Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres

Abstract

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in 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 C 2 . With the other orientation our methods do not resolve the question, but do give rise to an example of a contractible, boundaryirreducible 4-manifold that admits no Stein structure with either orientation, though its boundary has Stein fillings with both orientations.

Mazur-type manifolds with $L$-space boundaries

“…Remark 4. Recall that a well-known equivalent phrasing of the smooth 4-dimensional Poincaré Conjecture is that every contractible manifold with boundary S 3 is diffeomorphic to B 4 (see [23,Remark 4.8] and related discussion after Question 1.2 in [17]). Theorem 1 touches on this, in that it shows that whenever S 3 bounds a contractible manifold M of Mazur-type, then M is diffeomorphic to B 4 .…”

Symplectic hats

“…There are two such trefoils, with rotation numbers ±1, corresponding to two non-isotopic, conjugate contact structures on Σ(2, 3, 7) = S 3 −1 (T 2,3 ). Since there are exactly two tight contact structures on Σ(2, 3, 7) [46], one of these two contact structure is the canonical one, and the corresponding handlebody is a Stein filling W with Q W = −1 .…”

On contact type hypersurfaces in 4-space

“…However, there are many examples of homology spheres that embed smoothly in R 4 but do not carry any symplectically fillable contact structure ξ having θ(ξ) = −2, such as −M p for p ≥ 2, where M p is the Seifert rational homology sphere considered in Section 5.2 (see [61,Lemma 21]). Irreducible integer homology spheres not carrying fillable structures with θ = −2 include −Σ(2, 3, 12n + 1) for n ≥ 1 [38], though for n ≥ 3 is is unknown if these manifolds admit smooth embeddings in R 4 .…”

“…Besides Brieskorn spheres that do not satisfy B2, or those for which the classification of contact structures has been obtained and shows that no fillable structure has the correct homotopy class to be filled by a homology ball, the only direct evidence for Conjecture 1.3 was given in [38]. There it was shown that a certain contractible domain in C 2 , having boundary the Brieskorn manifold Σ (2,3,13), is not diffeomorphic to a Stein domain.…”