# Group actions, corks and exotic smoothings of R^4

## TL;DR

This paper explores the complex structure of exotic smoothings of R^4, revealing uncountably many diffeotopy classes and group actions, and connects cork theory to these phenomena.

## Contribution

It provides the first detailed analysis of diffeotopy groups of exotic R^4 smoothings and relates cork twisting to exotic smooth structures.

## Key findings

- Uncountably many isotopy classes of self-diffeomorphisms for each exotic smoothing.
- Explicit group actions realize these diffeomorphisms.
- Cork twisting is shown to be equivalent to twisting on exotic R^4 under broad conditions.

## Abstract

We provide the first information on diffeotopy groups of exotic smoothings of R^4: For each of uncountably many smoothings, there are uncountably many isotopy classes of self-diffeomorphisms. We realize these by various explicit group actions. There are also actions at infinity by nonfinitely generated groups, for which no nontrivial element extends over the whole manifold. In contrast, every diffeomorphism of the end of the universal R^4 extends. Our techniques apply to many other open 4-manifolds, and are related to cork theory. We show that under broad hypotheses, cork twisting is equivalent (up to blowups) to twisting on an exotic R^4, and give applications.