Spectral sequences, Massey products and homology of covering spaces
Yongqiang Liu, Laurentiu Maxim, Botong Wang
TL;DR
The paper analyzes the relationship between the differentials in a Papadima--Suciu-type equivariant spectral sequence for covering-space homology and higher-order Massey products. By constructing a Massey-type spectral sequence and a dg-model C^*(X,η,∞), it shows that the eigenvalue-1 part of the homology of infinite and finite cyclic covers is controlled by Massey products and the Aomoto complex, yielding concrete bounds on Betti numbers and local-system ranks. It extends Pajitnov's eigenvalue-1 results to arbitrary field coefficients, providing computable bounds for mod-$p$ Betti numbers of prime-power covers and for ranks of rank-one local-system cohomology, with sharper estimates when higher Massey products are nontrivial. The work specializes to hyperplane arrangement complements, where vanishing Massey products imply combinatorial determinacy of mod-$p$ Betti numbers, thus linking covering-space topology, Massey products, and arrangement combinatorics. Overall, it offers a unified framework to bound and understand covering-space homology via higher-order Massey products, particularly over finite fields, and highlights cases where topology is governed by combinatorial data.
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.