Poincare duality is a fundamental tool in the study of manifolds. We give two generalizations of Poincare duality for equivariant (co)homology (here we mean the (co)homology of the Borel construction). The first one is given by introducing a second equivariant (co)homology theory denoted by DH stating that for a closed oriented smooth manifold M with a smooth and orientation preserving action of a finite group there is an isomorphism DH^{G}_{m-k}(M) \to H^{k-dim(G)}_{G}(M) and DH^{k}_{G}(M) \to H^{G}_{m-k-dim(G)}(M). The second one states that under the same conditions there is a long exact sequence, ...\to TH^{k-1}_{G}(M) \to H^{G}_{m-k-dim(G)}(M) \to H^{k}_{G}(M) \to TH^{k}_{G}(M) \to .., where TH^{*}_{G} stand for equivariant Tate cohomology. We use this to give a construction for equivariant Tate cohomology suitable for compact Lie groups. This construction is geometric. It is given as a bordism theory using a certain generalization of manifolds called stratifolds which were defined by Kreck. In the second part of the talk we will introduce a product in the homology of BG - the classifying space of a compact Lie group G. This construction is geometric and simple. We will see some results indicating the vanishing of this product for many groups, but not for all groups. In case were G is finite, this product is equal to the product induced by the cup product in negative Tate cohomology.
Donnerstag, den 11. Juli 2013 um 14.00 Uhr, in INF 288, HS5 Donnerstag, den 11. Juli 2013 at 14.00, in INF 288, HS5
Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. Banagl