Spectral sequences, Massey products and homology of covering spaces

TL;DR

This paper explores the relationship between spectral sequences, Massey products, and homology in covering spaces, providing new computational tools and bounds for Betti numbers and cohomology ranks, especially in hyperplane arrangements.

Contribution

It demonstrates that differentials in a spectral sequence are computed by Massey products and extends results relating Alexander modules to Massey products over arbitrary fields.

Findings

Differentials are computed by higher order Massey products.

Provides upper bounds for mod p Betti numbers of cyclic covers.

Shows vanishing Massey products imply combinatorial determination of Betti numbers.

Abstract

We revisit the equivariant spectral sequence considered by Papadima-Suciu, and show that all its differentials are computed by higher order Massey products. As a first application, we extend to arbitrary field coefficients results of Pajitnov relating the size of Jordan blocks for the eigenvalue 1 part of the Alexander modules to the length of nonvanishing Massey products in cohomology. We also give computable upper bounds for the mod p Betti numbers of prime power cyclic covers, and resp. for the ranks of the cohomology groups with coefficients in a prime order rank one local system. Under suitable conditions, these bounds are improvements of the ones obtained by Papadima-Suciu. We also specialize these results to the case of hyperplane arrangement complements, showing, e.g., that vanishing of higher-order Massey products implies that the mod p Betti numbers of prime p tower cyclic covers are combinatorially determined.