Jim Bryan
Furuta's ten-eighth'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 actions 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.