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Stein open subsets and topological pseudoconvexity

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Robert Gompf
UT Austin
Fri, 2019-09-20 12:00 - 12:50
MPIM Lecture Hall

We will examine the interplay between Eliashberg's theory for embedding Stein domains and Freedman's theory of topological 4-manifolds. The speaker showed (J. Top. 2013) that an open subset $U$ of a complex surface $X$ is smoothly isotopic to a Stein open subset iff the induced almost-complex structure on $U$ is homotopic to a Stein structure. Using Casson handles, he showed (J. Symp. Geom. 2005) that $U$ is topologically isotopic to a Stein open subset iff it is homeomorphic to the interior of a 2-handlebody. Using the full power of Freedman's capped towers and reimbedding, we can now conclude that any embedded 2-handlebody $H$ with locally flat boundary is topologically ambiently isotopic to one with Stein interior (as a subset of $X$). Furthermore, we can assume that the core 2-complex is smooth and totally real except at one point on each 2-cell, and that in the obvious mapping cylinder structure on H, the levels indexed by the Cantor set each cut out a Stein open subset. These open subsets frequently represent infinitely many diffeomorphism types. The levels are only topologically embedded (locally flat) 3-manifolds, but nevertheless inherit a natural notion of pseudoconvexity. They inherit some of the properties of smooth pseudoconvex 3-manifolds, such as a homotopy class of plane fields and a propensity to cut out Stein surfaces, yet they are more plentiful. They also provide a dual notion of topological pseudoconcavity of open complex surfaces, with uncountably many exotic smoothings of $\mathbb{R}^4$ admitting such structure.

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