  Let G=S^1, G=Z/p or more generally G be a finite p group, where p is an odd
prime number. If G acts on a space whose cohomology ring satisfies Poincare
duality (with appropriate coefficients k), we prove a mod 4 congruence between
the total Betti number of X^G and a number which depends only on the
k[G]-module structure of H^*(X;k). This improves the well known mod 2
congruences that hold for actions on general spaces.
