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The $ P 2^ 1$ Margolis Homology of Connective Topological Modular Forms

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The $ P 2^ 1$ Margolis Homology of Connective Topological Modular Forms 1 THE P2 MARGOLIS HOMOLOGY OF CONNECTIVE TOPOLOGICAL MODULAR FORMS PRASIT BHATTACHARYA, IRINA BOBKOVA, AND BRIAN THOMAS 1 Abstract. The element P2 of the mod 2 Steenrod algebra A has the property 1 2 1 F (P2) = 0. This property allows one to view P2 as a differential on H∗(X, 2) 1 for any spectrum X. Homology with respect to this differential, M(X, P2), 1 is called the P2 Margolis homology of X. In this paper we give a complete 1 calculation of the P2 Margolis homology of the 2-local spectrum of topological modular forms tmf and identify its F2 basis via an iterated algorithm. We 1 apply the same techniques to calculate P2 Margolis homology for any smash power of tmf . 1. Introduction 1 1 2. Action of P2 and the length spectral sequence 6 3. The reduced length 12 1 ∧r Z ×n 4. P2 Margolis homology of tmf and B( /2 )+ 20 References 23 Contents Convention. Throughout this paper we work in the stable homotopy category of spectra localized at the prime 2. 1. Introduction The connective E∞ ring spectrum of topological modular forms tmf has played a vital role in computational aspects of chromatic homotopy theory over the last two decades [Goe10], [DFHH14]. It is essential for detecting information about the chromatic height 2, and it has the rare quality of having rich Hurewicz image. arXiv:1810.05622v3 [math.AT] 1 Jun 2020 There is a K(2)-local equivalence [HM14] hG48 LK(2)tmf ≃ E2 where E2 is the second Morava E-theory at p = 2 and G48 is the maximal finite G hG48 subgroup of the Morava stabilizer group 2. The spectrum E2 can be used to build the K(2)-local sphere spectrum (see [BG18]). The homotopy groups of tmf approximate both the stable homotopy groups of spheres and the ring of integral modular forms. In many senses, tmf is the chromatic height 2 analogue of connec- tive real K-theory ko. Further, the homotopy groups of tmf are completely known [Bau08]. 1 Let us now recall the definition of the element P2 ∈ A. Milnor described the mod 2 dual Steenrod algebra A∗ as the graded polynomial algebra [Mil58, App. 1] A∗ =∼ F2[ξ1, ξ2, ξ3,... ], 1 2 PRASIT BHATTACHARYA, IRINA BOBKOVA, AND BRIAN THOMAS i where |ξi| =2 − 1. The Steenrod algebra A has an F2-basis dual to the monomial F 2s basis of A∗. The elements of the 2-basis of A which are dual to ξt are denoted s 0 s by Pt , and the elements Pt are denoted by Qt−1. When s<t, the elements Pt s 2 are exterior power generators, i.e. (Pt ) = 0. Thus, any left A-module K can be s regarded as a complex with differential given by the left multiplication by Pt (for s<t). This leads to the following definition. Definition 1.1 ([Mar83]). Let K be any left A-module and 0 ≤ s<t. Let L s Pt : K −→ K s s L s denote the left action by Pt . The left Pt Margolis homology group of K, M (K, Pt ), is defined as s Ker LP : K → K L Ps t M (K, t ) := L s . Im Pt : K → K s For a right A-module K, one can similarly define the right Pt Margolis homology group of K as s Ker RP : K → K R Ps t M (K, t ) := R s Im Pt : K → K R s s where Pt is the right action by Pt on K. s L ∗ s Notation 1.2. For a spectrum X, M(X, Pt ) will denote M (H (X), Pt ) or equiv- R s alently M (H∗(X), Pt ). Computations of Margolis homology underly many essential computations in ho- motopy theory. For example, Adams work on BP h1i cooperations [Ada74] relies on the computations of M(BP h1i, Qi) for i = 0, 1. Calculations like M(bo, Qi) for i = 0, 1 are essential ingredients in the work of Mahowald on bo-resolutions [Mah81]. More recently, Culver described BP h2i resolutions [Cul] by understand- ∧n ing M(BP h2i, Qi) for i = 0, 1, 2. Computation of M(tmf , Q2) is an essential ingredient in [BBB+b]. The element Qi is primitive for all i ∈ N. In other words, the comultiplication map ∆ on A sends Qi to (1.3) ∆(Qi)= Qi ⊗1+1 ⊗ Qi . Consequently, Qi acts on H∗(X) as a derivation, namely it follows the Leibniz rule Qi(xy)= Qi(x) · y + x · Qi(y), whenever X is a ring spectrum. The Leibniz rule implies the K¨unneth isomorphism [Mar83, Proposition 17, pg 343] ∼ M(X ⊗ Y, Qi) = M(X, Qi) ⊗M(Y, Qi) and hence, M(X, Qi) is an F2 algebra whenever X is a ring spectrum. As a result, computation of Qi Margolis homology and its description is often fairly straight- forward. s On the other hand, for s> 0, Pt is not a primitive element of A. In particular, 1 1 1 ∆(P2)= P2 ⊗1+ Q1 ⊗ Q1 +1 ⊗ P2 1 THE P2 MARGOLIS HOMOLOGY OF tmf 3 and its action on H∗(X) for a ring spectrum X, does not follow the Leibniz rule. Instead, we have 1 1 1 (1.4) P2(xy)= P2(x)y + Q1(x) Q1(y)+ x P2(y). 1 1 As a result, the product of two P2 cycles may not necessarily be a P2 cycle, hence 1 M(X, P2) may not admit any multiplicative structure even if X is a ring spectrum. 1 This is the main reason why the P2 Margolis homology calculations are significantly more complicated. Let us now consider the spectrum tmf . It is well-known ([HM14], [Mat16]) that F ∼ F 8 4 2 H∗(tmf ; 2) = 2[ζ1 , ζ2 , ζ3 , ζ4, ζ5,... ] ⊂ A∗ is a subalgebra of A∗. Here the elements ζi are the images of ξi under the antipode of the Hopf algebra A∗ (see Section 2). The right action of Qi is given by the formula (see [Cul, §2] for details) 2i+1 Qi(ζn)= ζn−i−1. Then, since the Qi are derivations, it can be easily seen that F 8 4 (1.5) M(tmf , Q0)= 2[ζ1 , ζ2 ] F 8 2 2 2[ζ1 , ζ3 , ζ4 ,... ] (1.6) M(tmf , Q1)= 4 4 hζ3 , ζ4 ,... i F 4 2 2 2[ζ2 , ζ3 , ζ4 ,... ] (1.7) M(tmf , Q2)= 8 8 8 . hζ2 , ζ3 , ζ4 ,... i ∧r 1 In this paper, we give a complete calculation of M(tmf , P2) for arbitrary r ≥ 1. In fact, the calculation for r> 1 follows from the case r = 1, because after forgetting the internal grading one can construct a non-canonical isomorphism (see Section 4) ∧r 1 ∼ 1 M(tmf , P2) = M(tmf , P2). For the case r = 1, we give an iterated algorithm (see Definition 3.16) that con- F 1 1 structs an 2-basis of M(tmf , P2). We give a complete description of M(tmf , P2) 1 in Theorem 3.18 which is the main result of this paper. Although M(tmf , P2) is 1 not an algebra, we notice that M(tmf , P2) is a module over an infinitely gener- ated exterior algebra S (see Lemma 3.1 for a description of S). Theorem 3.18 also 1 describes M(tmf , P2) as an S-module. The key tool we use is the length spectral sequence (2.14), which we define in Section 2. The length spectral sequence admits a d0 differential and a d2 differential and collapses at the E3 page. The Leibniz rule does hold for the d0, but not for d2. In order to work around this issue, we notice that the E2 page admits an action of S (i.e. d2 are S linear) and we use it to simplify the computation of E∞ = E3. We also notice that almost identical calculations lead to a complete description Z ×n 1 of M((B /2 )+, P2). The methods developed in this paper can be considered 1 as a blueprint for computations of Pt Margolis homology of a variety of other A-modules. ∧r 1 Our calculations of M(tmf , P2) have many applications, as the spectrum tmf has a wide range of applications, particularly in chromatic homotopy theory. First 4 PRASIT BHATTACHARYA, IRINA BOBKOVA, AND BRIAN THOMAS note that the cohomology of tmf , as a module over the Steenrod algebra A, is isomorphic to (see [HM14], [Mat16]) ∗ (1.8) H (tmf ; F2) =∼ A//A(2) where A(2) is the subalgebra of A generated by Sq1, Sq2 and Sq4. This, and a change of rings isomorphism, imply that the E2 page of the Adams spectral sequence converging to tmf ∗X (for a spectrum X) is s,t s,t ∗ F (1.9) E2 := ExtA(2)(H (X), 2). One can detect infinite families in the E2 page via the map s,t ∗ s,t ∗ q : Ext (H (X), F2) −→ Ext 1 (H (X), F2). A(2) Λ(P2) 1 The codomain of q can be understood by calculating M(X, P2). Note that s,t ∼ Ext 1 (F2, F2) = F2[h2,1], Λ(P2 ) where |h2,1| = (1, 6) and F 1 s,t ∗ F 2[h2,1] ⊗M(X, P2) ⊂ Ext 1 (H (X), 2) Λ(P2 ) accounts for all the elements with positive s filtration. This shows that the knowl- 1 edge of M(X, P2) is crucial in detecting patterns in the E2-page of (1.9). Motivation I - Towards homotopy groups of K(2)-local sphere.
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