Exotic smoothings via large R^4's in Stein surfaces

TL;DR

This paper explores the connection between exotic smooth structures on R^4 and Stein surfaces, constructing new examples of large exotic R^4's embedded in Stein surfaces and analyzing their impact on 4-manifold smoothing theory.

Contribution

It introduces the first known examples of large exotic R^4's embedded in Stein surfaces and extends Casson's Embedding Theorem for this purpose.

Findings

Constructed uncountably many diffeomorphism types using these R^4's

Maintained control over genus-rank function and Taylor invariant

Extended Casson's Embedding Theorem for closed 4-manifolds

Abstract

We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds. In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4-manifolds. Under sufficiently nice conditions, we show that using these R^4's as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.