On contact type hypersurfaces in 4-space

TL;DR

This paper uses Heegaard Floer homology to show certain 3-manifolds, like Brieskorn spheres, cannot be contact type hypersurfaces in R^4, revealing new topological constraints in symplectic geometry.

Contribution

It introduces an obstruction based on Heegaard Floer homology that prevents Brieskorn spheres from embedding as contact type hypersurfaces in R^4, advancing understanding of symplectic topology.

Findings

Brieskorn homology spheres cannot be contact type embeddings in R^4.

No rationally convex domain in C^2 has boundary diffeomorphic to a Brieskorn sphere.

Existence of contact 3-manifolds bounding Stein but not symplectically convex domains.

Abstract

We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic $R^4$. Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in $R^4$, a result that has bearing on conjectures of Gompf and Koll\'ar. This implies in particular that no rationally convex domain in $C^2$ has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains but not symplectically convex ones; in particular we find Stein domains in $C^2$ that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries.