Abstract: In this second talk we will discuss the idea of complexifying Clifford algebras and classifying them. We will give many examples of the Clifford algebra $Cl(s + t), s + t = n,$ on $R^n$ and see that these are actually matrices with entries from either $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Our discussions continues as we look at Clifford modules, which are representations of a Clifford algebra, and Clifford bundles. When $M$ is a Riemannian manifold with a metric $g$, the Clifford bundle of $M$ is the Clifford bundle generated by the tangent bundle $TM$.

Abstract: 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 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 (Poincaré) 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.