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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.

##### MSC:
 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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