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Handlebody construction of Stein surfaces. (English) Zbl 0919.57012
A Stein manifold is a complex manifold \( X \) that admits an “exhausting strictly plurisubharmonic function”, which is essentially characterized as being a proper function \( f: X \rightarrow\mathbb{R} \) that is bounded below and can be assumed a Morse function, whose level sets \( f^{-1} (c) \) are “strictly pseudoconvex” (away from critical points), where \( f^{-1} (c) \) is oriented as the boundary of the complex manifold \( f^{-1}((- \infty, c])\). Strict pseudoconvexity implies, and for \( X \) of real dimension \( 4 \) is equivalent to asserting, that \( f^{-1} (c) \) inherits a contact structure determining its given orientation. A compact complex \( X \) with boundary is called a Stein domain if it admits a strictly plurisubharmonic function such that the boundary \( \partial X \) is a level set. A Stein manifold or domain of complex dimension \( 2 \) is called a Stein surface (with boundary). Y. Eliashberg [Int. J. Math. 1, No. 1, 29-46 (1990; Zbl 0699.58002); Legendrian and transversal knots in tight contact 3-manifolds, in ‘Topological methods in modern mathematics’, 171-193 (1993; Zbl 0809.53033)]has characterized Stein manifolds in all dimensions in terms of differential topology. The author proves that Eliashberg’s theorem in high dimensions applies up to homeomorphisms to 4-manifolds, that is, an open, oriented topological 4-manifold \( X\) is (orientation-preserving) homeomorphic to a Stein surface if and only if it is the interior of a topological (or smooth) handlebody without handles of index \( > 2\), and if so, then any almost-complex structure can be so realized. The given handle structure will not necessarily come from a plurisubharmonic (or even smooth Morse) function on the Stein surface, however. As an example, \(\mathbb{C}\mathbb{P}^ 2 \) minus a point admits an uncountable family of diffeomorphism types of Stein exotic smooth structures, none of which admit proper Morse functions with finitely many critical points. It is shown that \(\mathbb{R}^ 4 \) admits uncountably many exotic smooth structures that can be realized as Stein surfaces.
A standard form for any handle decomposition obtained from a strictly plurisubharmonic function on a compact Stein surface is established. New invariants for distinguishing contact 3-manifolds are produced, including a complete set of invariants for determining the homotopy class of an oriented 2-plane field on an oriented 3-manifold. These invariants are readily computable for the boundary of a compact Stein surface presented in standard form. Several families of oriented 3-manifolds are examined, namely the Seifert fibered spaces and all surgeries on various links in \( S ^3, \) and in each case it is seen that “most” members of the family are the oriented boundaries of Stein surfaces.

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R70 Critical points and critical submanifolds in differential topology
57R65 Surgery and handlebodies
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