  Let G be either a finite cyclic group of prime order or S^1. We find new
relations between cohomology of a manifold (or a Poincare duality space) M with
a G-action on it and cohomology of the fixed point set, M^G. Our main tool is
the notion of Poincare duality on the Leray spectral sequence of the map M_G ->
BG. We apply our results to study group actions on 3-manifolds.
