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arxiv: 2008.02755 · v3 · pith:5JEMUPOOnew · submitted 2020-08-06 · 🧮 math.GT · math.CV· math.SG

On contact type hypersurfaces in 4-space

Thomas E. Mark , B\"ulent Tosun This is my paper

Pith reviewed 2026-05-24 14:14 UTC · model grok-4.3

classification 🧮 math.GT math.CVmath.SG
keywords Brieskorn homology spherescontact type embeddingsHeegaard Floer homologyStein domainssymplectically convexrational convexitycontact structures4-manifolds
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The pith

No Brieskorn homology sphere admits a contact type embedding in R^4

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Brieskorn homology spheres cannot arise as contact type hypersurfaces in standard symplectic R^4 by means of a Heegaard Floer homology obstruction. Sympathetic readers care because this constrains the possible boundaries of convex domains in C^2 and distinguishes Stein fillability from symplectic convexity relative to the ambient structure. It also supplies infinitely many contact three-manifolds that bound Stein domains without admitting symplectically convex realizations. The result bears on conjectures regarding such embeddings and their topological restrictions.

Core claim

Using an obstruction derived from Heegaard Floer homology, we prove that no Brieskorn homology sphere admits a contact type embedding in R^4. This has implications for conjectures of Gompf and Kollár and shows in particular that no rationally convex domain in C^2 has boundary diffeomorphic to a Brieskorn sphere. We also construct infinitely many examples of contact three-manifolds that bound Stein domains but not symplectically convex ones in the standard symplectic structure on C^2.

What carries the argument

Heegaard Floer homology obstruction applied to contact type hypersurfaces in R^4

If this is right

  • No Brieskorn homology sphere bounds a rationally convex domain in C^2.
  • Infinitely many contact 3-manifolds bound Stein domains but not symplectically convex domains in C^2.
  • Stein domains in C^2 exist that cannot be realized as Weinstein domains with respect to the ambient symplectic structure while keeping the boundary contact structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation between Stein and Weinstein structures may indicate deeper differences in how fillings interact with ambient symplectic forms.
  • Obstructions of this type could extend to other families of 3-manifolds with nontrivial Floer homology.
  • Such results might help classify which contact manifolds can appear in convex position in higher-dimensional symplectic spaces.

Load-bearing premise

The Heegaard Floer homology obstruction for contact structures extends to contact type embeddings in the standard symplectic R^4.

What would settle it

An explicit example of a Brieskorn homology sphere realized as a contact type hypersurface in R^4 would falsify the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper proves that no Brieskorn homology sphere admits a contact-type embedding into standard symplectic R^4 by deriving an obstruction from Heegaard Floer homology; this yields consequences for conjectures of Gompf and Kollár, shows that no rationally convex domain in C^2 has boundary diffeomorphic to a Brieskorn sphere, and produces infinitely many contact 3-manifolds that bound Stein domains but not symplectically convex ones in the ambient structure.

Significance. If the central claim holds, the result supplies a concrete topological constraint on contact-type hypersurfaces in R^4 and separates Stein-fillable structures from those that can be realized as Weinstein domains with respect to the standard symplectic form on C^2. The argument draws on externally established properties of Heegaard Floer homology rather than ad-hoc normalizations.

major comments (1)
  1. [Abstract and §1] Abstract and §1: the central claim requires that every contact structure induced by a contact-type embedding of a Brieskorn sphere into (R^4, ω_std) satisfies the HF vanishing condition used in the obstruction. The manuscript states that an obstruction is 'derived from' HF and then applied, but does not explicitly confirm that the outward-pointing Liouville condition with respect to ω_std forces the induced ξ into the obstructed class; this compatibility step is load-bearing for the embedding result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the central claim requires that every contact structure induced by a contact-type embedding of a Brieskorn sphere into (R^4, ω_std) satisfies the HF vanishing condition used in the obstruction. The manuscript states that an obstruction is 'derived from' HF and then applied, but does not explicitly confirm that the outward-pointing Liouville condition with respect to ω_std forces the induced ξ into the obstructed class; this compatibility step is load-bearing for the embedding result.

    Authors: We agree that the manuscript would benefit from an explicit statement of this compatibility. The obstruction derived from Heegaard Floer homology is formulated for contact structures on Brieskorn spheres that arise precisely as the boundaries of contact-type hypersurfaces in (R^4, ω_std); the outward-pointing Liouville vector field is part of the definition of contact type and ensures the induced ξ belongs to the class where the HF vanishing applies. Nevertheless, to make the logical step fully transparent, we will insert a short clarifying paragraph in §1 (and update the abstract if needed) that recalls how the Liouville condition places the contact structure in the obstructed class. This is a minor expository revision. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external HF obstruction to embeddings

full rationale

The paper derives its main result by applying a pre-existing obstruction from Heegaard Floer homology (an independent theory developed by Ozsváth-Szabó) to the induced contact structures on Brieskorn spheres arising from contact-type embeddings in (R^4, ω_std). The abstract and description indicate no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations; the passage from embedding to HF vanishing condition rests on standard properties of contact invariants rather than any reduction internal to the paper's own inputs or equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Heegaard Floer homology for contact 3-manifolds and the definition of contact type embeddings; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Heegaard Floer homology provides an obstruction to the existence of certain contact structures or embeddings
    The paper states that an obstruction derived from HF is used; this is a background fact from the literature on HF.

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