Diffeotopy groups of non-compact 4-manifolds

TL;DR

This paper investigates the diffeotopy groups of exotic smoothings of punctured 4-manifolds, revealing uncountably many smoothings with complex diffeotopy group structures and extending known results to non-smoothable cases.

Contribution

It extends previous results on diffeotopy groups of exotic ^4's to a broader class of 4-manifolds and punctured cases, including non-smoothable manifolds.

Findings

Uncountably many smoothings with uncountable diffeotopy groups.

Existence of smoothings with diffeotopy groups similar to _U.

Results hold for non-smoothable manifolds with multiple punctures.

Abstract

We provide information on diffeotopy groups of exotic smoothings of punctured 4-manifolds, extending previous results on diffeotopy groups of exotic $\mathbb{R}^4$'s. In particular, we prove that for a smoothable 4-manifold $M$ and for a non-empty, discrete set of points $S \subsetneq \mathring{M}$, there are uncountably many distinct smoothings of $M\smallsetminus S$ whose diffeotopy groups are uncountable. We then prove that for a smoothable 4-manifold $M$ and for a non-empty, discrete set of points $S \subsetneq \mathring{M}$, there exists a smoothing of $M\smallsetminus S$ whose diffeotopy groups have similar properties as $\mathcal{R}_U$, Freedman and Taylor's universal $\mathbb{R}^4$. Moreover, we prove that if $M$ is non-smoothable, both results still hold under the assumption that $|S| \ge 2$.