# Non-smoothable Z/p-Actions on Nuclei

# Non-smoothable Z/p-Actions on Nuclei

In this talk we show that for any smooth 4-manifold X homeomorphic to a Gompf nucleus N(2n) and any odd prime p, the standard Z/p-action given by rotation in the fibers of the Seifert-fibered boundary cannot extend smoothly to a Z/p-action over X (with one exceptional case), whereas in some cases these actions do extend topologically. In particular, we show that for each prime p>=5 and each n>=1 there exists a non-smoothable Z/p-action on N(2pn) extending the standard Z/p-action on its boundary. Furthermore these actions remain non-smoothable after arbitrarily many equivariant stabilizations with S^2 x S^2 of a certain type, showing that non-smoothable Z/p-actions do not satisfy a Wall stabilization principle with respect to this class of stabilizations. The proof makes use of invariants coming from Seiberg-Witten Floer K-Theory as well as some equivariant index theory.