Nonsmoothable involutions on spin 4-manifolds

TL;DR

This paper constructs locally linear pseudofree involutions on certain spin 4-manifolds that cannot be smoothed, highlighting differences between topological and smooth categories in four-dimensional topology.

Contribution

It demonstrates the existence of nonsmoothable involutions on specific spin 4-manifolds with particular intersection forms, expanding understanding of symmetries in 4-manifold topology.

Findings

Existence of nonsmoothable involutions on spin 4-manifolds with n ≡ 2 mod 4

Construction of locally linear pseudofree Z2-actions

Distinction between topological and smooth symmetries in 4D

Abstract

Let $X$ be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to $n(-E_8)\bigoplus mH$, where $H$ is the hyperbolic form. In this paper, we prove that for $n$ such that $n\equiv 2 ~{\rm mod} ~4$, there exists a locally linear pseudofree $\mathbb{Z}_2$-action on $X$ which is nonsmoothable with respect to any possible smooth structure on $X$.