Resonance varieties and Dwyer-Fried invariants

TL;DR

This paper explores the relationship between resonance varieties and Dwyer-Fried invariants in certain spaces, showing that for 'straight' spaces these invariants can be computed directly from resonance varieties, with implications for understanding regular covers.

Contribution

It identifies conditions under which Dwyer-Fried invariants are exactly determined by resonance varieties, extending previous containment results to equality for 'straight' spaces.

Findings

For 'straight' spaces, mbda-invariants are equal to the complement of Schubert varieties.

Resonance varieties can be used to compute mbda-invariants in these spaces.

Translated components in characteristic varieties complicate the computation in general.

Abstract

The Dwyer-Fried invariants of a finite cell complex X are the subsets \Omega^i_r(X) of the Grassmannian of r-planes in H^1(X,\Q) which parametrize the regular \Z^r-covers of X having finite Betti numbers up to degree i. In previous work, we showed that each \Omega-invariant is contained in the complement of a union of Schubert varieties associated to a certain subspace arrangement in H^1(X,\Q). Here, we identify a class of spaces for which this inclusion holds as equality. For such "straight" spaces X, all the data required to compute the \Omega-invariants can be extracted from the resonance varieties associated to the cohomology ring H^*(X,\Q). In general, though, translated components in the characteristic varieties affect the answer.