Seiberg-Witten Theory and Z/2^p actions on spin 4-manifolds

TL;DR

This paper strengthens Furuta's 10/8 theorem for spin 4-manifolds with Z/2^p actions, leading to new bounds and classifications using Seiberg-Witten theory and group actions.

Contribution

It introduces a method to improve signature bounds on spin 4-manifolds with Z/2^p symmetry and classifies involutions on rational cohomology K3 surfaces.

Findings

Strengthened bounds on signatures with Z/2^p actions

New genus bounds for classes with divisibility

Classification of involutions on rational cohomology K3's

Abstract

Furuta's ``10/8-th's'' theorem gives a bound on the magnitude of the signature of a smooth spin 4-manifold in terms of the second Betti number. We show that in the presence of a Z/2^p action, his bound can be strengthened. As applications, we give new genus bounds on classes with divisibility and we give a classification of involutions on rational cohomology K3's. We utilize the action of a twisted product of Pin(2) and Z/2^p on the Seiberg-Witten moduli space. Our techniques also provide a simplification of the proof of Furuta's theorem.