{"id":"f94e7d14-865f-4210-8fff-87d1736459b2","arxiv_id":"2412.20941","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In McDuff and torus bundle Liouville domains, every weakly exact Lagrangian torus is homotopic to a standard fibre, and any exact Lagrangian meeting two different boundary components has non-zero wrapped Floer cohomology.","lead":"This paper proves new non-vanishing and vanishing criteria for wrapped Floer cohomology and symplectic cohomology in Liouville domains, and classifies the weakly exact Lagrangian tori in the two standard families of non-Weinstein Liouville domains. A generalist should read it because it turns a simple count of boundary components into a topological test for non-trivial symplectic invariants.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.9's final inference is under-proved: free plus rank-one H_1 image does not imply cyclic; for T^2/Klein-bottle the missing argument is short, so the claim likely survives but needs repair.","rationale":"The reader's weakest-assumption analysis identifies the same load-bearing spot as I do: Lemma 3.9 is the step that produces the cyclic image subgroup needed to construct the infinite cyclic cover \tilde V ≅ T^*S^1 × C^* in the McDuff case. The printed proof establishes that G is free and that its H_1-image has rank at most one, but it does not explicitly exclude a rank-two free subgroup whose generators become homologous in H_1(Σ_g). That is a genuine logical gap, and it is load-bearing because Theorem A Part (2) depends on the cyclicity of G. However, the gap is repairable by a short algebraic argument: for a torus the image of π_1(L) is abelian, and for a Klein bottle the image of the central element A^2 gives a nontrivial central element in the free group G unless the image of A is trivial; in either case G is cyclic. I did not find a deeper obstruction in the covering argument or in the use of Theorem 3.3; those steps are consistent with the stated setup. The correct repair differs slightly from the reader's formulation, since B^2 is not central in the standard Klein-bottle presentation, but the intended argument works with A^2. Because the flaw is an omitted justification rather than a wrong conclusion, the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":33476,"tokens_out":20484,"duration_ms":222393,"concrete_test":"Re-derive the final paragraph of Lemma 3.9 without using the sentence 'hence it is either trivial or Z'. Instead prove directly: (i) for L = T^2, G = im(π_1(L)) is abelian, so G ≅ Z or is trivial; (ii) for L equal to a Klein bottle with presentation ⟨A, B | A B A^{-1} = B^{-1}⟩, the image of A^2 lies in the center of G, and nontriviality of that image forces G ≅ Z, while triviality forces A = 1 and hence G = ⟨B⟩. Then check that the standard counterexample of a rank-two free subgroup of [π_1(Σ_g), π_1(Σ_g)] is excluded by (i)/(ii). If cyclicity of G fails, the lift to the infinite cyclic cover used in the proof of Part (2) cannot be constructed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem A Part (2) rests on Lemma 3.9, whose final inference is not justified as written. The proof uses Jaco's theorem to conclude that the image G ⊂ π_1(Σ_g), g ≥ 2, is free, and after showing its image in H_1(Σ_g) has rank at most one, asserts that G is trivial or Z. This does not follow from those two facts alone: a free group F_2 can embed in a surface group with its generators inside the commutator subgroup, giving H_1-image rank zero while G is free of rank two. What is missing is the extra structure of π_1(L). For L = T^2, G is a quotient of Z^2 and hence abelian, and a free abelian subgroup of a surface group is cyclic. For L = K = ⟨A, B | A B A^{-1} = B^{-1}⟩, the element A^2 is central in π_1(K) (not B^2), so its image is central in G. If that image is nontrivial, the free group G has nontrivial center and is therefore cyclic; if it is trivial, torsion-freeness of π_1(Σ_g) gives A = 1 and G = ⟨B⟩. Thus the lemma is true, but the written proof omits an essential argument. Since the cyclic cover \tilde V ≅ T^*S^1 × C^* in §3.3 is produced from this cyclic subgroup, Theorem A Part (2) is conditional on this repair.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Liouville domains that are trivial bi-fillings, with emphasis on the McDuff and torus-bundle domains whose skeletons are smooth codimension-one hypersurfaces. It introduces the notion of a Liouville-Hamiltonian structure and uses it to describe the symplectic structure near such skeleta. The main results are: (i) Theorem A, which restricts the homotopy type of closed weakly exact Lagrangians in McDuff and torus-bundle domains to tori or connected sums of 2k copies of RP^2, and asserts that weakly exact tori become standard fibres or circle-bundle tori after passing to suitable finite covers; (ii) Theorems B and C, which give action-based criteria for non-vanishing of wrapped Floer cohomology of exact Lagrangians and of symplectic cohomology in terms of the number of boundary components, together with a decomposition of the kernel of the canonical Morse-to-Floer map; and (iii) Theorem D, which obstructs periodic orbits of the Liouville flow on a smooth skeleton from being contained in a smooth ball, via a vanishing criterion for wrapped Floer cohomology.","tokens_in":1619,"tokens_out":1570,"duration_ms":101010,"significance":"The results are significant if they hold. Theorem A is, to my knowledge, the first topological restriction on weakly exact Lagrangians in the main non-Weinstein Liouville bi-fillings, predicting that weakly exact tori are homologically standard after finite covers. Theorems B and C are clean and potentially widely applicable: they give purely topological conditions for non-vanishing of wrapped Floer cohomology and symplectic cohomology, and the proof via action filtrations and neck-stretching is largely self-contained. The paper is transparent about its reliance on external results, including the author's published theorem [DR] and preprints [CLMM], [HS], and [HCK]; no fitted parameters or circular dependencies are apparent. The main caveat is the proof gap in Lemma 3.9, which underpins Theorem A(2); the missing argument appears short and repairable, so the central claims are likely sound.","major_comments":[{"comment":"The final inference of Lemma 3.9 is not justified. The proof shows that the image G of π_1(L) in π_1(Σ_g) has rank-one image in H_1(Σ_g) and, by Jaco's theorem, that G is free; it then concludes that G is trivial or Z. This does not follow: a free group of rank two can embed in a surface group with its generators in the commutator subgroup, giving H_1-image of rank zero. What is missing is an appeal to the specific algebraic structure of π_1(L). For L = T^2, the image G is abelian, and a hyperbolic surface group contains no Z^2, so G is cyclic. For L a Klein bottle, the image is virtually abelian and torsion-free, hence cyclic; equivalently, the central element A^2 (with the presentation ⟨A,B | ABA^{-1}=B^{-1}\rangle) maps to a central element of G, forcing G cyclic if nontrivial, and if trivial, torsion-freeness gives A=1 and G=⟨B⟩. Because the cyclic cover used in the proof of Theorem A(2) is produced from this subgroup, the classification of weakly exact tori in McDuff domains is conditional on this repair. I recommend adding the omitted argument; the claim itself appears correct.","section":"Section 3.3, Lemma 3.9"},{"comment":"The proof of Lemma 4.1 ends with 'The effect of this wrapping on the action, i.e. the primitive of the pull-back of λ to the Lagrangian, is a computation that we leave to the reader.' This computation is load-bearing: the lemma is used to justify that the Lagrangian can be assumed to have a primitive that is arbitrarily C^0-small in the interior and constant on each boundary component, which underpins the action estimates for the wrapped Floer generators in Subsection 4.1.3 and hence the kernel-containment statements in Theorems B and C. The missing computation should be supplied explicitly, or else a reference to a published argument should be provided.","section":"Section 4.1.3, Lemma 4.1"}],"minor_comments":[{"comment":"The sentence 'In fact, in the case L = T^2 the rank is also one, since otherwise we could conclude that f* : H*(Σ_g) → H*(L) is surjective ... which contradicts the fact that f* is a morphism of unital rings' is not explained and is not convincing as written. Surjectivity of f* on cohomology does not by itself contradict being a unital ring morphism. The later abelian-subgroup argument (or a degree/transfer argument) would be cleaner and more rigorous.","section":"Section 3.3, Lemma 3.9"},{"comment":"There is a typo: 'One we have managed to construct' should read 'Once we have managed to construct.'","section":"Section 3, proof of Theorem 3.2"},{"comment":"The notation for the McDuff domain alternates between (U*Σ, α_g − Θ_g) and (D*Σ_g \\ O_0Σ, λ_can + η). A short glossary relating the two descriptions would help the reader, since the proof of Theorem A switches between them.","section":"Section 2.5.1 and Section 3.3"},{"comment":"The statement that for n ≥ 3 the complement of Darboux balls is contactomorphic to the complement of a finite set is used without a reference or proof. Since this is a standard consequence of convexity of Darboux balls, a reference to e.g. Giroux or Eliashberg would suffice.","section":"Section 5, Corollary 5.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on several recent preprints, including [CLMM], [HS], [HCK], and on the author's own prior results. The editor may wish to verify that the cited statements in these preprints are independently established or have by now appeared in refereed venues, especially [CLMM, Lemma 4.7 and 4.8] and [HS, Theorem 1.5]. This is a routine verification concern, not an indictment of the paper's novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe paper is worth taking seriously. It delivers three things: (i) Theorems B and C, which give clean topological conditions for non-vanishing of wrapped Floer cohomology and symplectic cohomology when the Lagrangian or boundary meets multiple components; (ii) a new organising notion, Liouville-Hamiltonian structure, with a normal form and a neat lemma showing closed examples cannot be stable Hamiltonian; and (iii) Theorem A, the homotopy classification of weakly exact Lagrangian tori in McDuff and torus-bundle domains. The non-vanishing theorems are proven with an action-filtration argument that is mostly self-contained, and the neck-stretching lemma in the appendix is a useful tool. The dependence on [CLMM], [HCK], [HS], and [Hus] is real but mostly as black boxes with public or published results; that is not a flaw by itself, just a limitation on how much a referee can verify in isolation.\n\nThe soft spot the reader flagged is genuine. Lemma 3.9 concludes the image G of pi_1(L) in pi_1(Sigma_g) is trivial or Z from freeness plus rank-one H_1 image. Those two facts alone do not imply cyclicity; a free F_2 can sit inside the commutator subgroup. The stress-test repair is right: for L = T^2 the image is abelian, hence cyclic in a surface group; for a Klein bottle, A^2 is central, and a nontrivial central element forces a free subgroup to be cyclic. That is a short missing argument, not a broken theorem, but the proof as written should not stand. Lemma 4.1 defers an action computation to the reader; that is minor, but the computation should appear.\n\nTheorem D's earlier mistake was caught and is acknowledged; the current assumptions look coherent. I would not hold that against the paper.\n\nWho is this for? Symplectic topologists working on non-Weinstein Liouville domains and wrapped Fukaya categories. They will want the non-vanishing criteria and the classification step; the Liouville-Hamiltonian material may also be useful to people connecting Anosov flows and symplectic fillings.\n\nRecommendation: send it to peer review, with instructions that the referee specifically check Lemma 3.9 and Lemma 4.1. If the missing group-theoretic argument and the computation are supplied, I would expect acceptance without substantial further changes.\n\nBest,\n[You]","headline":"New non-vanishing criteria and a Lagrangian classification that mostly hold up; Lemma 3.9 has a short but real gap that should be fixed before publication.","tokens_in":34301,"tokens_out":2060,"would_cite":true,"duration_ms":21802,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D40","53D12","53D35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that closed weakly exact Lagrangian surfaces in McDuff and torus-bundle Liouville domains are only tori or connected sums of real projective planes, and that every weakly exact torus becomes a standard torus after passing…","keywords":["weakly exact Lagrangian","Liouville bi-filling","McDuff domain","torus bundle domain","wrapped Floer cohomology","symplectic cohomology","Liouville-Hamiltonian structure","closed Lagrangian classification"],"falsifier":"Exhibit a weakly exact Lagrangian torus in a McDuff domain whose induced fundamental-group map to the base surface is a rank-two free group while the induced first-homology map has rank one; that would contradict Lemma 3.9 and break Theorem A, and checking whether such a map can be realized is a concrete search.","tokens_in":33266,"feed_emoji":"","tokens_out":10083,"duration_ms":93664,"temperature":0.7,"pith_summary":"The paper tries to show that weakly exact Lagrangian surfaces in the two main families of non-Weinstein Liouville domains—McDuff domains and torus-bundle domains—are topologically standard. If correct, the only closed weakly exact Lagrangians are tori and connected sums of copies of the real projective plane, and every weakly exact torus becomes a standard fibre or circle-bundle torus after a finite cover. This matters because non-Weinstein Liouville domains have resisted the generation results that make wrapped Fukaya categories computable for Weinstein domains. The paper also gives general non-vanishing criteria for wrapped Floer cohomology and symplectic cohomology, and uses them to rule out periodic orbits of certain Liouville flows inside smooth balls.","feed_headline":"Weakly exact tori in McDuff and torus-bundle domains are standard","feed_subtitle":"After a finite cover each such torus is isotopic to a standard fibre or circle-bundle torus.","key_machinery":"The paper's main structural tool is the Liouville-Hamiltonian structure, a triple $(M,\\eta,\\beta)$ on an odd-dimensional hypersurface for which $d\\eta$ is maximally non-degenerate, $\\ker\\eta \\supset \\ker d\\eta$, and $\\ker\\beta \\cap \\ker d\\eta = \\{0\\}$; it encodes the symplectic form near a hypersurface tangent to the Liouville flow via $d(\\eta + s\\beta)$ and recovers the dynamics through a Liouville vector field $\\zeta$ and a characteristic vector field $C$ with $\\beta(C)=1$. The key dynamical threshold is $d\\beta(C,\\zeta)>-1$, the linear contact-deformation condition, which makes the Liouville flow repelling along the skeleton. For the classification, the load-bearing mechanism is to lift a weakly exact Lagrangian to infinite covers of the domain, where a McDuff or torus-bundle domain becomes a complement of a symplectic section in $T^*\\tilde\\Sigma$ with $\\tilde\\Sigma = \\mathbb{R}^2$ or $\\mathbb{R}\\times S^1$, and then to apply the classification of weakly exact Lagrangians in $T^*T^2$ together with the non-existence of such Lagrangians in cotangent bundles of open surfaces. The passage from infinite covers back to finite covers is completed by a finite-cover separation result for surface-group covers.","core_discovery":"The central claim is Theorem A: in a McDuff domain, a nontrivial $S^1$-bundle over $\\Sigma_g \\times I$ of Euler number $2g-2$ with $g\\ge 2$, or in a torus-bundle domain, a Lagrangian $T^2$-fibration over $S^1 \\times I$, every closed weakly exact Lagrangian $L^2$ is either a torus or a connected sum of $2k$ copies of $\\mathbb{RP}^2$ with $k\\ge 2$. A weakly exact torus is incompressible, and after a suitable finite cover it is Hamiltonian isotopic to a standard fibre in the torus-bundle case, or isotopic through weakly exact Lagrangians to a circle-bundle torus in the McDuff case; when $L$ is exact, the McDuff isotopy can be taken Hamiltonian to an exact circle-bundle torus. The paper further establishes that symplectic cohomology is non-vanishing whenever the boundary has at least two components, that a connected exact Lagrangian with boundary in two different boundary components has non-vanishing wrapped Floer cohomology, and that a four-dimensional Liouville domain with a smooth, codimension-one, repelling skeleton has no periodic Liouville orbit contained in a smooth ball.","pith_inferences":["If the missing step in Lemma 3.9 is supplied, the same covering strategy would adapt to weakly exact Lagrangians in other trivial bi-fillings whose skeletons come from hyperbolic dynamics, since the only inputs are cotangent-bundle classification and surface-group restrictions.","The non-vanishing criteria suggest a generation principle: in a Liouville domain with several boundary components, a Lagrangian that connects different boundary components is Floer-theoretically visible, so wrapped Fukaya categories of non-Weinstein domains may be generated by such connecting Lagrangians rather than by cocores.","The condition $d\\beta(C,\\zeta)>-1$ converts a symplectic-topological vanishing statement into a dynamical one; one could test whether small perturbations of hyperbolic bi-fillings preserve the absence of contractible periodic orbits, which the paper does not address."],"forward_implications":["Every weakly exact Lagrangian torus in a McDuff or torus-bundle domain is incompressible, so it cannot be null-homotopic or represent a class that dies in the ambient fundamental group.","The only closed weakly exact Lagrangian surfaces that can occur in these domains are tori and connected sums of $2k$ copies of $\\mathbb{RP}^2$ with $k\\ge 2$; Klein bottles are ruled out.","In the torus-bundle case, a sufficiently large finite cover contains a Hamiltonian isotopy from the lifted torus to a standard Lagrangian fibre, so the torus becomes standard after passing to that cover.","In the McDuff case, an exact Lagrangian torus becomes Hamiltonian isotopic to an exact circle-bundle torus in a finite cover, while a merely weakly exact torus is isotopic to a circle-bundle torus through weakly exact Lagrangians.","If a Liouville domain has at least two boundary components, its symplectic cohomology is non-zero, and any connected exact Lagrangian with boundary in two different boundary components has non-zero wrapped Floer cohomology; consequently, in a four-dimensional domain with a smooth repelling skeleton, no periodic Liouville orbit lies in a smooth ball."],"supporting_citations":[{"why":"supplies the lift criterion for Lagrangian tori and Klein bottles in torus-bundle domains and the description of circle-bundle tori in McDuff domains","marker":"[CLMM]"},{"why":"excludes closed weakly exact Lagrangians in cotangent bundles of open surfaces, forcing the relevant lift to the $T^*T^2$ cover","marker":"[LS]"},{"why":"classifies weakly exact Lagrangians in $T^*T^2$ up to Hamiltonian isotopy to standard fibres, providing the standard-form isotopy","marker":"[DR]"},{"why":"gives the freeness of infinite-index subgroups of surface groups used in Lemma 3.9","marker":"[Jac]"},{"why":"provides the finite-cover separation result that turns an infinite cover into a finite $k$-fold cover while preserving embedded isotopies","marker":"[Sco]"},{"why":"excludes Lagrangian Klein bottle embeddings in the relevant cotangent and uniruled manifolds","marker":"[She]"},{"why":"supplies the Pontryagin-square and self-intersection formula restricting the possible Euler characteristics of closed Lagrangians","marker":"[Aud]"}],"fun_headline_variants":["Weakly exact tori turn standard in McDuff and torus-bundle domains","In McDuff and torus-bundle domains, weakly exact tori are standard","McDuff and torus-bundle domains: weakly exact tori standard","Weakly exact tori classified: standard in McDuff and torus bundles","Weakly exact Lagrangians in McDuff and torus-bundle domains are tori or RP^2 sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification of weakly exact tori in McDuff domains rests on the lemma that a map from a torus or Klein bottle to a closed surface of genus at least two has fundamental-group image either trivial or infinite cyclic; as written, the proof shows the image is free with rank-one homology but does not rule out a rank-two free subgroup.","fun_headline_variants_meta":{"raw":{"variants":["Weakly exact tori turn standard in McDuff and torus-bundle domains","In McDuff and torus-bundle domains, weakly exact tori are standard","McDuff and torus-bundle domains: weakly exact tori standard","Weakly exact tori classified: standard in McDuff and torus bundles","Weakly exact Lagrangians in McDuff and torus-bundle domains are tori or RP^2 sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001382,"raw_usage":{"total_tokens":5612,"prompt_tokens":978,"completion_tokens":4634,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":4523}},"tokens_in":594,"tokens_out":4634,"duration_ms":32178,"temperature":1.0,"reasoning_tokens":4523,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:08:37.670549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a weakly exact Lagrangian torus in a McDuff domain whose induced fundamental-group map to the base surface is a rank-two free group while the induced first-homology map has rank one; that would contradict Lemma 3.9 and break Theorem A, and checking whether such a map can be realized is a concrete search.","supporting_citations":[],"review_version":1}