A Toda bracket convergence theorem for multiplicative spectral sequences

Abstract

Moss' theorem, which relates Massey products in the E r -page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable topological model category, we prove a general version of Moss' theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. The theorem has broad applications, e.g. to the computation of the motivic slice and motivic Adams spectral sequences.

The $\mathbb C$-motivic Adams-Novikov spectral sequence for topological modular forms

“…Toda brackets. For background on Massey products and Toda brackets, including statements of the May convergence theorem and the Moss convergence theorem, we refer readers to [Tod62], [May69], [Mos70] and also [Isa19], [BK21].…”

“…[BK21, Theorem 4.16] implies that c = h 2 2 , 2, h 1 in the mANss E 2 -page. Multiply by ∆ to obtain ∆c = h 2 2 , 2, h 1 ∆ = h 2 2 , 2, ∆h 1 .…”

Stable homotopy groups of spheres: from dimension 0 to 90

“…-We next discuss the Moss Convergence Theorem [50], which is the essential tool for computing Toda brackets in stable homotopy groups. See also [9] for a modern proof of the Moss Convergence Theorem that applies not only in classical stable homotopy theory but also to a wide variety of stable homotopy theories including C-motivic stable homotopy theory.…”

Stable operations and topological modular forms

“…In particular, working modulo 8 and c 4 , we see that x8, h 2 , gy 8,c4 8∆. The Moss convergence theorem (see [Mos70] for the original statement for the Adams spectral sequence and [BK21] for the adaptation to other multiplicative spectral sequences) for the descent spectral sequence of TMF 2 then shows our desired Toda bracket x8, ν, κy is congruent to r8∆s modulo 8 and c 4 . From this and Lm.10.1.4, we obtain the following chain of containments:…”

“…In particular, modulo 8 and all the elements in the basis B 96d 12 not of the form ∆ 8d 1 the Massey product x8, h 2 , gy is congruent to the singleton set of 8∆. The Moss convergence theorem (see [Mos70] for the original statement for the Adams spectral sequence and [BK21] for the adaptation to other multiplicative spectral sequences) shows our desired Toda bracket x8, ν, κ∆ 8d y is congruent to r8∆s modulo 8 and the elements of the complement B 96d 12 ¡ t∆ 8d 1 u. By Lm.10.1.4 and the above, we obtain the following containment of sets:…”