Symmetry groups of non-simply-connected four-manifolds

Abstract

Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $\chi(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically trivial, locally linear, effective action on $M$, then $G$ must be cyclic. With additional assumptions to ensure orientability of some components of the singular set (e.g. if $G$ acts by symplectic symmetries, or preserving a spin structure), we also rule out $C_2 \times C_2$ actions. The proofs use equivariant cohomology, localization, and a careful study of the first cohomology groups of the (potential) singular set.