On nonnegatively curved 4-manifolds with discrete symmetry

Abstract

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z (m) -action for a positive integer m a parts per thousand 61(7). Assume that Z (m) acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z (m) has at most one two-dimensional component, then M is homeomorphic to S (4), # (i) (l) =1S (2) x S (2), l = 1, 2, or # (j) (k) = 1 +/- CP (2), k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic chi(M) under the homological assumption of a Z (m) -action as above by using the Lefschetz fixed point formula.