Resonance varieties and Dwyer–Fried invariants

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 $H^1 (X, \mathbb{Q})$ which parametrize the regular $\mathbb{Z}^r$-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 $H^1 (X, \mathbb{Q})$. 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 $H^{\ast} (X, \mathbb{Q})$. In general, though, translated components in the characteristic varieties affect the answer.