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.

16 pages. An earlier version erroneously asserted that surfaces fixed by certain involutions were orientable, and that error invalidated a later proof. This version discusses the problem and corrects the error

Cited by in corpus (2)

• A survey of group actions on 4-manifolds
• Rank conditions for finite group Actions on 4-Manifolds