Geometriae Dedicata

, Volume 121, Issue 1, pp 89–111 | Cite as

Stein domains and branched shadows of 4-manifolds

Original Paper


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.


Stein domain Polyhedra Manifold Shadow 

Mathematics Subject Classifications (2000)

Primary 57N13 Secondary 32B28 


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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique AvancéeStrasbourg CedexFrance

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