Torus and \(\mathbb Z/p\) actions on manifolds. (Q1430401)

If \(G\) is either a finite cyclic group of prime order or \(S^1\) acting on a space \(M\) then the \(G\)-equivariant cohomology of \(M\) can be calculated from the Leray spectral sequence of the map \(M\times _G EG\to BG\). If \(G=S^1\) and \(M\) is a Poincaré duality space then the components of the second term of this spectral sequence satisfy Proincaré duality. The author shows that if \(M^{S^1} \neq \emptyset\) then each term of this spectral sequence satisfies a properly defined ''Poincaré duality.'' As a consequence of this fact he proves results relating the cohomology of any Poincaré duality space with a torus action or a finite cyclic group action of prime order, to the cohomology of the fixed point set of this action. He also discusses the consequences and the ramifications of these results in 3-dimensional topology.