Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres

2016/03/24 by Thomas E. Mark, Mark, Thomas E., Bülent Tosun +1 · 1 citation
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG) #math.CV #math.GT #math.SG

paper · pdf · doi:10.48550/arxiv.1603.07710

Main result modified due to an error; added Theorem 1.9 exhibiting a cork that does not admit a Stein structure with either orientation

arxiv created 2017/12/11 · arxiv updated 2017/12/12

Abstract

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in \mathbb C², with either orientation. A related question asks whether every compact contractible 4-manifold admits the structure of a Stein domain. We verify Gompf's conjecture, with one orientation, for a family of Brieskorn spheres of which some are known to admit a smooth embedding in \mathbb C². With the other orientation our methods do not resolve the question, but do give rise to an example of a contractible, boundary-irreducible 4-manifold that admits no Stein structure with either orientation, though its boundary has Stein fillings with both orientations.

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