Adams-Novikov Spectral Sequence Research Articles

Hodge theory for elliptic curves and the Hopf element ν$\nu$

AbstractWe show that the vector bundle on the moduli stack of elliptic curves associated to the 2‐cell complex is isomorphic to the de Rham cohomology sheaf of the universal elliptic curve . We use this to calculate the homotopy groups of the ‐quotient of by , called the spectrum of ‘topological quasimodular forms’, by relating its Adams–Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitting of the Hodge–de Rham filtration.

Successive spectral sequences

In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact couples is studied, together with examples where they arise naturally. As for ordinary spectral sequences we will see differentials and group extensions, however the real power comes from the appearance of natural isomorphisms between pages of differing indices. The constructions reveal finer invariants than ordinary spectral sequences, and they connect to other fields such as Fary functors and perverse sheaves. They are based on a natural index scheme, which allows us to obtain new results even in the standard case of Z \mathbb {Z} -filtered chain complexes, e.g. a useful criterion for a product structure for Grothendieck’s spectral sequences, and new paths to connect the first or second page to the limit. This turns out to yield the right framework for unifying several spectral sequences that one would usually apply one after another. Examples that we work out are successive Leray–Serre spectral sequences, the Adams–Novikov spectral sequence following the chromatic spectral sequence, successive Grothendieck spectral sequences, and successive Eilenberg–Moore spectral sequences.

The telescope conjecture at height 2 and the tmf resolution

Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo-resolutions. In this paper, we study the height 2 telescope conjecture at the prime 2 through the lens of tmf-resolutions. To this end, we compute the structure of the tmf-resolution for a specific type 2 complex Z. We find that, analogous to the height 1 case, the E 1 -page of the tmf-resolution possesses a decomposition into a v 2 -periodic summand, and an Eilenberg–MacLane summand which consists of bounded v 2 -torsion. However, unlike the height 1 case, the E 2 -page of the tmf-resolution exhibits unbounded v 2 -torsion. We compare this to the work of Mahowald–Ravenel–Shick, and discuss how the validity of the telescope conjecture is connected to the fate of this unbounded v 2 -torsion: either the unbounded v 2 -torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v 2 -parabolas and the telescope conjecture is false. We also study how to use the tmf-resolution to effectively give low-dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E ( 2 ) -local Adams–Novikov spectral sequence for Z collapses.

Explicit modular forms from the divided beta family

We compute modular forms known to arise from the order 5 generators of the 5-local Adams-Novikov spectral sequence 2-line, generalizing and contextualizing previous computations of M. Behrens and G. Laures. We exhibit analogous computations at other primes and conjecture formulas for some of the modular forms arising in this way at arbitrary primes ≥5.

A Note on Hopkins' Picard Groups of the Stable Homotopy Categories of $L_n$-Local Spectra

For a stable homotopy category, M. Hopkins introduced a Picard group as a category consisting of isomorphism classes of invertible objects. For the stable homotopy category of $L\_n$-local spectra, M. Hovey and H. Sadofsky showed that the Picard group is actually a group containing the group of integers as a direct summand. Kamiya and the author constructed an injection from the other summand of the Picard group to the direct sum of the $E\_r$-terms $E\_r^{r,r-1}$ over $r\ge 2$ of the Adams–Novikov spectral sequence converging to the homotopy groups of the $L\_n$-localized sphere spectrum. In this paper, we show in a classical way that the injection is a bijection under a condition.

Stable Stems

We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, eta, and nu, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences. We also compute the motivic stable homotopy groups of the cofiber of the motivic element tau. This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of tau are the same as the E2-page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.

Inverting the Hopf map

We calculate the η-localization of the motivic stable homotopy ring over C, confirming a conjecture of Guillou and Isaksen. Our approach is via the motivic Adams–Novikov spectral sequence. In fact, work of Hu, Kriz and Ormsby implies that it suffices to compute the corresponding localization of the classical Adams–Novikov E 2 -term, and this is what we do. Guillou and Isaksen also propose a pattern of differentials in the localized motivic classical Adams spectral sequence, which we verify using a method first explored by Novikov.

On the homotopy of Q(3) and Q(5) at the prime 2

We study modular approximations Q(l), l = 3,5, of the K(2)-local sphere at the prime 2 that arise from l-power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(l) and record Hill, Hopkins, and Ravenel's computation of the homotopy groups of TMF_0(5). Using these tools and formulas of Mahowald and Rezk for Q(3) we determine the image of Shimomura's 2-primary divided beta-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use low-dimensional computations of the homotopy of Q(3) and Q(5) to explore the role of these spectra as approximations to the K(2)-local sphere.

THE FIRST LINE OF THE BOCKSTEIN SPECTRAL SEQUENCE ON A MONOCHROMATIC SPECTRUM AT AN ODD PRIME

The chromatic spectral sequence is introduced in (8) to compute the E2-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The E1-term E s,t (k) of the spectral sequence is an Ext group of BP∗BP-comodules. There are a sequence of Ext groups E s,t (n s) for non-negative integers n with E s,t 1 (0) = E

The Adams–Novikov spectral sequence and Voevodsky’s slice tower

We show that the spectral sequence converging to the stable homotopy groups of spheres, induced by the Betti realization of the slice tower for the motivic sphere spectrum, agrees with the Adams-Novikov spectral sequence, after a suitable re-indexing. The proof relies on an extension of Deligne's d\'ecalage construction to the Tot-tower of a cosimplicial spectrum.

Stable motivic [formula omitted] of low-dimensional fields

Let k be a field with cohomological dimension less than 3; we call such fields low-dimensional. Examples include algebraically closed fields, finite fields and function fields thereof, local fields, and number fields with no real embeddings. We determine the 1-column of the motivic Adams–Novikov spectral sequence over k. Combined with rational information we use this to compute π1S, the first stable motivic homotopy group of the sphere spectrum over k. Our main result affirms Morel's π1-conjecture in the case of low-dimensional fields. We also determine π1+nαS for weights n∈Z∖{−2,−3,−4}.

A comparison of motivic and classical stable homotopy theories

Let k be an algebraically closed field of characteristic zero. Let SH(k) denote the motivic stable homotopy category of T-spectra over k and SH the classical stable homotopy category. Let c:SH -> SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism {\pi}_n(E)-> {\pi}_{n,0}c(E) for all spectra E. Fix an embedding of k into the complex numbers and let Re:SH(k) -> SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum has Betti realization which is strongly convergent. This gives a spectral sequence "of motivic origin" converging to the homotopy groups of the classical sphere spectrum; this spectral sequence at E_2 agrees with the E_2 terms in the Adams-Novikov spectral sequence.

A composite map in the stable homotopy groups of spheres

In 2001, J. Lin detected a non-trivial element in the stable homotopy group of the sphere spectrum S for at the prime greater than three. In the stable homotopy group of the Smith–Toda spectrum , X. Liu constructed an essential element for at the prime greater than three. Let denote the Spanier–Whitehead dual of the generator , which defines the -element . Let . In this paper, we show that the composite maps and are non-trivial, where and . In the Adams–Novikov spectral sequence, the maps and are represented by and , respectively. Here denotes the well-known element in .

The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime

AbstractThe chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the E2-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The E1-term of the spectral sequence is an Ext group of BP*BP-comodules. There is a sequence of Ext groups for nonnegative integers n with and there are Bockstein spectral sequences computing a module (n – s) from So far, a small number of the E1-terms are determined. Here, we determine the for p > 2 and n > 3 by computing the Bockstein spectral sequence with E1-term for s = 1, 2. As an application, we study the nontriviality of the action of α1 and β1 in the homotopy groups of the second Smith-Toda spectrum V(2).

ON THE DOUBLE TRANSFER AND THE f-INVARIANT

AbstractThe purpose of this paper is to investigate the algebraic double S1-transfer, in particular the classes in the two-line of the Adams–Novikov spectral sequence which are the image of comodule primitives of the MU-homology of ℂP∞ × ℂP∞ via the algebraic double transfer. These classes are analysed by two related approaches: the first, p-locally for p ≥ 3, by using the morphism induced in MU-homology by the chromatic factorisation of the double transfer map together with the f′-invariant of Behrens (for p ≥ 5) (M. Behrens, Congruences between modular forms given by the divided β-family in homotopy theory, Geom. Topol.13(1) (2009), 319–357). The second approach (after inverting 6) uses the algebraic double transfer and the f-invariant of Laures (G. Laures, The topological q-expansion principle, Topology38(2) (1999), 387–425).

Toda brackets and congruences of modular forms

definedin [Lau99]with valuesin Katz’sringofdivided congruences. Thef-invariantgeneralizes the classical e invariant of Frank Adams. The purpose of this work isan extension of Adams’ formulas [Ada66, 11] for the e-invariant of Toda bracketsto the second filtration:ehα,pι,βi = −pe(α)e(β)ehpι,α,βi = −pδe(α)e(β)ehα,β,pιi = −pδe(α)e(β)modulo the indeterminacy. Here, δ is a certain number which only depends on thedimensions.The e-invariant is not strong enough to detect elements in the second filtration.The reason is the following: the e-invariant is based on K-theory and in the K-based Adams-Novikov spectral sequence there aren’t any classes in the second lineother than classes in the image of J. In [Lau99] K-theory was replaced by ellipticcohomology theory of level N (see [Bry90][Fra92][Bak94]). It was shown that thecorresponding tertiary invariant f is injective on the 2-line. So, it is natural to askfor a precise relationship between the f-invariant and Toda brackets.The 2-line of the classical Adams-Novikov spectral sequence was computed forodd primes by Miller, Ravenel and Wilson in [MRW77] with the help of the chro-matic spectral sequence and by Shimomura for p = 2 in [Shi81]. In contrast to the1-line it is not cyclic. The number of generators is unbounded when considered asa function of the dimension. Its complicated structure makes it hard to determineclasses such as Massey products. The f-invariant simplifies the 2-line in two steps:first the complex orientation gives an injective map into the elliptic based E

Motivic invariants ofp-adic fields

AbstractWe provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL〈n〉 overp-adic fields. These spectra interpolate between integral motivic cohomology (n= 0), a connective version of algebraicK-theory (n= 1), and the algebraic Brown-Peterson spectrum (n= ∞). We deduce that, overp-adic fields, the 2-completeBPGL〈n〉 splits over 2-complete BPGL〈0〉, implying that the slice spectral sequence forBPGLcollapses.This is the first in a series of two papers investigating motivic invariants ofp-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.

The beta elementsβtp2∕rin the homotopy of spheres

In [1], Miller, Ravenel and Wilson defined generalized beta elements in the E2-term of the Adams-Novikov spectral sequence converging to the stable homotopy groups of spheres, and in [5], Oka showed that the beta elements of the form βtp2/r for positive integers t and r survives to the stable homotopy groups at a prime p > 3, when r ≤ 2p − 2 and r ≤ 2p if t > 1. In this paper, we expand the condition so that βtp2/r for r ≤ p2 − 3 survives to the stable homotopy groups.

Remarks on motivic homotopy theory over algebraically closed fields

AbstractWe discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

On β-elements in the Adams-Novikov spectral sequence

In this paper we detect invariants in the comodule consisting of β-elements over the Hopf algebroid (A(m+1), G(m+1)) defined in [4], and we show that some related Ext groups vanish below a certain dimension. The result obtained here will be extensively used in future work with Ravenel to extend the range of our knowledge for π*(T(m)) obtained in [4].