Group actions on Poincaré duality spaces

Let G = Z p for p prime and K = Z p , or let G = S 1 and K = g, and let G act on the compact space X. In this paper, we outline two proofs of the following: THEOREM. Suppose the compact G-space X is a Poincar duality space over K of formal dimension n. Then each connected component of the fixed point set is a Poincar duality space over K, and, if G ^ Z 2 , has formal dimension congruent to n mod 2. This solves affirmatively the conjecture of Su given in Let E G - B G be the universal bundle for G and let X G be the balanced product (X x E G )/G. The basic tools for both proofs are the fibre space X - X G - B G and the localization theorem of Borel ([1], [4]). In the case X is totally nonhomologous to zero in X G , Bredon has proven the Su conjecture [2], However, this condition can be replaced by the two lemmas below, and this constitutes our algebraic proof. The second proof involves applying the localization theorem to a Thorn space.