Spectral sequences, Massey products and homology of covering spaces

Abstract

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, respectively, 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.