Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres

TL;DR

This paper proves Gompf's conjecture that certain Brieskorn spheres cannot be pseudoconvex embedded in ${ m C}^2$ with one orientation, and provides examples of contractible 4-manifolds without Stein structures.

Contribution

It verifies Gompf's conjecture for a family of Brieskorn spheres with one orientation and constructs a contractible 4-manifold that admits no Stein structure in either orientation.

Findings

Gompf's conjecture is verified for a family of Brieskorn spheres with one orientation.

An example of a contractible, boundary-irreducible 4-manifold with no Stein structure in either orientation.

Boundary of the manifold admits Stein fillings with both orientations.

Abstract

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in ${\mathbb C}^2$, 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 ${\mathbb C}^2$. 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.