Resonance varieties and Dwyer-Fried invariants

Abstract

The Dwyer-Fried invariants of a finite cell complex X are the subsets Ω^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 Ω-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 Ω-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.

39 pages; to appear in "Arrangements of Hyperplanes - Sapporo 2009," Advanced Studies in Pure Mathematics

References in corpus (6)

• Algebraic invariants for right-angled Artin groups
• Toric complexes and Artin kernels
• Geometric and algebraic aspects of 1-formality
• The line geometry of resonance varieties
• On the irreducible components of characteristic varieties
• Homology of subgroups of right-angled Artin groups

Cited by in corpus (7)

• Characteristic varieties and Betti numbers of free abelian covers
• Around the tangent cone theorem
• The pure braid groups and their relatives
• Pure virtual braids, resonance, and formality
• Chen ranks and resonance varieties of the upper McCool groups
• Geometric and homological finiteness in free abelian covers
• Cohomology, Bocksteins, and resonance varieties in characteristic 2