Spectral sequences obtained from towers in the infinity category of spectra: décalage, exact couples and Massey products

Abstract

There are various ways of constructing spectral sequences from the infinity category of towers of spectra. Classically there is the approach via exact couples; we will also discuss the décalage functor as constructed by Hedenlund. We will show that this construction of spectral sequences from towers and the construction via exact couples give isomorphic spectral sequences. This can be proved by showing that both methods of can be related to another third construction method by Lurie, using recent work by Antieau. Furthermore, the décalage construction yields a functor of infinity operads and provides a way to construct multiplicative spectral sequences. Then we can define Massey products on such a multiplicative spectral sequence. Lastly, we will discuss a possible relation between these Massey products and Toda brackets on homotopy groups of spectra, analogous to Moss' theorem for the Adams spectral sequence.