Diffeotopy groups of non-compact 4-manifolds

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 $S⊊\mathring{M}$, there are uncountably many distinct smoothings of $M\setminus S$ whose diffeotopy groups are uncountable. We then prove that for a smoothable 4-manifold $M$ and for a non-empty, discrete set of points $S⊊\mathring{M}$, there exists a smoothing of $M\setminus S$ 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 $|S|\ge 2$.

arXiv comment19 pages, 7 figures. This is the revised version. The paper has been now accepted for publication in the Michigan Mathematical Journal