Homologically Trivial Smooth Involutions on K3 Surfaces

Abstract

We will show that any smooth involution on a K3 surface induces a non-trivial action on its homology. In fact, a closed spin 4-manifold $M$ with $H_1(M; \mathbf{Z}_2) = 0$ and sign $M \ne 0$ will be shown to admit no homologically trivial locally linear involutions. The proof uses only the $G$-signature theorem and the sublattices and branched coverings arguments.