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A conjecture due to Gompf asserts that no nontrivial Brieskorn homologysphere admits a pseudoconvex embedding in ${\mathbb C}^2$, with eitherorientation. A related question asks whether every compact contractible4-manifold admits the structure of a Stein domain. We verify Gompf'sconjecture, with one orientation, for a family of Brieskorn spheres of whichsome are known to admit a smooth embedding in ${\mathbb C}^2$. With the otherorientation our methods do not resolve the question, but do give rise to anexample of a contractible, boundary-irreducible 4-manifold that admits no Steinstructure with either orientation, though its boundary has Stein fillings withboth orientations.

Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres