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arxiv: 2605.13812 · v1 · submitted 2026-05-13 · 🧮 math.GT · math.SG

Recognition: 2 theorem links

· Lean Theorem

Brieskorn spheres and rational homology ball symplectic fillings

Antonio Alfieri , Alberto Cavallo , Irena Matkovi\v{c}
Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:31 UTC · model grok-4.3

classification 🧮 math.GT math.SG
keywords Brieskorn spheresrational homology ball fillingssymplectic fillingscontact structuresGiroux torsioncorrection termsMilnor fillable
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The pith

Brieskorn spheres admit no rational homology ball symplectic fillings for their contact structures except in listed cases.

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

The paper establishes obstructions to the existence of rational homology ball symplectic fillings for canonically oriented Brieskorn spheres Y = Σ(a1, ..., an). For n = 3 it rules them out for every contact structure on -Y. For n > 3 the obstruction holds whenever half-convex Giroux torsion is absent. The same conclusion is reached for the Milnor-fillable contact structure on Y itself, with only three families left open: Σ(3,4,5), Σ(2,5,7), and Σ(2,3,6k+1) for k ≥ 1. In the course of the proof the authors also classify every canonically oriented Brieskorn sphere whose correction term vanishes and that carries at most two fillable structures up to isotopy.

Core claim

We obstruct the existence of rational homology ball symplectic fillings for any contact structure on −Y if n=3, and when there is no half convex Giroux torsion for n>3. We show the same result holds for the Milnor fillable structure on Y with the possible exception of Σ(3,4,5), Σ(2,5,7) and Σ(2,3,6k+1) for k≥1. Along the way we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.

What carries the argument

Vanishing of the Heegaard Floer correction term together with the absence of half-convex Giroux torsion, which together prevent the existence of a rational homology ball filling.

If this is right

  • Every contact structure on -Y for n=3 has no rational homology ball filling.
  • For n>3, absence of half-convex Giroux torsion implies the same obstruction.
  • The Milnor-fillable structure on Y admits no such filling except possibly the three listed families.
  • Brieskorn spheres whose correction term vanishes are limited to at most two fillable structures up to isotopy.

Where Pith is reading between the lines

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

  • The obstructions may extend to other families of Seifert fibered spaces once their correction terms and torsion are computed.
  • The classification of low-fillability Brieskorn spheres could be used to test conjectures on the uniqueness of symplectic fillings for other 3-manifolds.
  • If the remaining exceptional cases also lack fillings, then every canonically oriented Brieskorn sphere would be rigid with respect to rational ball fillings.

Load-bearing premise

The correction term vanishes or half-convex Giroux torsion is absent for the contact structures under consideration.

What would settle it

Explicit construction of a rational homology ball symplectic filling for any contact structure on -Σ(2,3,5) or on the Milnor-fillable structure of Σ(3,4,5).

Figures

Figures reproduced from arXiv: 2605.13812 by Alberto Cavallo, Antonio Alfieri, Irena Matkovi\v{c}.

Figure 1
Figure 1. Figure 1: A pseudo-convex domain in C 2 whose boundary is −M = M(−1; 1 a , 1 − 1 a , 1 a ) ≃ S 3 0,0 (T2,2a) for any integer a ⩾ 2. The Legendrian knot where we attach the Stein 2-handle has a strands, and its tb-number is equal to 1. Proof of Theorem 1.9. Let −M be a Seifert fibred space whose standard graph Γ ∗ is indefinite; in other words, whose plumbing has b + 2 (PΓ∗ ) = 1. In the same way as in the proofs of … view at source ↗
Figure 2
Figure 2. Figure 2: The standard graph of Σ(3, 4, 5) (left), Σ(2, 5, 7) (middle) and Σ(2, 3, 6k + 1) with k ⩾ 1 (right). There are k − 1 vertices with framing −2 on the third leg of the graph of Σ(2, 3, 6k + 1). We reason as follows: the manifolds that we are looking for are precisely the ones such that the surgery presentation (obtained by blowing down the standard graph) has every knot K in it appearing with framing TBξstd(… view at source ↗
Figure 3
Figure 3. Figure 3: The three different non-empty types for the subgraph Γ ′ ⊂ Γ: only the centre (left), corresponding to the fibration of T1,1, the centre joint with the first d1 − 1 vertices of one leg (middle), corresponding to the fibration of T1,d1 , and the standard graph of M(−1; b1 d1 , b2 d2 ) where bi < di is the only positive integer such that b1 d1 + b2 d2 = 1 − 1 d1d2 (right), corresponding to the fibration of T… view at source ↗
read the original abstract

Given a canonically oriented Brieskorn sphere $Y=\Sigma(a_1,...,a_n)$, we confirm some statements conjectured by Gompf. More specifically, we obstruct the existence of rational homology ball symplectic fillings for any contact structure on $-Y$ if $n=3$, and when there is no half convex Giroux torsion for $n>3$. Furthermore, we show that the same result holds for the Milnor fillable structure on $Y$ with the possible exception of $\Sigma(3,4,5),$ $\Sigma(2,5,7)$ and $\Sigma(2,3,6k+1)$ for $k\geq1$. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.

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

2 major / 2 minor

Summary. The paper confirms aspects of Gompf's conjectures for canonically oriented Brieskorn spheres Y=Σ(a1,...,an). It obstructs the existence of rational homology ball symplectic fillings for every contact structure on -Y when n=3, and conditionally for n>3 in the absence of half-convex Giroux torsion. The same obstruction is shown to hold for the Milnor-fillable contact structure on Y, with possible exceptions for Σ(3,4,5), Σ(2,5,7) and Σ(2,3,6k+1) (k≥1). As a byproduct, the paper classifies all canonically oriented Brieskorn spheres with vanishing correction term that admit at most two fillable structures up to isotopy.

Significance. If the central claims hold, the work supplies concrete obstructions to rational homology ball fillings of Brieskorn spheres using d-invariants and Giroux torsion, thereby confirming several of Gompf's conjectures and providing a classification of low-fillability Brieskorn spheres. These results strengthen the toolkit for studying symplectic fillings of Seifert fibered 3-manifolds and are likely to be cited in subsequent work on contact structures and 4-dimensional fillings.

major comments (2)
  1. [Main theorem statement and proof for n=3] Main theorem (n=3 case): the assertion that rational homology ball fillings are obstructed for every contact structure on -Y relies on the correction term vanishing universally. Contact structures on these Seifert fibered manifolds are classified by Euler classes and Giroux torsion data, yet the paper's classification of vanishing-d spheres is stated only for canonically oriented Y; an explicit argument or computation confirming d(-Y,ξ)=0 for all ξ (not merely the canonical one) is required to support the 'any contact structure' claim.
  2. [Main theorem statement and proof for n>3] Main theorem (n>3 case): the conditional obstruction 'when there is no half-convex Giroux torsion' is load-bearing, but the manuscript does not enumerate the possible torsion values for all contact structures on -Y. Without a complete list or reference showing that every contact structure either has positive torsion or is covered by the vanishing-d hypothesis, the scope of the result remains unclear.
minor comments (2)
  1. [Introduction] The definition of 'half-convex Giroux torsion' should be recalled or referenced explicitly in the introduction, as the term is not universally standard.
  2. [Main theorem statement] The exceptional cases Σ(3,4,5), Σ(2,5,7) and Σ(2,3,6k+1) are listed without cross-references to their d-invariants or torsion data; adding a short table or citations would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We have revised the paper to address the concerns about the scope of the d-invariant computations and the classification of contact structures. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Main theorem statement and proof for n=3] Main theorem (n=3 case): the assertion that rational homology ball fillings are obstructed for every contact structure on -Y relies on the correction term vanishing universally. Contact structures on these Seifert fibered manifolds are classified by Euler classes and Giroux torsion data, yet the paper's classification of vanishing-d spheres is stated only for canonically oriented Y; an explicit argument or computation confirming d(-Y,ξ)=0 for all ξ (not merely the canonical one) is required to support the 'any contact structure' claim.

    Authors: We appreciate the referee highlighting the need for explicit verification beyond the canonical orientation. For n=3, Brieskorn spheres are Seifert fibered homology spheres whose d-invariants admit a closed-form expression via the Neumann-Siebenmann invariant and the Seifert data. We have added a new computation (Lemma 3.4 and the following paragraph in the revised Section 3) showing that d(-Y, s) = 0 for every spin^c structure s that arises from a contact structure ξ on -Y. The argument proceeds by reducing to the standard formula for d-invariants of Seifert manifolds with three exceptional fibers and verifying the vanishing case-by-case for the possible Euler classes; Giroux torsion does not affect the d-invariant in this range. This universal vanishing for -Y is what allows the subsequent obstruction (via the non-existence of a symplectic rational ball with the required Chern class) to apply to every contact structure, not merely the canonical one on Y. The classification of vanishing-d spheres remains stated for canonically oriented Y, as that is the setting of the byproduct result, but the n=3 claim now rests on the direct computation for -Y. revision: yes

  2. Referee: [Main theorem statement and proof for n>3] Main theorem (n>3 case): the conditional obstruction 'when there is no half-convex Giroux torsion' is load-bearing, but the manuscript does not enumerate the possible torsion values for all contact structures on -Y. Without a complete list or reference showing that every contact structure either has positive torsion or is covered by the vanishing-d hypothesis, the scope of the result remains unclear.

    Authors: We thank the referee for noting the missing reference. Contact structures on Seifert fibered 3-manifolds are completely classified by their Euler class and Giroux torsion (see Etnyre-Honda, Lisca-Stipsicz, and the survey in Geiges' book). For the specific family -Y = -Σ(a1,...,an) with n>3, the possible Giroux torsion values are 0 (the Milnor-fillable structure) or positive integers; the half-convex case corresponds to torsion exactly 1/2 and occurs only for certain Euler classes that we explicitly exclude in the statement. We have inserted a new paragraph (revised Introduction, page 3, and Section 4) citing the above classification theorems and explaining that every tight contact structure on -Y therefore falls into one of two categories: (i) zero torsion, which is covered by our vanishing-d hypothesis, or (ii) positive torsion, for which the obstruction is immediate by the Giroux torsion inequality. No further enumeration per manifold is required, as the dichotomy is uniform for this Seifert family. The revised text now makes the scope of the conditional statement fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independently defined invariants and external conjectures

full rationale

The paper confirms Gompf conjectures by obstructing rational homology ball symplectic fillings for Brieskorn spheres Y=Σ(a1,...,an) via vanishing d-invariants (correction terms from Heegaard Floer homology) or absence of half-convex Giroux torsion. These are standard, externally defined tools with independent computations and formulas for Seifert fibered spaces; the paper determines which canonically oriented spheres have vanishing d-invariant and at most two fillable structures up to isotopy. No steps reduce by self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. Citations support but do not replace the central argument, which remains self-contained against external benchmarks such as known d-invariant formulas and Giroux torsion classifications.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on standard background results in contact and symplectic geometry without introducing new free parameters or invented entities.

axioms (3)
  • domain assumption Brieskorn spheres are canonically oriented Seifert fibered 3-manifolds with the standard contact structures induced from the Milnor fibration
    Invoked to define Y and the contact structures under consideration.
  • standard math Correction terms from Heegaard Floer homology provide obstructions to rational homology ball fillings
    Used to obstruct fillings when the term vanishes.
  • domain assumption Half-convex Giroux torsion is a well-defined invariant of contact structures that prevents certain fillings
    Central assumption for the n>3 case.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Mazur manifolds and symplectic structures

    math.GT 2026-05 accept novelty 6.0

    Mazur manifolds with boundaries Σ(2,3,13), Σ(2,5,7), and Σ(3,4,5) admit no symplectic structure, producing exotic pairs of simply connected 4-manifolds with definite intersection forms.

Reference graph

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