Geometry & Topology seminar

Graham Denham, Western University

“Duality properties for abelian covers”

In parallel with a classical definition due to Bieri and Eckmann, we say an FP group G is an abelian duality group if Hp(G,Z[Gab]) is zero except for a single integer p = n, in which case the cohomology group is torsion-free. We make an analogous definition for spaces. In contrast to the classical notion, the abelian duality property imposes some obvious constraints on the Betti numbers of abelian covers.
While related, the two notions are inequivalent: for example, surface groups of genus at least 2 are (Poincar ́e) duality groups, yet they are not abelian duality groups. On the other hand, using a result of Brady and Meier, we find that right-angled Artin groups are abelian duality groups if and only if they are duality groups: both properties are equivalent to the Cohen- Macaulay property for the presentation graph. Building on work of Davis, Januszkiewicz, Leary and Okun, hyperplane arrangement complements are both duality and abelian duality spaces. These results follow from a more general cohomological vanishing theorem, part of work in progress with Alex Suciu and Sergey Yuzvinsky.