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Detecting Heegaard Floer homology solid tori

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Pith's one-line read A rational homology solid torus is a Heegaard Floer homology solid torus if and only if filling along its rational longitude yields a non-separating 2-sphere.

desk verdict A crisp new iff for Heegaard Floer solid tori, built from known machinery; the proof has one terse step that deserves referee pressure but nothing fatal. read the letter →

arxiv 2505.01217 v1 pith:OYVBMG57 submitted 2025-05-02 math.GT

Akram Alishahi , Tye Lidman , Robert Lipshitz This is my paper · ORCID
classification math.GT MSC 57K18
keywords HeegaardFloerhomologyborderedsolidtorusDehnfillingimmersedcurvesrationallongitudenon-separating2-sphereSeifertfiberedspaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a purely topological detection criterion for a class of bordered 3-manifolds known as Heegaard Floer homology solid tori (HFSTs): a rational homology solid torus is an HFST exactly when the Dehn filling along its rational longitude contains a non-separating 2-sphere. HFSTs are defined by an algebraic invariance, namely that their bordered Floer invariant is unchanged by Dehn twisting along the rational longitude, so the theorem turns a difficult algebraic condition into a single 3-manifold-topology check. The criterion explains and unifies known examples, including knot complements in reducible manifolds, and it yields a complete classification of Seifert fibered HFSTs.

What carries the argument

The load-bearing object is the immersed curve invariant $\widehat{dHF}(M)$: an immersed 1-manifold, possibly with local systems, in the punctured torus that packages the bordered Floer homology of a torus-boundary manifold. The decisive identity is Proposition 1.2, quoted from [HRW24, Proposition 7.11], which says $M$ is an HFST if and only if $\widehat{dHF}(M)$, after pulling tight, is supported in a neighborhood of the rational longitude $\lambda$. The proof then uses the twisted-coefficient pairing theorem to convert that support condition into vanishing of $H_*(\underline{\mathcal{S}} \boxtimes \widehat{CFD}(M))$, with the auxiliary module $\underline{\mathcal{S}}$ built from the 0-framed solid torus over $\mathbb{F}_2[[t,t^{-1}]]$, and the vanishing theorem of [AL19] that identifies this vanishing with the presence of a non-separating 2-sphere in $M(\lambda)$.

What would settle it

One concrete check: take a rational homology solid torus $M$ whose rational-longitude filling $M(\lambda)$ is a connected sum of lens spaces, so it contains a non-separating 2-sphere, and compute the immersed curve invariant $\widehat{dHF}(M)$ after pulling tight. The theorem predicts the curve is supported in a neighborhood of $\lambda$; if any strand escapes that neighborhood, the equivalence fails.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for a rational homology solid torus $M$ with rational longitude $\lambda$, $M$ is a Heegaard Floer homology solid torus if and only if the Dehn filling $M(\lambda)$ contains a non-separating 2-sphere. The proof passes through the immersed-curve reformulation of bordered Floer homology: $M$ is an HFST precisely when its immersed curve invariant $\widehat{dHF}(M)$, pulled tight, is supported in a neighborhood of $\lambda$. From that support condition the authors split the curve into components either non-parallel or parallel to $\lambda$, show each component pairs trivially with a model module over the Laurent series ring $\mathbb{F}_2[[t,t^{-1}]]$, and conclude that the twisted Floer homology of $M(\lambda)$ vanishes, which by a known criterion is equivalent to $M(\lambda)$ containing a homologically essential 2-sphere. The converse direction uses the surgery exact triangle with twisted coefficients to show that knot complements in reducible 3-manifolds with $b_1=1$ are always HFSTs.

Load-bearing premise

The load-bearing bridge is a previously established equivalence (cited as [HRW24, Proposition 7.11]) between being a Heegaard Floer homology solid torus and having the immersed curve invariant supported in a neighborhood of the rational longitude; if that equivalence has an unhandled edge case, the topological criterion in Theorem 1.1 would not follow.

Editorial extensions

If this is right

  • HFST status can be checked by a single Dehn filling: fill along the rational longitude and look for a non-separating 2-sphere.
  • Every HFST arises as the complement of a knot of infinite order in a closed reducible 3-manifold with $b_1=1$, so the class is exactly the class of such knot complements.
  • For Seifert fibered rational homology solid tori, the HFST condition is completely classified: the base orbifold is a Möbius band, with or without cone points, or the manifold is $D^2(0;p/q,-p/q)$.
  • Together with earlier L-space filling results, the theorem closes the remaining case where no filling is an L-space: there, HFSTs are precisely detected by the reducible rational-longitude filling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same surgery-exact-triangle mechanism may detect other algebraic invariance conditions: any bordered invariant that is unchanged under a rational-longitude twist should force a reducible filling, generalizing the HFST phenomenon to other Floer-theoretic settings.
  • The immersed-curve support condition suggests an algorithmic route: from a Heegaard diagram or surgery description, draw $\widehat{dHF}(M)$ and check whether it lies in a neighborhood of $\lambda$, giving a computable test for reducibility of $M(\lambda)$.
  • The Seifert classification leaves open the analogous question for graph manifolds; the reducible-filling criterion gives a concrete first obstruction to test there.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper proves a characterization of Heegaard Floer homology solid tori: a rational homology solid torus M is an HFST if and only if the Dehn filling of M along its rational longitude λ contains a non-separating 2-sphere. The forward direction is established in Section 3 via the surgery exact sequence with twisted coefficients and the Hanselman-Rasmussen-Watson characterization of HFSTs (Prop. 1.2). The reverse direction is proved in Section 4 using the immersed curve reformulation: assuming dHF(M) is supported near λ, the paper computes the twisted Floer homology of M(λ) as a tensor product of the solid torus module S with the bordered invariant, splitting according to components of the immersed curve. Section 5 applies Theorem 1.1 to classify Seifert fibered HFSTs.

Significance. The main theorem is an appealing and potentially very useful bridge between an algebraic condition (invariance under Dehn twists along the rational longitude) and a simple topological condition (a reducible Dehn filling). The proof is concise and relies on recent deep results rather than new technical machinery, which makes the paper accessible. If the gap in Proposition 4.1 is repaired, the result would be a significant contribution to the understanding of bordered Floer homology and L-space fillings. The paper also provides a clean classification of Seifert fibered HFSTs.

major comments (2)
  1. [Section 4, Proposition 4.1, Case 2] The proof that a component C_i homotopic to λ^j is regularly homotopic into a neighborhood of λ is under-specified. The argument uses the cover Σ corresponding to ⟨λ^j⟩ and asserts that C_i lifts to an embedded generator loop and that the preimage of λ is also an embedded generator loop, so any two such loops in the cylinder are regularly homotopic. However, the text does not explain how the 'through the basepoint z' clause in Prop. 1.2(3) is used, and if ν(λ) is not an annular neighborhood the topology of Σ could be different. The subsequent step applying Observation 2.3 to conclude ι1 [CFD(C_i) = 0 also requires the curve to be disjoint from the circle [0,1]×{0}, which a curve parallel to λ generally is not. Since both directions of Theorem 1.1 pass through Proposition 4.1, this is a load-bearing point that needs a detailed justification.
  2. [Section 4, Proposition 4.1, Case 1] The assertion 'By hypothesis, C_i is regularly homotopic to a curve which is disjoint from λ' is not immediate from the quoted hypothesis that dHF(M) is supported in a neighborhood of λ through the basepoint after pulling tight. The authors should explain why this neighborhood condition implies the existence of such a regular homotopy for components not homotopic to a power of λ; this is necessary for the admissibility and vanishing conclusion H_*(S⊠P) = 0.
minor comments (3)
  1. [Section 4, Case 2] The sentence 'Projecting that regular homotopy to the punctured torus gives (and stopping just before the end) gives the desired regular homotopy' contains a duplicated 'gives'.
  2. [Section 5, Möbius band case] The phrase 'an arc α in F which is the generator of H1(F,∂F) and geometrically dual to α' should read 'geometrically dual to η', since the arc is meant to be dual to the core curve η.
  3. [Lemma 2.2] The final sentence of the proof, 'the dimension of S⊠P is at least this large,' is unclear; it should specify which homology group's dimension is being compared.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof reduces HFST to a quoted external characterization (HRW24) and an independent AL19 vanishing criterion; the topological condition is a genuine new equivalent.

full rationale

Theorem 1.1 is established by two independent implications. The forward direction assumes M is an HFST, applies the quoted equivalence [HRW24, Prop. 7.11] to obtain the geometric condition that the immersed curve dHF(M) is supported in a neighborhood of the rational longitude λ, then Proposition 4.1 proves from that geometric condition alone that the twisted Floer homology dHF(M(λ); F2[[t,t^{-1}]]) vanishes; the final step to a non-separating 2-sphere is the independent theorem [AL19, Thm. 1.1]. The converse direction assumes M(λ) contains a non-separating 2-sphere, uses [AL19, Thm. 7.11] to get vanishing of the twisted Floer group, runs the surgery exact sequence to show dim dHF(M(µ+kλ)) is independent of k, and then applies the external equivalence [HRW24, Prop. 7.11(2)] to conclude M is an HFST. Neither direction assumes the theorem it proves: the HRW24 proposition is an external characterization of HFSTs in terms of immersed curves and filling dimensions, and the AL19 results are parameter-free theorems about vanishing of twisted Heegaard Floer homology in the presence of S^2×S^1 summands. The fact that AL19 shares two authors with the present paper does not make it circular, since it is a distinct, previously established result whose assumptions do not include Theorem 1.1 (hard rule 4). There are no fitted parameters, no ansatz smuggled through citations, and no uniqueness claim imported from prior work. The proof of Proposition 4.1 is terse about how the 'through the basepoint z' clause is used, but that is a matter of rigor and exposition, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is built on established theorems from the bordered Floer literature; the paper introduces no new entities or fitted parameters. The heaviest external inputs are the immersed curve correspondence of Hanselman-Rasmussen-Watson and the vanishing theorem of Alishahi-Lipshitz.

assumptions (5)
  • standard math Surgery exact sequence with twisted (Laurent series) coefficients for Heegaard Floer homology.
    Used in Proposition 3.1 to compare dim dHF(M(mu+k*lambda)) across k; cited as [OSz04a, Theorem 9.21] and [AP10, Theorem 3.1].
  • standard math Pairing theorem for bordered Floer homology with twisted coefficients.
    Equation (1) in Section 4, [LOT18, Theorem 9.44], identifies CF(M(lambda); F2[[t,t^{-1}]]) with S tensor CFD(M).
  • standard math Hanselman-Rasmussen-Watson immersed curve formulation, including Proposition 7.11 and the structural facts about train tracks and immersed curves.
    Proposition 1.2 supplies the working definition of HFST used throughout; Section 4 relies on [HRW24] for admissibility, unobstructedness, and the correspondence between type D structures and immersed curves.
  • standard math Vanishing criterion for twisted Floer homology and non-separating 2-spheres.
    [AL19, Theorem 1.1 and Theorem 7.11] is used in Proposition 4.1 to pass from vanishing of dHF(M(lambda); F2[[t,t^{-1}]]) to existence of a non-separating 2-sphere.
  • standard math Seifert fibered space facts: classification of reducible Seifert fibered spaces, Seifert structures on S^2 times S^1, and behavior of fiber fillings.
    Used in Theorem 1.3; cites [Hat, Proposition 1.12 and Theorem 2.3] and [Hei74, Proposition 2(b)].

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Pith. "Pith review of Detecting Heegaard Floer homology solid tori." pith.science (2026). https://pith.science/paper/OYVBMG57

@misc{pith2026250501217,
  author       = {Pith},
  title        = {Pith review of: Detecting Heegaard Floer homology solid tori},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYVBMG57}},
  note         = {Machine review of arXiv:2505.01217}
}
read the original abstract

We show that a rational homology solid torus is a Heegaard Floer homology solid torus if and only if it has a Dehn filling with a non-separating 2-sphere. Using this, we characterize Seifert fibered Heegaard Floer solid tori.

Figures

Figures reproduced from arXiv: 2505.01217 by the authors.

Figure 1
Figure 1. An example of the first case in the proof of Proposi￾tion 4.1. The longitude is dashed and the immersed curve is solid; this pair is admissible. v1 v2 v3 ρ12 ρ12 ρ12 v1 ⊗ p v1 ⊗ q v2 ⊗ p v2 ⊗ q v3 ⊗ p v3 ⊗ q t t t [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. An example of the second case in the proof of Proposi￾tion 4.1. Left: the immersed curve invariant, with conventions as in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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