A-infinity-algebras, spectral sequences and exact couples

Abstract

We study in this article a possible further structure of homotopic nature on multiplicative spectral sequences. More precisely, since Kadeishvili's theorem asserts that, given a dg (or A-infinity-)algebra, its cohomology has also a structure of A-infinity-algebra such that both become quasi-isomorphic, and in a multiplicative spectral sequence one considers the cohomology of dg algebras when moving from a term to the next one, a natural problem that arises is to study how this two possible structures intertwine. We give such a homotopic structure proposal, called A-infinity-enhancement of multiplicative spectral sequences, which could be of interest in our opinion. As far we know, this construction was studied only recently by S. Lapin, even though he did not state any definition. Seeing that the procedure considered by Lapin is rather complicated to handle, we propose an equivalent but in our opinion easier approach. In particular, from our definition we show that the canonical multiplicative spectral sequence obtained from a filtered dg (or A-infinity-)algebra, which could be viewed as the main example, has such an A-infinity-enhancement.