Duality properties for abelian covers

Grahan Denham, University of Western Ontario


In parallel with a classical definition due to Bieri and Eckmann, say an FP group G is an abelian duality group if H^p(G,Z[G^{ab}]) 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 direct 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 (Poincaré) duality groups, yet they are not abelian duality groups. Some other families of groups and spaces possess both properties, in which cases the unifying framework is the Cohen-Macaulay property of a certain subgroup lattice. This is joint work with Alex Suciu and Sergey Yuzvinsky.

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