Group actions and the topology of nonnegatively curved $4$-manifolds

Abstract

We consider nonnegatively curved $4$-manifolds that admit effective isometric actions by finite groups and from that draw topological conclusions about the manifold. Our first theorem shows that if the manifolds admits an isometric $Z_{p} \times Z_{p}$, for $p$ large enough that the manifold has Euler characteristic less than or equal to five. Our second theorem requires no hypothesis on the structure of the group other then that it be large but it does require the manifold to be $\delta$-pinched, in which case we can then again conclude that the Euler characteristic is less than or equal to five.