Non-smoothable $\mathbb{Z}/p$-actions on nuclei

Abstract

In this article we construct examples of non-smoothable $\mathbb{Z}/p$-actions on indefinite spin 4-manifolds with boundary for all primes $p\geq 5$. For example, we show that for each prime $p\geq 5$ and each $n\geq 1$ there exists a locally linear $\mathbb{Z}/p$-action on the Gompf nucleus $N(2pn)$ which is not smoothable with respect to any smooth structure on $N(2pn)$. Furthermore we investigate the behavior of these actions under two different types of equivariant stabilizations with $S^{2}\times S^{2}$, namely \emph{free} and \emph{homologically trivial} stabilizations -- in particular we show that our non-smoothable $\mathbb{Z}/p$-action on $N(2pn)$ remains non-smoothable after $2n-2$ free stabilizations, and after arbitrarily many homologically trivial stabilizations. We also show that free stabilizations satisfy a Wall stabilization principle in the sense that any non-smoothable $\mathbb{Z}/p$-action becomes smoothable after some finite number free stabilizations (under certain assumptions), whereas our aforementioned result implies that homologically trivial stabilizations do not satisfy this property. The proofs of these results use equivariant $\kappa$-invariants defined by the author in \cite{Mon22}, calculations of equivariant $\eta$-invariants for the odd signature and Dirac operators on Seifert-fibered spaces, as well as an analysis of the geometric $S^{1}$-action on the Seiberg-Witten moduli spaces of Seifert-fibered spaces induced by rotation in the fibers, which may be of independent interest.