Stein domains and branched shadows of 4-manifolds

Abstract

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact four-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev's shadows suitably "smoothed"; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting "many" positive and negative Stein fillable contact structures, and prove a four-dimensional analog of Oertel's result on incompressibility of surfaces carried by branched polyhedra.

A short introduction to shadows of 4-manifolds

“…In this section we summarize the results proved by the author in [4] (and which will be presented in a self-contained way in [5] and [6]) in this direction and based on the notion of branched shadow.…”

“…This notion of "shadow complexity" turns out to be intimately connected with the geometric properties of 3-manifolds and in particular with their hyperbolic structures; this fact was also used in [7] to exhibit and study a particular class of 3-manifolds having strong geometric properties. Moreover a refinement of the notion of shadow of a 4-manifold, called a "branched shadow", has been studied by the author in [6], to attack a combinatorial study of gauge invariants and complex convexity problems. Branched shadows allow one to encode homotopy classes of almost complex structures on 4-manifolds, which, on manifolds admitting a shadow, are in a natural bijection with the set of Spin c -structures.…”

Branched Shadows and Complex Structures on 4-Manifolds

“…A necessary and sufficient condition (see [10]) for a 4-manifold M to admit a shadow is that M is a 4-handlebody, that is M admits a handle decomposition without 3 and 4-handles. In particular, ∂M is a non empty connected 3-manifold.…”

Stein structures and holomorphic mappings

“…Stein domains obtained in this way will typically have nonsmooth boundaries in X and may be chosen to realize uncountably many distinct diffeomorphism types. In certain special cases when the 2-cells in M satisfy certain framing conditions, it is possible to find Stein thickenings of a C 0 -small smooth perturbation of M in X which even have the diffeomorphism type of a smooth handlebody with core M. In this direction see also [5].…”