Fixed point free circle actions and finiteness theorems

We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any T-k-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, lambda less than or equal to sec less than or equal to Lambda, and diameter, diam less than or equal to d, contains only finitely many diffeomorphism types depending only on n, lambda, Lambda and d. (ii) The set of simply connected n-manifolds (n less than or equal to 6) with lambda less than or equal to sec less than or equal to Lambda and diam less than or equal to d contains only finitely many diffeomorphism types depending only on n, lambda, Lambda and d.