classes through a BP 1 resolution of the sphere spectrum. Loosely speaking, this criterion claims thath thei obstructions to lifting to the second stage in this resolution are given by the image of the J-homomorphism, whereas further obstructions behave almost as in the classical case. The material in this paper is part of the author’s Ph.D. dissertation, written at the University of Rochester under the supervision of Professor Samuel Gitler. The author also wishes to thank the referee of the paper for pointing out a couple of mistakes in the original formulation of Definitions 2.4 and 3.1 and for a number of suggestions leading to a considerable improvement in the exposition of the material.
2. Preliminaries Unless otherwise specified, all spectra are localized at an odd prime p.Welet q,Z(p)and Fp stand for 2p 2, the ring of p–local integers and the field with p elements, respectively. The− (p–localization of the) spectrum for connective (real or complex) K–theory decomposes as a wedge of suspensions of a spectrum most commonly denoted by BP 1 ([3], [16]). For simplicity we use Kane’s notation ` for this spectrum [17]. Thus `hisi an associative and commutative ring spectrum whose 0 homotopy is a polynomial algebra over Z(p) on a variable v πq(`). Let i : S ` pr ∈ → denote the unit of ` and let S0 i ` ` be the canonical cofibration. For s 0let s → → s s 0 ≥ 0 ` denote the s–fold smash product of ` with itself and let Ss =Σ− ` (` =S = S0). The standard `–Adams resolution of a spectrum X is obtained by repeatedly smashing with i : S0 ` and taking fibers:
→
o o o X o S X S X 1 ∧ 2 ∧ ···
i X i S1 X i S2 X
∧ ∧ ∧ ∧ ∧
l X l S1 XlS2X ∧ ∧ ∧ ∧∧ The homotopy groups of these fibrations assemble into an exact couple whose associated spectral sequence is the `–Adams spectral sequence for X. The first term of this spectral sequence is given by st s (1) E1 (X; `)=πt s(` Ss X)=πt(` ` X) − ∧ ∧ ∧ ∧ and the first differential is πt(∆s), where ∆s is the composite
s s+1 s pr ` X s s +1 i ` X s +1 (2) ` ` X ∧ ∧ ` ` X = S 0 ` X ∧ ∧ ` ` X. ∧ ∧ −−−−→ ∧ ∧ ∧ ∧ −−−−→ ∧ ∧ Definition 2.1. Margolis showed [23] that a connective, locally finite (p–local) spec- trum X can be decomposed in a unique way, up to nonnatural equivalences, as a 0 wedge sum X KV (X) Xh i,whereV(X) is a locally finite, graded Fp vector ' ∨ space and KV (X) is the associated wedge of Fp Eilenberg-Mac Lane spectra, and 0 i where Xh i has no further such summands. We also define spectra Xh i (i>0) i i 1 2 and maps Xh i Xh − i (Adams projections) by requiring that Xh i 1 0 → 0 ··· → → Xh i Xh i be a minimal Fp–Adams resolution of Xh i. The homotopy types of these→ spectra and maps are again unique up to nonnatural equivalences [1, pg. 28]. Due to the lack of functoriality, choices will always be present in handling these constructions.
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Definition 2.2. For s 1letRs denote the set of s–tuples of positive integers. ≥ We will generally denote an element (n , ,ns) Rs simply by n, and define 1 ··· ∈ σ(n)= ni and ν(n!) = ν(ni!), where ν(a) denotes the maximal power of p dividing the integer a. P P The analysis of the groups in (1) was began by Mahowald for the two primary case. The odd primary version of his result is given by the following theorem.
Theorem 2.3 ([8], [17], [18]). Let s 1.Forn Rsthere is a spectrum Bn and 0 σ(n)q ≥ν(n!) ∈ s an equivalence (` Bn)h i Σ `h i. Moreover ` ` is homotopy equivalent ∧ ' ∧ to ` Bn. n Rs ∧ ∈W In terms of these splittings the maps in (2) have components ` Bn ` Bm (for X = S0) which neglecting the Eilenberg-Mac Lane summands∧ (in→ a way∧ to σ(n)q ν(n!) be made precise in the next section) are operations of the form Σ `h i σ(m)q ν(m!) → Σ `h i. Operations of this sort were studied by Lellmann, and we now de- scribe his results. Definition 2.4. Let b Z be a representative of a generator of the units in Z/p2,and consider the stable Adams∈ operation Ψb : ` `. According to [18] the operation b q → Ψ 1 lifts in a unique way through v :Σ ` `—the `-modulification of v πq(`)— defining− an operation φ : ` Σq` whose behavior→ in homotopy is given by∈ → r r r 1 p 1 (3) φ(v )=(k 1)v − , where k = b − . − Definition 2.5 ([13]). For nonnegative integers m n, define the Gaussian coeffi- n ≤ cient (( m)) and the Gaussian factorial n!! by n n 1 n m+1 ( k 1)(k − 1) (k − 1) (( n )) = − − ··· − , m (km 1)(km 1 1) (k 1) − − − ··· − n n 1 1 n!! = (( )) (( − )) (( )) , 1 1 ··· 1 where k is given as in (3). The formula ν(kn 1) = ν(n) + 1 [2, Lemma 2.12] − implies that these numbers lie in Z(p), and also give us the relations ν(n!!) = ν(n!) n n and ν(( m )) = ν ( m ). ν(n!) Definition 2.6. For a nonnegative integer n let Πn : `h i ` be the composite ν(n!) ν(n!) 1 0 → of the Adams projections `h i `h − i `h i = ` and the degree r ν(n!) → → ··· → map in `,wherer=n!!/p , which is a unit in Z(p). Lellmann showed that the th n n power φ of φ lifts through Πn, defining an operation nq ν(n!) (4) φn : ` Σ `h i. → r The effect in homotopy groups of each φn follows directly from (3): φn(v )= r n r n (( n )) ( k 1) v − .Sinceπ(`) is torsion free, it follows that no φn is a torsion operation.− Such operations∗ are quite manageable in view of the following result, which is a generalization of results in [18]. Theorem 2.7. Operations of the form σ(n)q ν(n!) σ(m)q ν(m!) (5) Σ `h i Σ `h i → are uniquely determined up to torsion operations by their effect in homotopy groups. Moreover, torsion operations of the form (5) factor as
σ(n)q ν(n!) α β σ(m)q ν(m!) Σ `h i KV Σ `h i, → →
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where V is a locally finite, graded Fp vector space and KV is the associated wedge of Fp Eilenberg-Mac Lane spectra. Using the techniques in [19] one can compute enough of the effect in homotopy 0 groups of the maps ∆s in (2) (for the case X = S ), so that 2.3 and 2.7 give a description of the components for ∆s (Theorem 2.9 below), up to operations factoring through Fp Eilenberg-Mac Lane spectra.
Definition 2.8. Let n =(n,... ,ns) Rs. A successor of n is any (s + 1)-tuple 1 ∈ m Rs+1 of the form m =(n1,... ,ne 1,j,ne j, ne+1,... ,ns)forsome1 e s ∈ − n−e ≤ ≤ and 0
0 We now extend the description of ∆s given in Theorem 2.9 above (X = S )to a more general type of spectra X. From now on HFp will denote the Fp Eilenberg- MacLanespectrum. Definition 3.1. A(p–local) connective, locally finite spectrum X is said to be (`, HFp)–prime ([19], [26]) if the Fp–Adams spectral sequence for ` X converges ∧ to ` (X) and collapses from its E2 term. We will require in addition that the ∗ q 0 q 0 composite (Σ ` X)h i , Σ ` X ` X (` X)h i yields a monomorphism in homotopy groups,∧ where→ the∧ first→ and∧ third→ maps∧ are the wedge inclusion and projection respectively, and the middle one is induced by the map v :Σq` `in 2.4. →
Remark 3.2. When X is an (`, HFp)–prime spectrum, all Adams projections i+1 i (` X)h i (` X)h i induce monomorphisms in homotopy groups, for a nontriv- ∧ → ∧ i+1 i ial homotopy class in (` X)h i mapping trivially into (` X)h i would produce by ∧ ∧
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definition a nontrivial differential dr in the Fp–Adams spectral sequence for ` X ∧ (r would be at least two, since d1 differentials are trivial when minimal resolutions are used). In particular the last condition in 3.1, together with the considerations i in 3.4 below, implies that Xh i is (`, HFp)–prime whenever X is. This extra con- dition allows an elementary proof of Lemma 3.3 as well as a conceptually cleaner interpretation (Proposition 3.10) of the ` regular complex (X) (to be defined in C 3.5). Spheres and stunted lens spaces are examples of (`, HFp)–prime spectra.
Throughout the rest of the paper X will always denote an (`, HFp)–prime spec- trum. Such spectra have the following property.
Lemma 3.3. Let K be a locally finite wedge of Fp Eilenberg-Mac Lane spectra. r α i Then any map Σ K (` X)h i induces trivial homomorphisms in homotopy groups. → ∧ q Proof. We can assume K = HFp.Sincethemapv:Σ` `in 2.4 is of positive classical Adams filtration, Margolis’ theorem [23, Theorem→ 3] implies that the com- r α i 0 q 0 posite Σ K (` X)h i (` X)h i (Σ− ` X)h i is null homotopic, where the middle map→ is∧ the Adams→ projection∧ → and the∧ last one is a suitable suspension of that in 3.1. The result follows from the hypothesis that the last two maps in this composite are injective in homotopy groups.
j 0 Remark 3.4. The wedge inclusion (` X)h i , ` X induces a map from the ∧ 0 → ∧ 1 minimal Fp–Adams resolution of (` X)h i into the Adams resolution `h i 0 ∧ ···→ ∧ X `h i X. This map of resolutions can be chosen so that each resulting map → i ∧ i i 0 i (` X)h i, `h i X agrees with the wedge inclusion of (`h i X)h i into `h i X. ∧ → ∧ π 0 ∧ ∧ Similarly the wedge projection ` X (` X)h i induces a corresponding map between the above resolutions; moreover∧ → choices∧ can be made so that each resulting i i i i 0 map `h i X (` X)h i agrees with the wedge projection of `h i X onto (`h i X)h i ∧ → ∧ i i i ∧ ∧ and so that each composite (` X)h i, `h i X (` X)h i is the identity. Thus i 0 ∧ i → ∧ → ∧ i 0 i in particular (`h i X)h i =(` X)hi.Likewise (` Xh i)h i =(` X)hi. ∧ ∧ ∧ ∧ To simplify notation we write Margolis’ decomposition 2.1 for the spectrum s s s s s s 0 ` ` X as ` ` X KV (X) Y (X), so that Y (X)=(` ` X)h i and ∧ s ∧ ∧ ∧ ' ∨ ∧ ∧ KV (X) is the locally finite wedge of Fp Eilenberg-Mac Lane spectra associated to s the graded Fp vector space V (X). Since any smash product KV X is again a KV –type spectrum [24, Proposition 6.6], it follows from 2.3 and 3.4∧ that s σ(n)q ν(n!) (6) Y (X) Σ (` X)h i, ' ∧ n R _∈ s and equation (1) gives st s st (7) E (X; `)=πt(Y (X)) V (X); 1 ⊕ moreover the differential in E1(X; `), as given in (2), is invariant on V (X) in view of 3.3 and (6). Definition 3.5. The `–regular cochain complex (X) is defined to be the quotient C complex of E1(X; `) by the subcomplex V (X). st s Thus (X)=πt(Y (X)), and the differential in (X) is induced by the compos- ite C C j s s +1 Y s(X) , ` ` X ∆s ` ` X π Y s+1(X), where j and π are the wedge → ∧ ∧ −→ ∧ ∧ →
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inclusion and projection respectively and ∆s is given in (2). By (6) this differential has components of the form
σ(n)q ν(n!) σ ( m ) q ν ( m !) (8) π Σ (` X)h i π Σ ( ` X ) h i ∗ ∧ −→ ∗ ∧ for n Rs and m Rs , each of which is induced by the corresponding composite ∈ ∈ +1 σ(n)q ν(n!) j σ(n)q ν(n!) j Σ (` X)h i , Σ `h i X , ` Bn X ∧ → ∧ → ∧ ∧ j s s +1 , ` ` X ∆s ` ` X (9) → ∧ ∧ −→ ∧ ∧ π π σ ( m ) q ν ( m !) ` B m X Σ ` h i X −→ ∧ ∧ −→ ∧ π σ ( m ) q ν ( m !) Σ ( ` X ) h i, −→ ∧ where each map j and π is either a wedge inclusion or projection given by 2.3 or 3.4. Theorem 3.6. a) With the notation of 2.8, if m is a successor of n, (8) is e σ(n)q induced by ( 1) Σ Πn,m. − b) If mr+1 = nr for r =1,... ,s, (8) is induced by a map compatible through Adams projections with the σ(n)q–suspension of the composite
φm X 0 1 ∧ m q ν ( m !) m q ν(m !) (` X)h i , ` X Σ 1 ` h 1 i X Σ 1 (` X)h 1 i, ∧ → ∧ −−−→ ∧ → ∧ where the first and last maps are the wedge inclusion and projection respec- tively. c) In any other case (8) is trivial. Proof. In the notation of 2.9 we see that the composite in (9) can be rewritten as
j (πϕi) X σ(n)q ν(n!) σ(n)q ν(n!) ∧ σ ( m ) q ν ( m !) Σ (` X)h i , Σ `h i X Σ ` h i X ∧ → ∧ −−→ ∧ π σ(m)q ν(m!) Σ (` X)h i, → ∧ which by 3.3 induces in homotopy the same morphism as the composite obtained by replacing the middle map above by Dn,m X. The result follows from 2.9 and the considerations in 3.4. ∧ Definition 3.7. Let R be a commutative ring with a decreasing filtration F iR F i+1R satisfying F iR F jR F i+jR. We consider filtered graded R–modules⊇ i · i+1⊆ i j i+j M and M 0 with F M F M and F R F M F M. A homomorphism ⊇ · ⊆ i i f : M M 0 is said to be filtration preserving if f(F M) F M 0. The tensor → ⊆ product filtration makes M M 0 a filtered R module (see for instance (20) in the next section). A filtered graded⊗ coalgebra over R is a graded coalgebra Γ which is also a filtered graded R module in such a way that both structural maps ε :Γ R and ∆ : Γ Γ Γ are filtration preserving. A filtered graded right comodule→ over Γ is a graded→ right⊗ comodule M over Γ which is also a filtered graded R module in ϕ such a way that the structural map M M Γ is filtration preserving. Filtered graded left comodules are defined analogously.→ ⊗
Example 3.8. On the ring Z(p) consider the (nonnegative) filtration defined by i a F Z(p) = x Z(p) ν(x) i (ν(x)=ν(a) ν(b)ifx= ,a,b Z,where { ∈ | ≥ } − b ∈ ν(a)andν(b) are as in 2.2). Define a filtered graded coalgebra Γ over Z(p) by k jq Γ =0ifk 0(mod q)andΓ is a Z(p)–free module with generator tj of filtration 6≡
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ν(j!). The coproduct ∆ is defined by ∆(ti)= tr ts and the counit by − r+s=i ⊗ ε(t0) = 1. For any spectrum X, ` (X) becomes a filtered graded right comodule ∗ P n over Γ with coaction Ψ : ` (X) ` (X) ΓgivenbyΨ(z)= n φ (z) tn, ∗ → ∗ ⊗ 0 ⊗ where φ : ` Σq` is defined in 2.4 and where the filtration in ` (X≥) is the usual → P∗ Fp–Adams filtration (Ψ is filtration preserving in view of 2.6). If in addition X 0 is an (`, HFp)–prime spectrum, then π ((` X)h i) is also a filtered graded right ∗ ∧ 0 comodule over Γ. In fact, by 3.3 the wedge projection ` X (` X)h i yields 0 ∧ → ∧ an epimorphism ` (X) π ((` X)h i) of filtered graded right comodules. In the ∗ →n ∗ ∧ 0 quotient, the action of φ on an element z π ((` X)h i) will still be denoted n ∈ ∗ ∧0 by φ (z). In particular, the Γ coaction on π ((` X)h i) is described by the same formula as the corresponding one on ` (X). ∗ ∧ ∗ Note 3.9. Let Γ be as above. For filtered graded right and left Γ comodules M and N respectively, let CΓ(M,N) denote the cobar complex [28, A1.2.11]. Taking the s tensor product filtration on each Cs (M,N)=M Γ⊗ N, we obtain a corre- Γ ⊗ ⊗ sponding filtration on the complex CΓ(M,N), which we call the Adams filtration on i CΓ(M,N). For i Z let AF CΓ(M,N) be the subcomplex consisting of elements ∈ in CΓ(M,N) having Adams filtration at least i. As shorthand, we will denote by i i 0 AF (X) the complex AF CΓ(π ((` X)h i), Z(p)). ∗ ∧ Theorem 3.10 ([19], p =2). There is an isomorphism (X) AF 0(X) of filtered complexes. C ' ν(n!) 0 Proof. For n Rs let Πn :(` X)h i (` X)h i be the composite of the ∈ ν(n!)∧ →ν(n!)∧1 0 Adams projections (` X)h i (` X)h − i (` X)h i and the ∧ 0 → ∧ ν(→n!) ··· → ∧ degree r map in (` X)h i,wherer=n1!! ns!! p− (see 2.8). By 3.2 and (6) these maps yield isomorphisms∧ ··· (10) st(X)= πν(n!) (` X), C t σ(n)q ∧ n R − M∈ s f where πn(` X) stands for the subgroup of πn(` X) consisting of the homotopy ∧ ∧ f classes of classical Adams filtration at least f (when f =0,πn(` X) should be 0 ∧ s,t interpreted as πn((` X)h i). In view of 2.6 and 3.6 the differential (X) s+1,t(X) extends to∧ a map C → C 0 0 (11) πt σ(n)q((` X)h i) πt σ(m)q((` X)h i) − ∧ −→ − ∧ n R m R M∈ s ∈Ms +1 0 0 whose component πt σ(n)q((` X)h i) πt σ(m)q((` X)h i) is trivial, unless: − ∧ → − ∧ a) m is a successor of n, in which case it is multiplication by ( 1)e where e is as in 2.8; or − b) mr+1 = nr,r=1,... ,s, in which case it is induced by the composite 0 m q m q 0 (` X)h i , ` X Σ 1 ` X (Σ 1 ` X)h i, where the first and third∧ maps→ are the∧ wedge→ inclusion∧ and→ projection∧ respectively, and the mid- m m q dle one is induced by φ 1 : ` Σ 1 `. → 0 Identification of an element z πt σ(n)q((` X)h i) with the corresponding ∈ − ∧ s,t 0 z[tn1 tns] C (π ((` X)h i); Z(p)) |···| ∈ Γ ∗ ∧ (where n =(n1,... ,ns)) sets an isomorphism 0 s,t 0 h : πt σ(n)q((` X)h i) CΓ (π ((` X)h i); Z(p)) − ∧ ' ∗ ∧ n R M∈ s
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4. The weight spectral sequence Throughout the rest of the paper Γ will be the filtered coalgebra defined in 3.8. In the last section the regular complex (X) is identified with the subcomplex of 0 C CΓ(π ((` X)h i), Z(p)) consisting of the elements of nonnegative Adams filtration. In particular,∗ ∧ (X) is a (nonnegative) filtered cochain complex and its cohomology groups couldC be analyzed via the spectral sequence associated to this filtration. 0 This approach is effective when the Γ coaction on π ((` X)h i) is trivial (which in general is not the case). The analysis of the (2–primary)∗ ∧ general situation was completed in [19] by considering a different filtration on (X). In this section we give a description of the first term of the resulting spectralC sequence in the odd primary case. We begin by fixing some notation. For a filtered graded Γ–comodule M (M = Z(p))wedenotebydthe differential in CΓ(M; Z(p)). Only the differential 6 0 in CΓ(Z(p), Z(p)) will be denoted by ∂.WhenM=π((` X)h i), the restriction of ∗ ∧ d to (X) will be denoted by d too. A cobar product [tn1 tns]inCΓ(Z(p),Z(p)) C |···| will be denoted by [tn], where n =(n1,... ,ns). Definition 4.1. For a filtered graded Γ–comodule M we define the total degree, the homological degree and the weight filtration degree of an element m[tn] s ∈ C (M; Z(p))asr+qσ(n), s and σ(n) respectively, where m Mr and σ(n)isasin Γ ∈ 2.2. The weight filtration degree of m[tn] will be denoted by WF(m[tn]).
As usual the differential in CΓ(M; Z(p)) depends both on the coalgebra Γ (that is, ∂) and the Γ coaction on M (see for instance [28, A1.2.11]). In particular, when 0 M = π ((` X)h i), 3.8 gives ∗ ∧ e(m) ∂[tn]= ( 1) [tm], m − (12) X n0 d(z[tn]) = φ (z)[tn tn]+z∂[tn], 0 | n 1 X0≥ where the summation on the right hand side of the first formula runs over the set of all successors m Rs+1 of n with m =(n1,... ,ne(m) 1, j(m), ne(m) j(m), ∈ − − ne(m)+1,... ,ns), 1 e(m) s and 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2638 JESUS´ GONZALEZ´ WSS is that of [12]: WFσst (13) Eσst(X)= , 0 WFσ+1,s,t σst σ st σst where WF = WF (X) (X). Thus elements in Er (X) have representatives x WFσst whose differential∩C dx lands in WFσ+r,s+1,t, so that the grading for the differentials∈ is σst σ+r,s+1,t (14) δr : E (X) E (X). r → r As in 4.1, the indices σ, s, t are called the weight filtration degree, the homological degree and the total degree respectively. 0 Remark 4.3. The total degree in CΓ(π ((` X)h i); Z(p)) does not change under the differential d, and therefore the WSS for∗ X∧ is actually a set of spectral sequences, σst one for each total degree t 0. Moreover an element z[tn] WF of weight ≥ ∈ 0 filtration degree σ t/q must have “coefficient” z πr((` X)h i) with r 0 (see 4.1). Since X is a connective spectrum (in the∈ sense that∧ π (X)=0for small enough) we see that at a given total degree t the weight filtration∗ on (X)is∗ bounded (uniformly on the homological degree s) and therefore the correspondingC spectral sequence collapses at a finite stage (which depends on the total degree taken). In particular, the WSS for X does converge to the cohomology groups of (X). C In view of (10) the subcomplex WFσ+1 in (13) is a direct summand of WFσ. This yields an obvious trigraded isomorphism (X) E0(X)(notofcomplexes), where the extra grading in (X) correspondingC to' the weight filtration degree. C Moreover, in view of (12) the differential in E0(X) differs from that in (X)only 0 C by the Γ–coaction on π ((` X)h i)—in E0(X) this coaction is not present. These ∗ ∧ 0 remarks apply also to CΓ(π ((` X)h i); Z(p)) instead of (X) to produce the tri- graded isomorphism ∗ ∧ C 0 0 (15) CΓ(π ((` X)h i); Z(p)) E0CΓ(π ((` X)h i); Z(p)). ∗ ∧ ' ∗ ∧ Again, this is not an isomorphism of complexes; however the right hand side in (15)—that is, the quotient object associated to the weight filtration on the cobar 0 complex CΓ(π ((` X)h i); Z(p))— is isomorphic as a complex to ∗ ∧ 0 π ((` X)h i) CΓ(Z(p), Z(p)). ∗ ∧ ⊗ Since the above isomorphisms preserve Adams filtration, 3.10 gives Proposition 4.4. As a trigraded cochain complex, E0(X) is isomorphic to the 0 subcomplex of π ((` X)h i) CΓ(Z(p), Z(p)) consisting of elements of nonnegative Adams filtration.∗ ∧ ⊗ Definition 4.5. The spectral sequences associated to the Adams filtration on the complexes E0(X)andCΓ(Z(p),Z(p)) will be denoted by E0r(X),δ0r and Er,∂r respectively. { } { } The grading of these spectral sequences is taken as in (13), although this time we have four gradings: Adams and weight filtration degrees as well as homological and total degrees. From (14) we have that δ0, and therefore all differentials δ0r, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2639 h i do not change weight filtration nor total degrees; consequently we omit these two gradings in E0r(X)andδ0r takes the form is i+r,s+1 (16) δ r : E (X) E (X), 0 0r → 0r where i and s stand for Adams filtration and homological degrees respectively. Similar considerations hold for Er,∂r . We study the spectral sequence E r(X) { } { 0 } at the end of this section, after a careful analysis of the spectral sequence Er . { } As in (15) above, given a filtered object A,weletE0Adenote the associated quotient object. For instance, using the filtrations in Z(p) and Γ defined in 3.8 we form the associated graded ring E0Z(p) and the associated bigraded coalgebra E0Γ over E0Z(p).NotethatE0Z(p)can be identified with the polynomial ring Fp[x], where x is of filtration degree 1 and corresponds to multiplication by p in Z(p).On the other hand, E0Γ is bigraded by F iΓk (17) E Γik = , 0 F i+1Γk which is zero unless k = jq (j 0) and i ν(j!), in which case it is isomorphic ≥ i+ν(≥−j!) k k to Fp with generator represented by p tj Γ . Note that each E0Γ∗ is a ∈ graded module over the (graded) ring Fp[x], and this is the structure taken in the graded tensor product E Γ E Γ: 0 ⊗Fp[x] 0 k k0 k00 (18) E Γ E Γ ∗ = E Γ∗ E Γ∗ , 0 ⊗Fp[x] 0 0 ⊗Fp[x] 0 k +k =k 0 M00 where the Adams filtration degree i in a given E Γ k0 E Γ k00 is the image of 0 ∗ Fp[x] 0 ∗ the obvious morphism ⊗ i0k0 i00k00 k 0 k 00 (19) E Γ E Γ E Γ ∗ E Γ ∗ . 0 ⊗Fp 0 −→ 0 ⊗ F p [ x ] 0 i i i 0+M00= Observe also that the bigraded object associated to the tensor product filtration on Γ Γ, ⊗ (20) F i(Γ Γ)k = F i0 Γk0 F i00 Γk00 , ⊗ ⊗ k k k i i i ! 0+M00= 0+X00= agrees with E Γ E Γ as described in (18) and (19). Moreover the filtration 0 ⊗Fp[x] 0 preserving map ∆ : Γ Γ Γ induces a corresponding map E0∆:E0Γ E(Γ Γ) = E Γ E→Γ which⊗ takes the form → 0 ⊗ 0 ⊗Fp[x] 0 (21) E0∆(tj )= tm tj m, ⊗ − where the sum runs over all indices m suchX that the Adams filtration of tm tj m j ⊗ − equals that of tj, or equivalently ν( m ) = 0. This determines the coalgebra structure of E0Γ. Now by definition E0 = E0CΓ(Z(p), Z(p)), and the above remarks generalize to give an isomorphism of graded Fp[x]–complexes E0 CE0Γ(Fp[x], Fp[x]) (see also the proof of [28, A1.3.9]). In particular we get ' (22) E H∗(E Γ). 1 ' 0 Definition 4.6. For i 0letΓi be the bigraded (by total degree and Adams fil- ≥ k tration degree) Fp[x]–coalgebra defined through total degree by Γi = 0 except for License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2640 JESUS´ GONZALEZ´ i k k = jp q (0 j p 1), in which case Γi is Fp[x]–free on a generator τjpi of ≤ ≤ − i 1 Adams filtration degree j(1 + p + +p − ). The coproduct is given by − ··· j (23) ∆i(τjpi )= τmpi τ(j m)pi . ⊗ − m=0 X Since the lowest total degree element (other than τ0)ineachΓi appears in a large total degree (as i gets large), we can form the tensor product i 0 Γi (over ≥ the graded ring Fp[x]). N Proposition 4.7. As a bigraded Fp[x]–coalgebra, E0Γ is isomorphic to i 0 Γi. ≥ The next lemma is well known, and a proof can be found in [29]. N Lemma 4.8. Let 0 m j be integers with corresponding p–adic decompositions k ≤ ≤ k j = jkp and m = mkp .Then k 0 k 0 ≥ ≥ P 1 P a) ν(m!) = p 1 (m mk), − − k 0 j P≥ b) ν( m )=0if and only if mk jk k 0, j ≤ ∀ ≥ c) ν( )=ν(j) ν(m)provided m mk jk k 0. On the other hand, by (23) the diagonal in i 0 Γi takes the form≤ ∀ ≥ ≥ N ∆(τj p0 τj pl) 0 ⊗···⊗ l = (τm p0 τm pl) (τ(j m )p0 τ(j m )pl), 0 l 0− 0 l− l 0 k l ⊗···⊗ ⊗ ⊗···⊗ m≤ ≤j Xk≤ k and the result follows. Definition 4.9. For i 0letai,bi+1 CΓ(Z(p), Z(p)) be the elements given by ≥ ∈ ai =[tpi]andbi+1 = ∂ai+1. Remark 4.10. By (12), bi+1 is a sum of elements of the form [tk tpi+1 k](0 In view of 4.7 we can rewrite (22) as E1 i 0 H∗(Γi). The cohomology ' ≥ of each Γi is known to be a tensor product of an exterior algebra with generator N i represented in CΓi (Fp[x], Fp[x]) by [τp ] and a polynomial algebra with generator represented by [τpi τ(p 1)pi ]+ +[τ(p 1)pi τpi ] (see for instance the proof of [7, Theorem 4.1], where| the− dual situation··· − is described).| Thus 4.10 gives (24) E (E(αi) P (βi )) , 1 ' ⊗ +1 i 0 O≥ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2641 h i where the tensor product is taken over Fp[x], and αi and βi+1 are represented in CΓ(Z(p), Z(p)) by the elements ai and bi+1 respectively. Since a0 is a ∂–cycle whereas by definition ∂ai = bi (i 0), 4.10 implies that a represents a permanent +1 +1 ≥ 0 cycle in Er and that βi+1 is the ∂1-boundary of αi+1 in E1 (compare with 5.5 and the calculation{ } of differentials in section 5). In particular, the decomposition E1 = E(α0) i 0(E(αi+1) P (βi+1)) is one of chain complexes, and since all ⊗ ≥ ⊗ modules involved are Fp[x]–free we obtain N (25) E2 = E(α0). Since E(α0) is all concentrated in homological degree one, the spectral sequence Er collapses from its second term. The only nontrivial extensions in E are given { } ∞ by multiplication by x Fp[x], and we recover the well known cohomology of Γ as ∈ the exterior algebra (over Z(p)) on a generator represented by [t1] CΓ(Z(p), Z(p)) (see [7, Theorem 4.1] or [27, Proposition 7.4]). ∈ We use the information above to study the spectral sequence E r(X) .Since { 0 } all terms in the spectral sequence Er are Fp[x]–free, we have that the spectral { } 0 sequence associated to the Adams filtration in π ((` X)h i) CΓ(Z(p), Z(p)) agrees ∗ 0 ∧ ⊗ with E0(π ((` X)h i)) Er, 1 ∂r (where tensor products are taken over Fp[x]; { ∗ ∧ ⊗ ⊗ } see the remarks in (17) through (20)). Moreover, since X is (`, HFp)–primeweget 0 0 (26) E0 π ((` X)h i) = ExtAp (H∗((` X)h i; Fp); Fp) ∗ ∧ ∧ (which is an Fp vector space), where Ap is the mod p Steenrod algebra (we only remark that the gradings in (26) are shifted as usual: in the standard Extst-grading, s contributes to the Adams filtration degree whereas t s—the homotopy degree— − contributes to our total degree). By 4.4 it follows that E00(X) is the subcomplex 0 of E0(π ((` X)h i)) E0 consisting of elements of nonnegative Adams filtration ∗ ∧ ⊗ degree; moreover by (16) the differential 1 ∂0,aswellasitsrestrictionδ00,donot change Adams filtration degree, and we obtain⊗ 0 Proposition 4.11. E01(X) is the subcomplex of E0 π ((` X)h i) E1 consist- ing of elements of nonnegative Adams filtration degree.∗ ∧ ⊗ Remark 4.12. The next differential δ01 increases the Adams filtration degree by one, so that by 4.11 classes in E02(X) arise in two ways: In positive Adams filtration 0 degrees, E01(X)andE0(π ((` X)h i)) E1 have the same homology, while in zero ∗ ∧ ⊗ 0 Adams filtration degree all cycles in E0(π ((` X)h i)) E1 (whether or not they are 1 ∂ boundaries) produce homology∗ classes∧ in E (⊗X). By (25) the first type ⊗ 1 01 of elements are the positive Adams filtration multiples of 1 and α0 (scalars taken 0 in E0(π ((` X)h i)) and Fp[x]), whereas the second type of elements are 1 and ∗ ∧ α0 themselves with zero Adams filtration coefficients (if any), as well as all 1 ∂1 0 ⊗ boundaries of elements in E0(π ((` X)h i)) E1 having Adams filtration 1. ∗ ∧ ⊗ − Definition 4.13. A sequence I =(e0,e1,m1,e2,m2,...) of nonnegative integers, almost all zero and such that ei 0,1 i 0, is called admissible if the following conditions hold: ∈{ }∀ ≥ a) I =(0,0,0,...), (1, 0, 0, 0,...), 6 b) mj > 0 ek =1forsomek=1,... ,j. ⇒ I Under such conditions, define the admissible monomials ϕ CΓ(Z(p), Z(p))and ∈ φI E by ϕI = ae0 ae1 bm1 ae2 bm2 and φI = αe0 αe1 βm1 αe2 βm2 . ∈ 1 0 1 1 2 2 ··· 0 1 1 2 2 ··· License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2642 JESUS´ GONZALEZ´ Proposition 4.14. An Fp[x]–basis for the quotient E1/(∂1-boundaries) is given by I all admissible monomials φ together with 1 and α0; these last two elements are ∂1-cycles with ∂1 injective on the submodule generated by the admissible monomials φI . Proof. Since ∂ αi = βi i 0, the ∂ –boundary of a nonadmissible monomial 1 +1 +1 ≥ 1 is a sum of nonadmissible monomials, whereas the ∂1–boundary of an admissible monomial is the sum of a nonadmissible monomial plus perhaps some admissible ones. Moreover every nonadmissible monomial other than 1 and α0 is considered in this way once and only once. The result follows. Thus, besides the multiples of 1 and α0, the second type of elements in E02(X) mentioned at the end of 4.12 arise as cycles in E01(X) of the form I 0 (27) γ∂1φ E0 π ((` X)h i) E1, ∈ ∗ ∧ ⊗ 0 I I where γ E0(π ((` X)h i)), γφ is of Adams filtration degree 1andφ is an ∈ ∗ ∧ − I admissible monomial (we omit the tensor product symbol in elements like γ ∂1φ I ⊗ and γ φ ). Since multiplication by x Fp[x] increases Adams filtration degree by ⊗ I I ∈ 1, we get that xγ∂1φ = δ01(xγφ ) is a boundary in E01(X). Hence multiplication by x is trivial over the elements described in (27). Note also that these elements lie in homological degrees at least 2. The following description of E02(X)results. Proposition 4.15. In homological degrees s 2 multiplication by x is trivial, and ≥ an Fp–basis for E02(X) is given by the elements described in (27) with γ running 0 over an Fp–basis of E0 π ((` X)h i) . On the other hand, in homological degrees ∗ ∧ 0 s 1, E02(X) consists of E0 π ((` X)h i) E(α0) (tensor product over Fp[x]). ≤ ∗ ∧ ⊗ The next differentials δ0r (r 2) increase Adams filtration degree at least by 2, and since all elements in s ≥1 are permanent cycles and those in s 2have ≤ ≥ zero Adams filtration degree, we get that E r(X) collapses from its second term, { 0 } so that 4.15 also gives a description of E0 (X). Moreover, by the remarks below ∞0 (25), the spectral sequence E0(π ((` X)h i)) Er converges to the cohomology 0 { ∗ ∧ ⊗ } of π ((` X)h i) CΓ(Z(p), Z(p)), and the above analysis implies that E0r(X) ∗ ∧ ⊗ 0 { } converges to E1(X). Representatives in (X) CΓ(π ((` X)h i); Z(p))forthe ∗ C ⊆ ∧ 0 described homology classes in E1(X) are as follows: For a given c π ((` X)h i)let 0 ∈ ∗ ∧ γ E0(π ((` X)h i)) be the associated element; then c 1andca0 represent γ 1and ∈ ∗ ∧ I · · I · I γ α0 respectively and (for suitable γ as in (27)) c∂ϕ represents γ∂1φ (here c∂ϕ · 0 is thought as an element of CΓ(π ((` X)h i); Z(p)) via the graded isomorphisms in ∗ ∧ (15)). Since multiplication by p Z(p) corresponds to multiplication by x Fp[x], ∈ ∈ the considerations after (27) take care of the extensions in E0 (X)andweendup ∞ with the following description of E1(X) analogous to that in [19] at the prime two. Theorem 4.16. The first term in the WSS for X, E1(X), is given in homological 0 degrees s 1 as π ((` X)h i) E(α0). In homological degrees s 2, E1(X) is ≤ ∗ ∧ ⊗ ≥ Fp–free on generators represented by elements I 0 c∂ϕ (X) CΓ(π ((` X)h i); Z(p)), ∈C ⊆ ∗ ∧ 0 where c π ((` X)h i) runs over a set of representatives for an Fp basis of ∈ ∗ 0 ∧ I I E0 π ((` X)h i) , ϕ CΓ(Z(p); Z(p)) is admissible and cϕ is of Adams filtration 1. ∗ ∧ ∈ − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2643 h i Remark 4.17. In view of 4.10, the Adams filtration degree of ∂ϕI is higher by one I than that for ϕ , so that (26) and 4.16 suggest writing E1(X) in homological degrees s 2as ≥ AF (ϕI ) 1 0 I Ext− − (H∗((` X)h i; Fp); Fp) ∂ϕ , Ap ∧ · I admissible M where the degree shown in the Ext functor corresponds to Adams filtration and the factor ∂ϕI is used to suggest the representative taken in (X). · C i 5. The weight spectral sequence for Rh i Let B be the suspension spectrum for the p–localization of the classifying space for the symmetric group on p letters, and consider the cofibration (28) B S 0 R, −→ −→ where the first arrow represents the Kahn–Priddy map (see [4]). In [22] it is shown that for p =2,b0 Rsplits as a wedge of suspensions of Z(2) Eilenberg-Mac Lane spectra. The methods∧ can easily be carried over at odd primes to produce a splitting nq (29) ` R Σ HZ(p), ∧ ' n 0 _≥ where HZ(p) stands for the Z(p) Eilenberg-Mac Lane spectrum. In particular, ` R is a ring spectrum whose homotopy ` (R) is given as a polynomial ring ∧ ∗ over Z(p) on a generator w `q(R) of zero Adams filtration. Using the clas- sical Adams spectral sequence∈ together with a standard change–of–rings isomor- phism (see for instance [25]), one readily obtains the following chart for ` (R): ∗ 6 6 6 6 6 t t t t t i+1 t t t2 t3 t4 wwi wi wi i t ti t t t i 1 − t t t t t t t t t t q q q q q q q q q q 2 q q q q q qqq 1 t t t t t ww2 w3 w4- t t t t t 0t qt 2qt 3qt 4qt as well as the fact that the map S0 R in (28) induces in ` homology a zero Adams filtration inclusion sending vn into→ pnwn. In particular,∗ this inclusion together with the action of φ on ` (S0) (see (3)) gives the corresponding action of φ on r 1 r ∗ r 1 r r 1 k 1 ` (R):φ(w)= (k 1)w − = c(( 1 )) w − ,wherec= − , which is a unit in ∗ p − p Z(p). Iterating, we obtain n r n r r n +1 r n n r r n (30) φ (w )=c (( )) (( − )) w − = c n !!(( )) w − . 1 ··· 1 n i 0 From 3.2 and 3.4 the corresponding chart for π ((` Rh i)h i) can be pictured by deleting elements of Adams filtration less that i∗ in the∧ chart above. As suggested License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2644 JESUS´ GONZALEZ´ m i 0 there, wi will denote the (zero Adams filtration) generator of πmq((` Rh i)h i)— m i m ∧ thus wi = p w in `mq(R). The isomorphisms in (15) imply that the cobar complex i 0 CΓ(π ((` Rh i)h i); Z(p))isZ(p)–free on generators ∗ ∧ m (31) wi [tn], where m 0andn Rs as in 2.2. Defining the (zero Adams filtration) ≥ ∈ s 0 elements ϑ Γand[ϑ ] ≥C ( ; )byϑ =n!!cnt and [ϑ ]=[ϑ ϑ ] n n Γ Z(p) Z(p) n n n n1 ns ∈ S∈ i |···|m if n =(n1,... ,ns), we see from 3.10 that (Rh i)isZ(p)–free on generators wi [ϑn]; C m moreover from the easy to check formula n!!(m n)!!(( n )) = m !! together with (12) − i and (30), we get the following expression for the differential in (Rh i): C m m n m m − 0 d(wi [ϑn]) = (( n )) w i [ ϑ n ϑ n ] 0 0 | n =1 (32) X0 n + ( 1)e(( e )) w m [ ϑ ϑ ϑ ϑ ]. j i ne 1 j ne j ne+1 − − 1 e s − ···| | | | |··· ≤ ≤ 0 Definition 5.1. Let Σ denote the filtered graded coalgebra free over Z(p) on gen- mq erators θm Σ of zero filtration and diagonal given by the formula ∆θm = m ∈ (( j )) θ j θ m j . 0 j m ⊗ − ≤P≤ As in 3.9 we extend the above filtration on Σ to one in CΣ(Σ; Z(p))andrefertoit as the Adams filtration. Also, as in 4.1 and its remarks, we have a weight filtration in this complex. The next result follows easily from the considerations above. m Lemma 5.2. The correspondence wi [ϑn] θm[θn] defines a chain isomorphism i ↔ (Rh i) CΣ(Σ; Z(p)) preserving both Adams and weight filtrations. C ' 00 In particular, the obvious inclusion Z(p) , CΣ (Σ; Z(p)) gives rise to an aug- mented complex which as usual is contractible→ (see for instance the proof of [28, i A1.2.12]). Thus although the cohomology of (Rh i) reduces to a single copy of Ci Z(p), a complete description of the WSS for Rh i will be crucial in our analysis of 0 i the WSS for (S )h i (in the next section). Another consequence of 5.2 is that the i WSS for Rh i is independent of i, so that in this section we will only consider the case i =0. Definition 5.3. To keep with the notations in 4.9 and 4.13 we define the following zero Adams filtration elements in CΓ(Z(p); Z(p)): for j 0weletuj =[ϑpj]= j pj 1 j+1 pj+1 ≥ p!!c aj , vj+1 = p (p )!!c bj+1 and for an admissible sequence I =(e0,e1,m1, I e0 e1 m1 e2 m2 I e2,m2,...)weleth =u0 u1 v1 u2 v2 ...=c(I)ϕ ,wherecis as in (30) and ej +mj pj!! cpj c(I)= with m0 =0. pmj j 0 Y≥ Remark 5.4. In terms of this notation, 4.16 gives the following description for E1(R): In homological degrees s 1, E1(R)=` (R) E(u0), whereas for s 2, ≤ ∗ ⊗ 1 m I ≥ E1(R)isFp–free on generators represented in (R) by the elements p w ∂h with C I I admissible and m 0(since∂uj+1 = pvj+1 we see that ∂h has Adams filtration ≥ 1 1 and is divisible by p in CΓ(Z(p); Z(p)), so that in the notation above the factor p affects ∂hI rather than wm). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2645 h i Remark 5.5. WSS-differentials δr are computed by finding optimal representatives for the classes in E1(R), that is, representatives whose differential d in (R)in- creases the weight filtration as much as possible. Although the differentialC of the representatives described in 5.4 increases weight filtration by 1, we can usually do 3 3 2 3 better; for instance when p =3theelementz=w ∂u2 (( 1 )) w [ ϑ 1 ] u 2 (( 2 )) w [ ϑ 2 ] u 2 1 − − is divisible by 3 in (R) (recall ∂u2 =3v2)and 3zrepresents the same element 1 3 C in E1(R)as 3w∂u2 does (they have weight filtration 9); moreover the formula d(w3u )=((3)) w 2 [ ϑ ] u +((3)) w [ ϑ ] u +[ϑ ]u w3∂u implies d(z)=d([ϑ ]u )= 2 1 1 2 2 2 2 3 2 − 2 3 2 ∂(u1u2). Since these elements are divisible by 3 in (R) and this complex has no 1 1 C torsion, we obtain d( 3 z)= 3∂(u1u2), which has weight filtration 12. This means 1 3 that 3 w ∂u2 survives to an element in E3(R) potentially yielding the δ3–differential 1 3 1 3 w ∂u2 3 ∂(u1u2). Such a differential might fail to hold if either one of the el- ements involved7→ dies in a previous stage of the WSS, in which case we should 1 3 proceed to find either a lower WSS-differential hitting 3 w ∂u2 ora(necessarily better) representative for this element having a d-differential with a higher weight filtration. Notation 5.6. Throughout the rest of the paper unless otherwise specified, I will stand for an admissible sequence and k(I) will denote the first nonnegative integer j such that ej =1,whereI=(e0,e1,m1,e2,m2,...). If no confusion arises, k(I) will simply be denoted by k. m I Definition 5.7. Define the elements Dj w h (R)(j=0,1), by the formulas m I m I m I ∈C d(w h )=D0w h +D1w h and m I m m j I D1w h = (( j )) w − [ ϑ j ] h . ν(m)=0 Xj The next result will produce as in 5.5 a family of differentials in the WSS for R. Recall from 4.1 that the weight filtration of an element x is denoted by WF(x). Lemma 5.8. For ν (m) m I m I m m p ν ( m ) I dD w h = dD w h = d (( ν ( m ) )) w − u h 0 − 1 − p ν ( m ) m m p ν ( m) I = (( ν ( m ) )) w − ∂ ( u h ) . − p ν ( m ) m I m I Remark 5.9. In view of 5.4 both D0w h and w ∂h have Adams filtration 1 and are divisible by p in (R) (uniquely divisible since (R)isZ(p)–free). Moreover, C C they agree modulo elements of weight filtration larger than WF(∂hI) in view of 1 (12), and so their p –multiples represent the same element in the WSS for R,which by 5.8 survives to the term Epν(m) (R), potentially producing (up to units) the 1 m I 1 m pν(m) I δ ν(m) –differential w ∂h w − ∂(u h ) (in view of 5.8, the target p p 7→ p ν(m) of such a differential represents a permanent cycle in the WSS). As in 5.5 this License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2646 JESUS´ GONZALEZ´ differential could still fail in case, for instance, the permanent cycle is killed by an earlier differential (we will see this is not the case). Note finally that the hypothesis I ν(m) in the WSS for R; however, u0u0u1 is not an admissible monomial, in fact (24) and 1 the methods in the previous section imply that 3 w∂(u0u0u1) produces a trivial 1 2 class in E1(R), so that 3 w ∂(u0u1)survivestoE2(R), and we next indicate how 2 it produces a δ2–differential. Let z = w ∂(u0u1) w[ϑ2]∂u1. Since the second 1− summand of z has weight filtration 5, we see that 3 z represents in the WSS for R 1 2 thesameelementas 3w∂(u0u1) does (recall ∂u1 =3v1 and ∂u0 = 0). Moreover, from (12) and (32) d(z)=d(w2∂(uu)) d(w[ϑ ]∂u ) 01 − 2 1 =((2)) w [ ϑ ] ∂ ( u u )+[ϑ ]∂(u u ) [ϑ ϑ ]∂u +((2)) w [ ϑ ϑ ] ∂u 1 1 0 1 2 0 1 − 1| 2 1 1 1 | 1 1 = (( 2 )) w [ ϑ ] u ∂u [ϑ ]u ∂u [ϑ ϑ ]∂u +((2)) w [ ϑ ϑ ] ∂u − 1 1 0 1 − 2 0 1 − 1| 2 1 1 1 | 1 1 =([ϑ2ϑ1] [ϑ1ϑ2])∂u1 (recall that [ϑ1]=u0) −3 |1 − | 3 3 =3((1)) − ( ∂ [ ϑ 3 ])v1 note that (( 1)) = (( 2 )) = unit (∂u )v · 1 1 = unit ∂(u v ), · 1 1 so that d( 1 z)=unit 1 ∂(u v ). Thus 1 ∂(u v ) represents a permanent cycle in 3 · 3 1 1 3 1 1 the WSS (u1v1 is admissible), and since WF(u0u1)=4andWF(u1v1) = 6, we get the δ –differential 1 w2∂(u u ) 1 ∂(u v ). 2 3 0 1 7→ 3 1 1 Definition 5.11. For a positive integer m let c(m) be defined by the relations 0 < c(m) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2647 h i Definition 5.14. a) For an admissible sequence I with hI starting in the form I h = ukuk+dH (where H is a monomial in u’s and v’s), define the derivative I 0 I0 of I by h = uk+1vk+dH.NotethatI0is again admissible. Define also the I0 element g CΓ(Z(p); Z(p)) by the relation (recall ∂uj+1 = pvj+1) ∈ 1 I0 (33) ∂(uk uk dH)=vk uk dH g . p +1 + +1 + − b) Define the elements D0j uk+1 CΓ(Z(p); Z(p))(j=0,1) by the formulas ∈ vk+1 = D00uk+1 + D01uk+1 and p 1 − 1 p k +1 D01uk+1 = jpk [ ϑ ( p j ) p k ϑ jpk ]. − p − | j=1 X m I c) Define the elements D0 w [ϑa]h (R)(j=0,1) by the formulas j ∈C m I m I m I d(w [ϑa]h )=D0w [ϑa]h +D10w [ϑa]h , m I m m j I a m I D10w [ϑa]h = j w − [ ϑ j ϑ a ] h j w [ ϑ j ϑ a j ] h . | − | − ν(m)=0 ν ( a )=0 Xj Xj d) For I as in a) and m such that ν(m)=k(I)=kand c(m)=p 1, define the m I − elements Dj w h (R)(j=0,1) by ∈C p 1 1 − k 1 k m I m λp − m ( λ 1)p I ∆k Djw h = ( λ 1)pk pk D j0 w − − [ϑλpk ]h − am − λ=1 X k j m (p 1)p I ∆k +( 1) w − − (D uk )h − , − 00 +1 I ∆ I I ∆ where h − k is the admissible monomial given by h = ukh − k . Lemma 5.15. For m and I as in 5.14 d), the following equation holds in (R) C modulo elements of weight filtration larger than WF(gI0): k m I m (p 1)p I0 dD0w h =w − − ∂g . I ∆ I0 k+1 Proof. Let σ = WF(h − k), so that WF(g )=p +σ. In what follows, equalities modulo weight filtration larger than pk+1 + σ are denoted by the symbol “ ”. For 0 <λ k k m λp 1 m ( λ 1)p I ∆k (( ( λ 1)pk )) (( pk )) − D 10 w − − [ϑλpk ]h − − p λ − k k k m λp 1 m ( λ 1)p m ( λ + µ 1)p I ∆k − − k k (( ( λ 1)pk )) (( pk )) − (( µpk )) w − − [ϑµp ϑλp ]h − ≡ − | µ=1 X λ 1 − k k k m λp 1 λp m ( λ 1)p I ∆k (( ( λ 1)pk )) (( pk )) − (( µpk )) w − − [ϑ(λ µ)pk ϑµpk ]h − . − − − | µ=1 X k k p 1 m λp 1 m ( λ 1)p I ∆k − k Therefore the component of λ=1(( ( λ 1)pk )) (( pk )) − D 10 w − − [ϑλp ]h − in − weight filtration rpk + σ (2 r p)is ≤P ≤ r 1 − k k k m λp 1 m ( λ 1)p m ( r 1)p I ∆k − − k k (( ( λ 1)pk )) (( pk )) − (( (r λ)pk )) w − − [ϑ(r λ)p ϑλp ]h − , − − | λ − X=1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2648 JESUS´ GONZALEZ´ minus, when r r 1 − k k k m rp 1 rp m ( r 1)p I ∆k (( ( r 1)pk )) (( pk )) − (( µpk )) w − − [ϑ(r µ)pk ϑµpk ]h − , − − | µ=1 X and any other components lie in weight filtration larger than pk+1 + σ.Thuswe get by 5.13 b) p 1 − k k m λp 1 m ( λ 1)p I ∆k (( ( λ 1)pk )) (( pk )) − D 10 w − − [ϑλpk ]h − − λ (34) X=1 p 1 − k m (p 1)p I ∆k aλmw − − [ϑ(p λ)pk ϑλpk ]h − . ≡ − | λ X=1 Multiplying by 1 and taking differentials, we get am p 1 1 − k k m λp 1 m ( λ 1)p I ∆k d (( ( λ 1)pk )) (( pk )) − D 0 w − − [ϑλpk ]h − am − λ X=1 p1 1 − k k m λp 1 m ( λ 1)p I ∆k d (( ( λ 1)pk )) (( pk )) − D 10 w − − [ϑλpk ]h − ≡− am − λ X=1 p1 − k +1 k 1 p m ( p 1)p I ∆k d (( λpk )) w − − [ϑ(p λ)pk ϑλpk ]h − ≡− p − | λ X=1 k m (p 1)p I ∆k d w − − (D uk )h − ≡ 01 +1 k m (p 1)p I ∆k d w − − (vk D uk )h − , ≡ +1 − 00 +1 m I m (p 1)pk I ∆ m (p 1)pk I ∆ so that d(D0w h ) d(w − − vk+1h − k ) w − − ∂ vk+1h − k , which by 5.14 a) implies≡ the result. ≡ m I m I Remark 5.16. By 5.4 both D0w h and w ∂h have Adams filtration 1 and are divisible by p in (R); moreover they agree (up to units) modulo elements of weight filtration larger thanC WF(hI) (from its definition, the lowest weight filtration term m I in D0w h is carried by 1 1 1 m I ∆k m I ∆k m I D0w [ϑpk ]h − = D0w [ϑpk ]h − = D0w h , am am am which, as in 5.9, agrees up to units with wm∂hI modulo weight filtration larger than I 1 WF(h )); thus their p –multiple represent (up to units) the same element in the WSS for R, which by 5.15 survives to the term E(p 1)pk (R), potentially producing − k 1 m I 1 m (p 1)p I0 up to units the δ(p 1)pk –differential p w ∂h p w − − ∂g .In5.20below − 7→ k 1 m (p 1)p I0 we make the arrangements needed to identify the target p w − − ∂g of this differential in terms of the basis for E1(R) given in 5.4. Recall from the remarks after (31) that (R)isZ(p)–free, so that inverting p C produces a monomorphism (R) , (R) Qof chain complexes. C →C ⊗ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2649 h i Definition 5.17. Let m 0. For 0 j<ν(m+ 1) define cj Q and elements Aj, ≥ ≤ ∈ Bj (R) Q(depending on m too) by ∈C ⊗ 1 1 c = m +1 , d(wm+1 1) = A + B j ν(m+1) j 1 pj+1 ν ( m +1) j j j p − − p − · and 1 m +1 m +1 r B = w − [ϑ ]. j pν(m+1) j r r − r pj+1 ≥X Note that cj is a unit in Z(p) in view of 4.8. m+1 m+1 m Lemma 5.18. a) d(w 1) = (( 1 )) w u 0 modulo elements of weight filtra- tion larger than 1. · b) For 0 j<ν(m+1), Aj lies in (R) and its differential agrees with 1 m≤+1 pj+1 C j+1 cj w − ∂uj modulo elements of weight filtration larger than p . − p +1 Proof. Everything follows by its definitions and 4.8, except for the formula in b), which is a consequence of the fact that d(Aj )= d(Bj) and that the lowest weight 1 m +1 m +1 p−j+1 filtration term in Bj is ν(m+1) j (( j+1 )) w − uj+1. p − p Remark 5.19. Recall from 5.4 that in homological degrees s 1, E1(R)isZ(p)– m ε ≤ free with basis the elements w u0 (m 0,ε =0,1). By 5.18 a) we get up to ≥m+1 ν(m+1) m units the WSS-differential for R : δ1(w 1) = p w u0. Moreover for 1 ·m+1 pj+1 0 j<ν(m+ 1), 5.18 b) implies that w − ∂uj is represented by a ≤ p +1 d–boundary in (R), thus producing a permanent cycle in the spectral sequence. jC m Since Aj and p w u0 represent up to units the same element in the WSS for R, j m 5.18 b) also shows that p w u0 survives to Epj+1 1(R), potentially producing up − j m 1 m+1 pj+1 to units the δpj+1 1-differential p w u0 p w − ∂uj+1. − 7→ Remark 5.20. Before describing in full the WSS for R we need to make some mi- nor adjustments in the way we detect elements in E1(R) arising in homological degrees s 2, that is, of the form 1 wm∂hI (see 5.4). Such elements were analyzed ≥ p through the auxiliary spectral sequence E0r(R),δ0r together with the basis of { } I E1/(∂1 boundaries) consisting of the admissible monomials φ .Sinceδ0(and − m therefore all differentials δ0r) are independent of the coefficients w ` (R)(see (15) and the remarks before it), the methods in the previous section work∈ ∗ perfectly m well if we use a different basis of E1/(∂1 boundaries) for each such coefficient w . This is done in order to identify in a natural− way the targets of the differentials suggested by 5.16. In detail, let m 0. For an admissible monomial hJ starting as J ≥ J h = uk+1vk+d ... with ν(m) k(J), consider the element g as defined in 5.14 a) (note that such an admissible≥J is the derivative of a uniquely defined admissible J sequence I) and define the corresponding element Φ E1 by replacing the u’s and v’s appearing in the expression for gJ by α’s and β∈’s respectively, so that gJ J J J represents the c(J)–multiple of Φ in E1 (recall from 5.3 that h = c(J)ϕ ,sothat in the expression “c(J)–multiple” we must take into account that E1 is an Fp[x]– module with x corresponding to multiplication by p). From its definition ΦJ φJ is − a (possibly empty) sum of monomials each of which by (24) is trivial in E1 (if d =1) or is an admissible monomial starting with a double “α”. Thus replacing each such J J φ by the corresponding Φ produces a new basis for E1/(∂1 boundaries), and as explained above, this is the basis we use for the coefficient w−m in the analysis of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2650 JESUS´ GONZALEZ´ the previous section. The result is a new Fp basis for E1(R) in homological degrees s 2, which can be described as follows (as usual, I is an admissible sequence): ≥ 1 m I I. Elements of the form p w ∂h with ν(m) 1 m I 1 m pν(m) I 1 m I a) δpν(m) p w ∂h = pw − ∂(uν(m)h ) for p w ∂h of type 5.20 I. ν(m) 1 m I 1 m (p 1)p I0 1 m I b) δ(p 1)pν(m) p w ∂h = pw − − ∂g for p w ∂h of type 5.20 III. − e m+1 e+ν(m+1) m c) δ1(p w 1) = p w u0 for e, m 0. j ·m 1 m+1 pj+1 ≥ d) δpj+1 1(p w u0)= pw − ∂uj+1 for 0 j<ν(m+1). − ≤ These differentials wipe out all elements in the WSS, except for the multiples of 0 w 1, which produce a copy of Z(p) in the cohomology of (R). · C Remark 5.22. In order to analyze differentials in the WSS for R as in 5.5, it will be convenient to make a summary of the representatives suggested by 5.8, 5.15 and 5.18 for the basis described in 5.20: 1 m I 1 m I i) A representative p w ∂h of type 5.20 I is replaced by p D0w h ; the corre- 1 m pν(m) I sponding representative p w − ∂(uν(m)h ) of type 5.20 II is replaced by 1 m I p dD0w h (see 5.9). 1 m I 1 m I ii) A representative p w ∂h of type 5.20 III is replaced by p D0w h ; the cor- ν(m) 1 m (p 1)p I0 responding representative p w − − ∂g of type 5.20 IV is replaced by 1 m I p dD0w h (see 5.16). a m+1 iii) Multiples p w 1 E1(R) tautologically represent themselves. Multiples j m · ∈ j ν(m+1) m+1 p w u E (R) are represented (up to units) by d(p − w 1) if 0 ∈ 1 · j ν(m +1)andby Aj if 0 j<ν(m+1),andinthelastcasethe ≥ 1 m+1 pj+1 ≤ representative p w − ∂uj+1 of type 5.20 V is replaced by d(Aj )(see 5.19). Proof of 5.21. We have already seen that the use of the representatives in 5.22 sug- gests such differentials. Note that a) sets a 1-1 correspondence between elements of type 5.20 I and elements of type 5.20 II; likewise, b) sets a 1-1 correspondence be- tween elements of type 5.20 III and elements of type 5.20 IV. Moreover every element j+1 1 m 1 m0+1 p on the right of d) is of type 5.20 V, and by rewriting p w ∂uk = pw − ∂uj+1 k with j = k 1andm0 =m+p 1 we see that every element of type 5.20 V − − appears as an element on the right of d). Similarly the (zero Adams filtration Z(p)- multiples of the) elements involved in c) and on the left of d) take into account License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2651 h i once and only once every element in E1(R) of homological degree s 1 (except for 0 ≤ the Z(p) multiples of w 1). Therefore there is no room for the anomalous behavior described at the end of· 5.5 for any more differentials. i 6. The weight spectral sequence for Sh i i In this section we obtain a description of the cohomology of (Sh i)(where i 0 i i C Shi=(S )hi) from our detailed understanding of the WSS for Rh i. As remarked at the beginning of section 5, the map S0 R in (28) induces Adams–filtration i 0 → i 0 preserving inclusions π ((` Sh i)h i) , π ((` Rh i)h i)whichcanbepicturedin the following charts: ∗ ∧ → ∗ ∧ 6 6 6 6 6 6 6 6 6 6 6 6 3 s s s s s s s s s s s s qq 2 s s s s s s q s s s s s s - 1 s s qqq s s s s s s qqqs s s s qqq - £ - 0s qs (i 1)q s iqs s ¢ 0 s q s (i 1)q s iq s s s − − s s s s s s s s s s c c c c c c c c c p p p p p p p p c c c c c c c c c c c c c c c As in 3.2, Adams projections induce inclusions i+1 0 i 0 π ((` Sh i)h i) , π ((` Sh i)h i) ∗ ∧ → ∗ ∧ and i+1 0 i 0 π ((` Rh i)h i) , π ((` Rh i)h i), ∗ ∧ → ∗ ∧ which in turn produce inclusions of chain complexes i+1 - i i+1 0 - i 0 (Rh i) (Rh i) CΓ(π ((` Rh i)h i); Z(p)) CΓ(π ((` Rh i)h i); Z(p)) C £ C ∗ ∧ £ ∗ ∧ 6 ¢ 6 6 ¢ 6 i+1 - i i+1 0 - i 0 (S¢¡h i) (S¢¡h i) CΓ(π ((` S¢¡h i)h i); Z(p)) CΓ(π ((` S¢¡h i)h i); Z(p)) C C ∗ ∧ ∗ ∧ ¢£ ¢£ i 0 i 0 and since π ((` Sh i)h i) π ((` Rh i)h i) has zero Adams filtration we see from 3.10 that ∗ ∧ → ∗ ∧ i i i 0 (35) (Sh i)= (Rhi) CΓ(π ((` Sh i)h i); Z(p)). C C ∩ ∗ ∧ 0 By the coefficient of a monomial in CΓ(π ((` X)h i); Z(p))wemeanthe 0 ∗ ∧ π((` X)h i) portion appearing on the right hand side of the second isomorphism ∗ ∧ i 0 in (15). Thus determining whether or not an element in CΓ(π ((` Rh i)h i); Z(p)) i 0 ∗ ∧ belongs to CΓ(π ((` Sh i)h i); Z(p)) reduces to a question about coefficients, which, in view of the charts,∗ ∧ is settled by a m i 0 (36) p wi π ((` Sh i)h i) if and only if a + i m ∈ ∗ ∧ ≥ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2652 JESUS´ GONZALEZ´ i a0 m and in particular we see from (31) that (Sh i)isZ(p)–free on generators p wi [ϑn] C i 0 where a0 =max 0,m i ν(n!) . Observe that the equation π ((` Sh i)h i)= i 0 { −0 − } ∗ ∧ π((` Rh i)h i) ` (S ) implies the following refinement of (35): ∗ ∧ ∩ ∗ i i 0 (37) (Sh i)= (Rhi) CΓ(` (S ); Z(p)). C C ∩ ∗ i i On the other hand, the inclusion (Sh i) , (Rhi) induces a map of spectral i i C →C sequences Er(Sh i) Er(Rhi) , which is injective for r = 1 in view of 4.16. As i { }→{ } i with Rh i, we change the given basis in E1(Sh i) to obtain the following description of these groups: i Description 6.1. In homological degrees s 1, E1(Sh i)isZ(p)–free on generators max 0,m i m ε ≤ p { − }wi u0 (ε =0,1), whereas for s 2itisFp–free on generators repre- ≥ m m sented by the elements of types I through V in 5.20 (replacing w by wi in view i i 0 of 5.2) lying in (Sh i), or equivalently whose coefficient lies in π ((` Sh i)h i). In C 1 m I 1 m I ∗ ∧ view of (36) and 5.3, a typical element p wi ∂h (or p wi ∂g ) represents a class in i E1(Sh i) if and only if (38) ν(I)+i 1 m − ≥ where ν(I) stands for ν(c(I)). 1 m I i Remark 6.2. We just observed the equivalence between p wi ∂h (Shi)and 1 m i 0 ∈C c(I)wi π((` Sh i)h i). In fact, since every summand in the expression (12) p ∗ 1 m∈I ∧ for d( p wi h ) comes either from the 1 ∂ portion of d, which is independent of the 1 m ⊗ coefficient p c(I)wi , or from the action of the powers of φ on this coefficient, it fol- i 0 1 m lows that each such summand lies in CΓ(π ((` Sh i)h i); Z(p)), provided c(I)wi ∗ p i 0 1 m I ∧ i 0 ∈ π ((` Sh i)h i). In particular, Djwi h CΓ(π ((` Sh i)h i); Z(p))(j=0,1) as ∗ p ∗ ∧ 1 m I i ∈m I ∧ i 0 long as wi ∂h (Shi). Likewise if cwi [ϑn]h lies in CΓ(π ((` Sh i)h i); Z(p)) p ∗ ∈C m I ∧ for some c Z(p),thensodoescDj0 wi [ϑn]h (j =0,1). ∈ 1 m I 1 m pν(m) I Lemma 6.3. Let ξ = p wi ∂h ,ζ=pwi− ∂(uν(m)h)and assume ξ is an element of type 5.20 I. i 1 m I a) If ξ represents a class in E1(Sh i) then so does ζ; moreover p D0wi h lies in i (Sh i). C i 1 m I m I b) If ζ represents a class in E1(Sh i) then both p dD0wi h and D1wi h lie in i 1 m I i 1 (Sh i), and if in addition i 1 then D w h lies in (Sh − i). C ≥ p 1 i C Proof. The first part in a) follows directly from (38); moreover under the hypothesis 1 m I i 0 in a), 6.2 gives D0wi h CΓ(π ((` Sh i)h i); Z(p)), but from its definition p ∗ 1 m I i ∈ ∧ D w h (Rhi), so that the second part in a) follows from (35). As for p 0 i ∈C b), given 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2653 h i 1 m I i 0 dD1wi h , which by (39) lies in CΓ(π ((` Sh i)h i); Z(p)), we obtain from p ∗ − 1 m I i ∧ (35) p dD0wi h (Shi). Finally the last assertion follows from the fact that 1 m I ∈Cm I i 1 p D1wi h = D1wi 1h (Rh− i), together with (37) and (39). − ∈C k 1 m I 1 m (p 1)p I0 Lemma 6.4. Let ξ = p wi ∂h ,ζ=pwi − − ∂g and assume ξ is an element of type 5.20 III (so that ν(m)=k(I)=k). i 1 m I a) If ξ represents a class in E1(Sh i) then so does ζ; moreover p D0wi h lies in i (Sh i). C i 1 m I m I b) If ζ represents a class in E1(Sh i) then both p dD0wi h and D1wi h lie in i 1 m I i 1 (Sh i), and if in addition i 1 then D w h lies in (Sh − i). C ≥ p 1 i C Proof. From its definition k+1 k k (40) ν(I0) ν(I)=ν(p !) ν(p !) 1=p 1 0, − − − − ≥ so that the first part in a) follows from (38). On the other hand, the hypothesis k 1 m (p 1)p I ∆k in a) implies that the summand p wi − − (D00uk+1)h − in the definition of 1 m I i D w h lies in (Sh i)andthat p 0 i C 1 k m (λ 1)p I ∆k i 0 wi − − [ϑλpk ]h − CΓ(π ((` Sh i)h i); Z(p)) p ∈ ∗ ∧ 1 for λ =1,... ,p 1 (in the first case the factor p affects D00uk+1, which has Adams − 1 filtration 1 and is divisible by p in CΓ(Z(p); Z(p)), and in the second case, p affects I ∆k h − , which, although of Adams filtration zero, is divisible by p in CΓ(Z(p); Z(p))), k 1 m (λ 1)p I ∆k i 0 i so D0wi − − [ϑλpk ]h − lies in CΓ(π ((` Sh i)h i); Z(p)) by 6.2, in (Rh i) p ∗ i ∧ 1 m I C i by definition of D0 and therefore in (Sh i) by (35). Thus D w h (Shi) 0 C p 0 i ∈C from its definition. The proof of b) is similar to that in 6.3, but this time we need to take into account the cancellations in (34). Since dD0 = dD1,wehave 1 m I m I i − m I to show that both p dD1wi h and D1wi h lie in (Sh i). Let D1wi h = A k C − m (p 1)p I ∆k B,whereB=wi − − (D00uk+1)h − ,andforλ=1,... ,p 1letMλ = k − m (λ 1)p I ∆k wi − − [ϑλpk ]h − ,sothat p 1 − k 1 m λp 1 A= (( ( λ 1)pk )) (( pk )) − D 10 M λ . a − m λ X=1 In view of (40) the hypothesis in b) is 2+i+ν(I)+pk m (p 1)pk,which readily implies − ≥ − − k k 1 m (p 1)p I ∆k 1 m (p 1)p I ∆k i B, w − − vk h − , w − − vk ∂h − (Shi), p i +1 p i +1 ∈C (41) k 1 m (p 1)p I ∆k i 1 and w − − vk h − (Sh− i)ifi1 p i +1 ∈C ≥ k 1 I ∆k m (p 1)p (the factor p affects D00uk+1 in the first case, ∂h − in the third and wi − − = k m (p 1)p 1 pwi −1 − in the fourth). In particular, we only need to show that A, p dA −i 1 i 1 ∈ (Sh i)aswellas p A (Sh− i)fori 1. From (34) the lowest weight filtration C ∈C ≥ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2654 JESUS´ GONZALEZ´ component of A is given by p 1 1 − k m (p 1)p I ∆k aλmwi − − [ϑ(p λ)pk ϑλpk ]h − am − | λ X=1 p 1 − k +1 k 1 p m ( p 1)p I ∆k − − k k = (( λpk )) w i [ϑ(p λ)p ϑλp ]h − p − | λ=1 X k m (p 1)p I ∆k = wi − − (D01uk+1)h − − k m (p 1)p I ∆k = wi − − (D00uk+1 vk+1)h − k − m (p 1)p I ∆k = B w − − vk h − , − i +1 i 1 i 1 which lies in (Sh i), and its p –multiple lies in (Sh − i) in view of (41). Since C I ∆ C k+1 this component has weight filtration WF(h − k)+p ,weget,moduloweight filtration larger than this value, k 1 1 m (p 1)p I ∆k dA = d B w − − vk h − p p − i +1 k 1 1 m (p 1)p I ∆k = dB w − − ∂(vk h − ) p − p i +1 k 1 1 m (p 1)p I ∆k = dB w − − vk ∂h − p − p i +1 1 (recall that vk+1 = p ∂uk+1 is a ∂–cycle of homology degree 2), where the last ele- i ment in the equalities above lies in (Sh i) in view of (41). To complete the proof we C I ∆ have to show that the components in weight filtration larger than WF(h − k)+ k+1 1 i 1 p of both A and p dA lie in (Sh i) and that the p –multiple of such compo- i 1 C nents in A lie in (Sh − i). In view of (37) it suffices to prove that the corre- C 1 1 0 sponding components of D10 Mλ and dD10 Mλ lie in CΓ(` (S ); Z(p)) (indeed, by p p ∗ 1 1 i n n their definitions D10 Mλ, p dD10 Mλ = p dD0Mλ (Rhi), and since wi = pwi 1, − 1 i 1 − ∈C p D10 Mλ (Rh− i)). ∈C I ∆ k+1 The summands of weight filtration larger than WF(h − k)+p in the def- k m (λ 1)p r I ∆k inition of D10 Mλ are, up to units, of the form wi − − − [ϑr ϑλpk ]h − with k 1 0 | r>(p λ)p ,andtheir –multiples lie in CΓ(` (S ); Z(p))since − p ∗ k 1+i+ν(r!) + ν((λp )!) + ν(I ∆k) − − ≥ 1+i+ν(((p λ)pk)!) + ν((λpk)!) + ν(I) ν(pk!) ≥− − − = 1+i+ν(I)+(p 1)ν(pk!) − − = 2+i+ν(I)+pk − m (p 1)pk (by the hypothesis in b)) ≥ − − = m (λ 1)pk (p λ)pk − − − − m (λ 1)pk r. ≥ − − − On the other hand a summand of D0 Mλ is, up to units, of the form S 1 k k m (λ 1)p r I ∆k m (λ 1)p i) w − − − [ϑr ϑ k ]h − with ν − − =0, or i | λp r m λ pk ( 1) I ∆k ii) wi − − [ϑ(λ µ)pk ϑµpk ]h − with 0 <µ<λ, − | License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2655 h i and we have to check that all summands in the definition of 1 d of weight filtration p S I ∆k k+1 0 larger than WF(h − )+p lie in CΓ(` (S ); Z(p)). This is verified by a direct ∗ calculation as above. In fact, since the power of p provided by [ϑn] to the coefficient i 0 n in π ((` Rh i)h i) agrees with the corresponding power provided by (( s )) [ ϑ n s ϑ s ], we see∗ that∧ the above calculation works in all cases except for when is of− type| i) above and the summand of 1 d to be considered is of the form S p S k k 1 m ( λ 1)p r m ( λ 1)p r s I ∆k (42) (( − − − )) w − − − − [ϑs ϑr ϑ k ]h − . p s i | | λp We complete the proof by giving full details for this situation. If r>(p λ)pk, the previous calculation certainly does work. So we assume r (p λ)pk−.Since m (λ 1)pk k ≤ − ν − − = 0, 4.8 forces r = µp with 0 <µ p λ. Moreover, our restriction r ≤ − on the weight filtration means r + s>(p λ)pk or s>(p λ µ)pk,thus − − − m (λ 1)pk r k 1+ν( − −s − )+i+ν(s!) + ν(r!) + ν((λp )!) + ν(I ∆k) − k − 1+i+ν(s!) + ν(r!) + ν((λp )!) + ν(I ∆k) ≥− − 1+i+ν(((p λ µ)pk)!) + ν((µpk)!) + ν((λpk)!) + ν(I) ν(pk!) ≥− − − − = 1+i+ν(I)+(p 1)ν(pk!) − − = 2+i+ν(I)+pk − m (p 1)pk (by the hypothesis in b)) ≥ − − = m (λ 1)pk (p λ)pk m (λ 1)pk r s, − − − − ≥ − − − − 0 which exhibits (42) as an element in CΓ(` (S ); Z(p)). ∗ Definition 6.5. Let em =max0,m i and em0 =max0,m+1 i ,sothat { − } { − } i e0 =em+ηm,whereηm =0ifm+1 iand ηm = 1 otherwise. By 6.1, E (Sh i) m ≤ 1 is Z(p)–free in homological degrees s 1, with basis as follows: for s =0the e m+1 ≤ e m elements p m0 w (m 1) and for s = 1 the elements p m w u (m 0). We i ≥− i 0 ≥ will omit the index m in em, em0 and ηm if no confusion arises. j ν(m+1) m+1 i Lemma 6.6. a) For j max e, ν(m +1) , d(p − wi ) lies in (Sh i) ≥ { j m } C and agrees, up to units, with p wi u0 modulo weight filtration larger than 1. 1 m+1 pj+1 b) For e j<ν(m+1),both pwi − ∂uj+1 and Aj (as defined in 5.17 and ≤ s s i replacing w by w in view of 5.2) lie in (Sh i). i C j ν(m+1) m+1 i Proof. In a) it is clear that p − wi lies in (Rh i), and therefore so does its differential, which by 5.18 a) agrees, up to unitsC and modulo weight filtra- j m tion larger than 1, with p wi u0. Moreover the summands in the expression for j ν(m+1) m+1 j ν(m+1) m +1 m +1 r d(p − wi ) are of the form p − (( r )) w i − [ϑr]for1 r m+1. i 0 ≤ ≤ Such a summand lies in CΓ(π ((` Sh i)h i); Z(p)) if and only if j ν(m +1)+ m+1 ∗ ∧ − ν r + i + ν(r!) m +1 r, which, in view of the hypothesis j m i,holds provided ≥ − ≥ − (43) ν(m +1)+ν(m+1)+ν(r!) + r 1. − r ≥ m+1 If r ν(m + 1), (43) certainly holds; otherwise from 4.8 we obtain ν r = ν(m ≥+1) ν(r) and again (43) is clear. Thus a) follows from (35). For b) it − i suffices to prove Aj (Shi) in view of 5.18 b). However the summands in the ∈C j ν(m+1) m+1 expression for Aj appear as summands of d(p − wi ), which as above lie in License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2656 JESUS´ GONZALEZ´ i 0 i CΓ(π ((` Sh i)h i); Z(p)). On the other hand, Aj lies in (Rh i) in view of 5.18 b) so that∗ the∧ result follows from (35). C The next lemma is a straightforward consequence of 4.8. Lemma 6.7. For 1 t pj+1, ν((pj+1 t)!) + ν(t!) ν((pj+1 1)!). ≤ ≤ − ≥ − Lemma 6.8. Let 0 j<ν(m+1).Then ≤ 1 m+1 pj+1 i i a) if w − ∂uj represent a class in E (Sh i) then d(Aj ) (Shi). p i +1 1 ∈C Assume in addition that j 1; then ≥ 1 m +1 m +1 r b) d(Aj ) pd(Aj 1)= pν(m+1) j d (( r )) w i − [ϑr] , and − − − pj r 1 m +1 m +1 r j+1 (44) d (( )) w − [ϑr] , 1 r 1 m +1 m +1 r m +1 r t (( )) (( − )) w − − [ϑ ϑ ] ν(m+1) j r t i t r p − | with 1 t m +1 r and t + r>pj+1. Such elements lie in the cobar complex ≤ ≤ i 0 − CΓ(π ((` Sh i)h i); Z(p)), since ∗ ∧ m +1 m +1 r j ν(m +1)+ν(( )) + ν (( − )) + i + ν ( t !) + ν(r!) − r t j ν(m +1)+ν(( m +1)) + i + ν ( t !) + ν(r!) ≥ − r = j ν(r)+i+ν(t!) + ν(r!) (by 4.8, since r 1 m +1 m +1 r d(Aj)= d(Bj)= d (( )) w − [ϑr] . − −pν(m+1) j r i − r pj+1 ≥X Finally for pj r License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2657 h i m+1 j j ν( r )+i+ν(r!) = j + i + ν((r 1)!) i + ν(p !) >m+1 p m+1 r. 1 m +1 m +1 r − ≥ i 0− ≥ − Therefore ν(m+1) j (( r )) w i − [ϑr] lies in CΓ(π ((` Sh i)h i); Z(p)), and since p − ∗ ∧ m+1 i ν( r ) ν(m +1) j, it also lies in (Rh i). Thus c) follows from (35) and part a). ≥ − C i We are now in position to give a full description of the WSS for Sh i. Recall i i from the remarks before 6.1 that E (Sh i) E (Rh i). 1 ⊆ 1 i Theorem 6.9. Let y be an element in E1(Sh i). i a) Suppose a differential y z holds in Er(Rh i) (as described in 5.21). Then i 7→ i z E1(Sh i) and the same differential holds in Er(Sh i). ∈ i b) Suppose a differential x y holds in Er(Rh i).Thenyis a permanent cycle i 7→ in Er(Sh i). i Proof. Pick representatives as in 5.22 for classes starting in E1(Sh i). Lemmas 6.3, 6.4, 6.6 and 6.8 a) show that the analysis in 5.9, 5.16 and 5.19 of the differentials in i 5.21 applies also for elements in Er(Sh i), because the chosen representatives do lie i in (Sh i). On the other hand Lemmas 6.3 b), 6.4 b), 6.6 a) and 6.8 a) show that C i i any class in E1(Sh i) that is the target of a differential in Er(Rh i) is represented i by a d–cycle in (Sh i) and therefore survives as a permanent cycle (perhaps as a C i i δr–boundary) in the WSS for Sh i. Since this takes each class in E1(Sh i) (except for the Z(p) in weight filtration degree zero) into account once and only once, we get as in the proof of 5.21 that there is no room for the anomalous behavior described in 5.5 for any more differentials. i Example 6.10. The following may help to understand the differentials in Er(Sh i)in low homological degrees. We use the notation introduced in 6.5. The δ1 differentials e0 m+1 e0+ν(m+1) m ν(m+1)+η e m σ,0,t i p wi p wi u0 = p p wi u0 yield E2 (Sh i)=0for(σ, t) = 7→ 1,1,(m+1)q i ν(m+1)+η e m 6 (0, 0) and E (Sh i)=Z/p with generator represented by p wi u0. The classes with representatives pjwmu for e j<ν(m+ 1) produce higher i 0 ≤ differentials of type 5.21 d), whereas those with max e, ν(m+1) j i In the next result, the cohomology of (Sh i) is denoted by H∗(i). C Corollary 6.11. a) In homological degrees s 2 the only possibly nontrivial i ≤ cohomology groups of (Sh i) are given by 0,0 C H (i)=Z(p), • H1,(m+1)q(i)=Z/pam+1 for m i,and • 2,(m+1)q max 0,a µ +1≥ H (i)=Z/p { m− m } for m max i +1,p 1 , • ≥ { − } where am =min m i, ν(m+1) and µm is the integral part of the logarithm in base p of (p {1)(−m +2 i). } − − i b) In homological degrees s 3 the cohomology groups of (Sh i) are Fp-free on generators represented by≥ cycles of the form C License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2658 JESUS´ GONZALEZ´ 1 dD wmhI with hI and m as in 5.20 I and 1+i+ν(I) Remark 6.12. The bidegree (s, t) of each element in 6.11 b) is explicitly determined as follows: Let s(I)andσ(I) denote the homological and weight filtration degrees of the admissible monomial hI .Thus s(I)= (ej +2mj) j 0 X≥ and j σ(I)= (ej +mj)p , j 0 X≥ where I =(e0,e1,m1,e2,m2,...)(m0= 0). Since the operators d, D0 and D0 preserve total degree and increase the homological degree by one, we see that an 1 m I 1 m I element p dD0wi h or p dD0wi h in 6.11 b) has total degree t = q(m + σ(I)) and homological degree s = s(I)+2. Corollary 6.13. Hst(i)=0if s 3 and t s<(p2 p 1)s + q(i p +2). ≥ − − − − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2659 h i Proof. Let s and t be the homological and total degrees, respectively, of an element 1 m I 1 m I p dD0wi h or p dD0wi h in 6.11 b). From 6.12 and the lower bound for m in 6.11 b) we must have (45) t s q(i + ν(I)+σ(I)) s(I) 2, − ≥ − − so that 5.3 gives j j q(ν(I)+σ(I)) = q (ej + mj )ν(p !) mj + (ej + mj )p − j 0 j 0 j 0 X≥ X≥ X≥ j+1 j+1 = q ejν(p !) + mj (ν(p !) 1) − j 0 j 1 X≥ X≥ j+1 j+1 = 2ej(p 1) + 2mj(p p). − − j 0 j 1 X≥ X≥ 1 m I In the case of an element p dD0wi h , we know from 5.20 I that ν(m) j+1 j+1 2 2ej(p 1) + 2mj(p p) (p p) (ej +2mj)+ , − − ≥ − ∇ j 0 j 1 j 0 X≥ X≥ X≥ where =2(p 1) + 2(p2 1) 2(p2 p)=4(p 1), and (45) becomes ∇ − − − − − t s qi +(p2 p)s(I)+ s(I) 2 − ≥ − ∇− − =(p2 p 1)(s 2) + qi + 2=(p2 p 1)s + D, −− − ∇− − − where D = qi + 2 2(p2 p 1) = q(i p +2). ∇− − − − − The corollary implies that in a chart with (t s, s) coordinates, the groups st 2 − 1 H (i) lie on the right of the line with slope (p p 1)− and s-intersection at 2 1 − − q(p i 2)(p p 1)− (which is about 2 for i = 0). By 6.12 every single element − − − − of H∗(i) can be explicitly located in such a chart. As a way of example we sketch below the corresponding chart for p =3,i=0andt s 110. The square at the − ≤ n origen represents a copy of Z(p),anumbernrepresents a cyclic group Z/p and 1 m I a (resp. ) represents a copy of Fp coming from an element p dD0wi h (resp. 1 • m I◦ I p dD0wi h ), in which case we indicate the corresponding admissible monomial h giving rise to the element (the exponent m is determined by 6.11 b), except for the classes with (t s, s)=(49,3) and (57, 3), which have m =4andm=6 respectively). − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2660 JESUS´ GONZALEZ´ 9 u0u1v 22 1 uv9 21 11 uuv8b 20 0r11 19 uv8 11b 18 uuv7 0r11 17 uv7 11b 16 uuv6 011 ruv5v uv6 12 15 11 1 buuv3v 14 uuv5 0112 011 ruv4vr uv5 12 13 1 1uuv4u 1 012 buuv3vb1 4 012 12 uuv 1uv4u 011 12 4ruv3vrb1 uv 123 11 11 1uuvu b2b011r2 3 u0u1vv2 10 uuv 1uv3u 011 12 3ruv2vrb1 uv 122 9 11 1uuvu b b011r2 2 uuvv 8 uuv 0112uv2u2 0r11 r1b12uv 2 uvv 12 7 u1v 112 1b bu0u1v1ru2 uuv uuv 6 011 012uvu r r1b12u0u2v2ru1u2v2 uv uv 5 11 12uuu uv b b012r22br uu uu 4 01 02uuuu r r12b12 r u uu 3 1b22brb 2 11r rr 1 1 112112113112112113112112114 0 5 101520253035404550556065707580859095100105110 By 3.5 there is a short exact sequence of complexes (46) 0 V (S0) E (S0; `) (S0) 0 → → 1 →C → 0 which produces a long exact sequence involving E2(S ; `)—the second term in the `–Adams spectral sequence for S0. It is shown in [15] that the vanishing line given 0 by 6.13 can be extended to a similar one in E2(S ,`). The homotopical applications described in the introduction of the paper are based on a combination of such a vanishing line and the results in the next and final section. 7. Liftings through `–resolutions i+1 i Recall from the beginning of section 6 that the Adams projections Sh i Sh i i+1 i → produce monomorphisms of complexes (Sh i) (Shi). The induced morphism C →C in cohomology will be denoted by A : H∗(i +1) H∗(i) (as in 6.11, H∗(j)is j → shorthand for H∗( (Sh i))). C Lemma 7.1. A : Hs(i) Hs(i 1) is trivial if s 3 and i 1. → − ≥ ≥ i s Proof. Representatives in (Sh i) for the nontrivial classes in H (i) are of the form 1 m I 1 m CI 1 m I 1 m I p dD0wi h = p dD1wi h and p dD0wi h = p dD1wi h for suitably chosen m − − i 1 and I (6.11 b)). These elements map into boundaries in (Sh − i)inviewofthe last assertions in 6.3 b) and 6.4 b). C Lemma 7.2. In homological degree s =2and total degree t =(m+1)q,the ν(m+1)–fold composite Aν(m+1) : Hst(i) A H st(i 1) A A H st(i ν(m +1)) −→ − −→···−→ − is trivial. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2661 h i k st Proof. Representatives in (Sh i) for the nontrivial classes in H (k)(sand t as C 1 m+1 pj+1 in the lemma) are given by the elements ej = p wi − ∂uj+1 for µm 1 a −µ +1≤ j φ q τ (47) 0 π(m+1)q(`) π ( m +1)q(Σ `) π ( m +1)q 1(J) 0(m 0), → −→ −→ − → ≥ and by (3) the group on the right is cyclic of order ν(m + 1) + 1. On the other hand, by 3.6, the first component of the differential 0, (m+1)q 0 1,(m+1)q 0 nq ν(n!) (48) (S ) (S )= π Σ `h i C →C (m+1)q n>0 M is given by the map φ in (47). Thus the composition of the projection 1, (m+1)q(S0) q C 1, (m+1)q π(m+1)q(Σ `)andthemapτin (47) induce a well defined map λ : H (0) → 0 J(m+1)q 1(S ). Note from 6.11 that these two groups are isomorphic and in fact →λ is an isomorphism,− since according to 6.10, the representative for the generator 1, (m+1)q m m in H (0) is p w u0, which in terms of the sum in (48) corresponds to the q generator in the first summand π(m+1)q(Σ `). The topological consequences of the calculations in this paper depend on the next result, which is proven for the 2–primary situation in [11]. The proof goes without changes for odd primes. Recall that the Adams filtration of a homotopy class α is denoted by AF (α). f g Lemma 7.4. Suppose X Y Z is a cofibration of spectra and Z is a wedge −→ k −→ of suspensions of HFp and `h i. Then for any class α πn(Y ) mapping trivially ∈ under g, there is a class β πn(X) such that f (β)=αand AF (β) AF (α) 1. ∈ ∗ ≥ − For dimensional reasons the unit S0 i ` maps trivially under φ, defining a j −→ φ unique class S0 J such that i = κ j,whereJ κ ` Σq`is the cofibration in ◦ 7.3. −→ → → f1 f 2 f 3 Theorem 7.5. Let S 0 S S be the standard `–resolution of S0 ←− 1 ←− 2 ←− ··· ( 2) and suppose given a homotopy class αs πn(Ss). § ∈ a) If s =0and n>0, then there is a homotopy class α πn(S ) such that 1 ∈ 1 f1(α1)=α0 and AF (α1) AF (α0) 1. ≥ − f j b) If s =1,AF (α ) 1 and α maps trivially under the composite S 1 S 0 1 ≥ 1 1 −→ → J , then there is a homotopy class α πn(S ) such that f f (α )=f (α ) 2 2 1 ◦ 2 2 1 1 and AF (α ) AF (α ) 1. ∈ 2 ≥ 1 − c) If s =2and AF (α2) ε2, then there is a homotopy class α3 πn(S3) such that f f (α )=f (α≥)and AF (α ) AF (α ) ε .Hereε =1+∈ ν(n+2) 2 ◦ 3 3 2 2 3 2 2 2 if n +2 0(modq),andε =1otherwise.≥ − ≡ 2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2662 JESUS´ GONZALEZ´ d) If s 3 and AF (αs) εs, then there is a homotopy class αs πn(Ss ) ≥ ≥ +1 ∈ +1 such that fs fs (αs )=fs(αs),andAF (αs ) AF (αs) εs.Here ◦ +1 +1 +1 ≥ − εs=2if n + s 0(modq),andεs=1otherwise. ≡ 0 Proof. Since the positive dimensional homotopy groups of S are finite, α0 maps 0 i trivially under πn(S ) π n ( ` ), and so part a) follows immediately from 7.4. For b) consider the following−→ diagram with cofiber rows: pr f1 i 1 - - 0 - Σ− ` S1 S ` i S ?∧ 1 ` S ∧ 1 ?π ? 1 - q 1 - - Σ− ` φ Σ − ` J κ ` Here π is the wedge projection given by 2.3 (ΣS1 = `); moreover, the square on the left commutes in view of 2.10. Thus the dashed line agrees with j : S0 J. 1,n+1 0 → From (1) and (2), α1 maps into a d1–cycle z E1 (S ; `)=πn(` S1)inthe first term of the `–Adams spectral sequence. The∈ current hypothesis together∧ with the observations after 7.3 show that z maps in turn into a boundary in (S0)(see 1,n+1 0 C (46)). Therefore there is class e E1 (S ; `) carried by splitting Fp Eilenberg- ∈ 0,n+1 0 Mac Lane spectra, and a class w E (S ; `)=πn (`) such that ∈ 1 +1 (49) d w = z e. 1 − As indicated in the diagram just before (35), every positive dimensional homotopy class in π (`) must have positive classical Adams filtration. Thus AF (d1w) ∗ ≥ AF (w) > 0, and since AF (z) AF (α1) > 0 it follows that e = 0. Recall from (2) ≥ i S 0 1 pr ∧ 1 that the differential d in E (S ; `) is induced by the composite Σ− ` S 1 1 −→ 1 −−→ ` S . Letting α0 = α pr(w), (49) gives (i S )(α0 )=z d w=e=0.Thus ∧ 1 1 1 − ∧ 1 1 − 1 α0 lifts through f , and since f (α0 )=f (α) f (pr(w)) = f (α), part b) follows 1 2 1 1 1 − 1 1 from 7.4 once we verify that AF (α10 ) AF (α1). If n +1 0(modq), w lies in a trivial group and we are done. Assume≥ then n +1=rq. From6≡ 2.3, the r th wedge rq 1 ν(r!) summand in ` S1 is Σ − `h i, and from 3.6 the corresponding component of ∧ rq 1 ν(r!) d1w is φr(w). Therefore φr(w)=ρr(d1w)=ρr(z), where ρr : ` S1 Σ − `h i is the wedge projection, and in particular ∧ → (50) AF (φr(w)) = AF (ρr(z)) AF (z) AF (α ). ≥ ≥ 1 From (3) we see that φr(vr)=(kr 1) (k 1) 1, and since AF (vr)=r and ν(kj 1) = ν(j) + 1, we get that−φr increases··· − the· Adams filtration by ν(r!) − in πrq(`). Thus 2.6 implies that φr preserves filtration in πrq(`), and by (50) AF (w)=AF (φrw) AF (α ), which implies the required inequality AF (α0 ) ≥ 1 1 ≥ AF (α1), completing the proof of part b). To analyze the `–obstructions for liftings in the case s 2, we use lemmas ≥ 7.1 and 7.2. Let αs πn(Ss) with AF (αs) εs. As above, the element z = ∈ s, n+s 0 ≥ (i Ss)(αs) πn(` Ss)=E1 (S ; `), having positive Adams filtration, lies in ∧ ∈ ∧ 0 the Fp Eilenberg-Mac Lane–free portion of this group, that is, in πn((` Ss)h i)= s, n+s(S0). If n + s 0(modq), such a group is trivial in view of (6) and∧ (7), and Cin this case c) and d)6≡ follow directly from 7.4. For n + s 0(modq)wenotethat ≡ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REGULAR COMPLEX IN THE BP 1 –ADAMS SPECTRAL SEQUENCE 2663 h i s, n+s a s a 0 (Sh i)=πns (`` Shi)hi C + ∧ ∧ 0 h i σ(n)q ν(n!) a =πns Σ `h i Sh i + ∧ nR ! _∈s a h i σ(n)q ν(n!) = πn s Σ `h i + n R ! _∈ s a = πn (` Ss)h i ∧ a in view of 3.4 and 2.3 (in that order). Thus in fact z (Shi), where a = ∈C b AF (αs) εs.Forb=a εs+ 1, 7.1 and 7.2 imply that, as an element in (Sh i), z produces≥ a trivial homology− class; and, as in the case for s = 1, thisC gives a s 1,n+s b s, n+s b class w E1− (Sh i;`) such that z d1w maps trivially into (Sh i). As remarked∈ in 2.1, these constructions are− not functorial; however choicesC can be made so to have the following commutative diagram of complexes (see (46)), where b 0 vertical maps are “induced” by the Adams projection Sh i S → - b - b - b - 0 V (Sh i) E (Sh i; `) (Sh i) 0 1 C ? ? ? - - - - 0 V (S0) E (S0; `) (S0) 0 1 C 0 It follows that w and z,aselementsinE1(S ;`), satisfy the relation z = d1w (as 0 elements in E1(S ; `),wand z have Adams filtration at least b). We can now proceed as for the case s = 1. In terms of the Adams resolution Ss+1 fs+1 ? i S ∧ s - Ss ` Ss YPP ∧ f PPηs s ? PP - Ss 1 ` Ss 1 − i Ss 1 ∧ − ∧ − the differential d is given by the composite (i Ss) ηs (compare with (2)), so 1 ∧ ◦ that the element α0 = αs ηs(w) maps trivially under i Ss and has AF (α0 ) s − ∧ s ≥ min AF (αs),AF(ηs(w)) b=a εs+ 1. The final conclusion follows from an application{ of 7.4 to the}≥ cofibration− on top of the above diagram. Theorem 7.5 is the odd primary version of [11, Theorem 5.1]. 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