Exotic smoothings via large ℝ4’s in Stein surfaces

Bennett, Julia

doi:10.2140/agt.2016.16.1637

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

Julia Bennett

Algebraic & Geometric Topology 16 (2016) 1637–1681

DOI: 10.2140/agt.2016.16.1637

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

Mathematical Subject Classification 2010

Primary: 57N13

Secondary: 57R55

References

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

Volume 16, issue 3 (2016)