Exotic smoothings via large $\mathbb{R}^4$'s in Stein surfaces

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.