Combinatorial covers and vanishing of cohomology

Abstract

Abstract. We use a Mayer-Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen-Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients, and obtain new generalizations.

Action dimensions of some simple complexes of groups

“…We will use these central subarrangements to construct free abelian subgroups of . The same construction of these free abelian groups is given in . Lemma For any , injects into .…”

“…Given G ∈ Q(A), let A G := {H ∈ A | H G} be the induced central subarrangement of hyperplanes containing G. We will use these central subarrangements to construct free abelian subgroups of π 1 (M (A)). The same construction of these free abelian groups is given in [22]. Of course, if the central arrangement A is reducible, then the center of π 1 (M (A)) has rank greater than one (take central elements from each factor).…”

Local Systems on Complements of Arrangements of Smooth, Complex Algebraic Hypersurfaces

“…Finally, let T A X be the hyperplane arrangement in the tangent space to Y at a point in the relative interior of X, guaranteed by our hypothesis on the intersection of hypersurfaces. One of the main tools we will need in this note is a spectral sequence developed in [DSY16], which we summarize in the next theorem.…”

“…It has long been recognized that complements of complex hyperplane arrangements satisfy certain vanishing properties for homology with coefficients in local systems. We revisited this subject in our joint work with Sergey Yuzvinsky, [DSY16,DSY17], in a more general context.…”

Homology, lower central series, and hyperplane arrangements

“…[12]). There exist basepoint-preserving maps r Y :M(A Y ) → M(A) such that j Y • r Y id relative to x 0 .…”