#### Volume 16, issue 3 (2016)

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

### Julia Bennett

Algebraic & Geometric Topology 16 (2016) 1637–1681
##### 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.

##### Keywords
exotic smooth structures, open $4$–manifolds, Stein surfaces
Primary: 57N13
Secondary: 57R55
##### Publication
Received: 25 November 2014
Revised: 20 September 2015
Accepted: 29 September 2015
Published: 1 July 2016
##### Authors
 Julia Bennett Department of Mathematics University of Texas at Austin Austin, TX 78712 United States