Stratifolds And Equivariant Cohomology Theories

Abstract

(co)homology is an important tool for studying spaces with a group action. These (co)homology groups are defined via the Borel construction. For discrete group they can also be defined in terms of homological algebra. One can also define Tate (co)homology for spaces with a finite group action in a similar way. There is a natural transformation from equivariant cohomology to Tate cohomology. In such a situation one naturally asks two questions:<br /> 1) Can one say something about the kernel and cokernel of this map?<br /> 2) Can one define Tate cohomology groups for spaces when the group acting is not finite but a compact Lie group?<br /> To both questions we give an answer in this thesis. The answer to the first question is given by defining a third (co)homology theory called backwards (co)homology and an exact sequence relating all three (co)homology theories. This new theory is a straightforward generalization of the construction of equivariant (co)homology and Tate (co)homology in terms of homological algebra. Of course, this only works for finite groups.<br /> The answer to the second question is given in the terms of stratifolds and bordism theories of stratifolds. We construct a long exact sequence which generalizes the sequence discussed above to the case of compact Lie groups.<br /> A third question that concerns us is equivariant Poincare duality. Poincare duality does not hold in equivariant (co)homology, even in the case of a point. We show that Tate cohomology is an obstruction for Poincare duality, which means that it holds if and only if Tate cohomology vanishes which is the case if and only if the action is free. We show that equivariant cohomology is Poincare dual to backwards homology, and equivariant homology is Poincare dual to backwards cohomology.<br /> The geometric point of view can be used for computations. We give a simple example how one should attack such a problem. In the last part of the thesis we give a geometric interpretation of the product in negative Tate cohomology, again in terms of stratifolds.