{"id":"b1938e1c-b3ca-4fc8-b9d0-edf9cb09831d","arxiv_id":"2411.12144","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"This overview explains the Legendrian surgery isomorphism between wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras, with consequences for symplectic homology, Hochschild invariants, and partially wrapped Floer theories.","lead":"An overview of Eliashberg's Legendrian surgery framework, explaining how wrapped Floer cohomology of co-core disks is quasi-isomorphic to the Chekanov-Eliashberg dg-algebra of attaching spheres. The paper surveys consequences for symplectic homology, Hochschild invariants, and partially wrapped Floer theories.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.7's proof omits the passage from action-truncated isomorphisms to the untruncated quasi-isomorphism: Lemma 3.1 chooses handle size after the action bound, so the stated full isomorphism needs a direct-limit or invariance step not written out.","rationale":"The reader identified Lemma 3.1 as the weakest assumption, and I agree the action-controlled Reeb chord/word correspondence is the base of the whole surgery isomorphism. However, the more precise soft spot is not the geometric content of Lemma 3.1 itself but the quantifier structure: the handle size is chosen after the action bound, so Lemma 3.6 only establishes isomorphisms on finite action truncations for sufficiently small handles. The paper then asserts the full untruncated quasi-isomorphism in Theorem 3.7 without spelling out how these truncations assemble. This is a proof-completeness gap rather than a known mathematical error; the theorem is a known result and the cited works almost certainly contain the missing details. I also checked the flagged circularity in Theorem 4.8: the sentence 'Since CO is an isomorphism' is a typo for 'Since OC is an isomorphism', since Corollary 4.5 gives OC as a quasi-isomorphism, so that issue is not a genuine circularity. Overall the reader's CONDITIONAL verdict is appropriate, and my concern does not change it.","tokens_in":11443,"tokens_out":13909,"duration_ms":161022,"concrete_test":"Independently verify the action-filtration proof of Theorem 3.7 in [7, Appendix B.2] and [10, Section 7], isolating the step where the a-truncated isomorphism for handles smaller than δ(a) is converted into an isomorphism of the untruncated complexes. If that step is absent, the proof needs an explicit direct-limit argument over handle size; a concrete one-handle check with Λ = Legendrian unknot in S^{2n-1} would reveal whether the truncation maps are compatible with inclusions as a increases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 3.7: Φ^CW is a quasi-isomorphism. Lemma 3.1 provides, for each action bound a, a handle size δ(a) such that Reeb chords of Γ of action < a correspond to composable words of Λ with total action < a. Lemma 3.6 then shows Φ^CW_1 is upper-triangular with ±1 diagonal on the action filtration. This proves an isomorphism on the a-truncated complexes only for handles of size < δ(a). For a fixed handle, Lemma 3.1 says nothing about chords of action ≥ a; hence the unfiltered Theorem 3.7 requires either a uniform δ for all a or a direct limit (shrink the handle, use invariance of CW^*(C) and CE^*(Λ), and pass to the limit). Neither argument appears in the paper; the phrase 'straightforward action filtration argument' in the proof of Theorem 3.7 glosses over exactly this point. This is a completeness gap in the proof as written, not evidence of falsehood; the isomorphism is a known result and [7,10] likely contain the missing details. The apparent circular 'Since CO is an isomorphism' in Theorem 4.8 is a typo for OC (Corollary 4.5), so it is not an independent correctness issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This overview paper describes Eliashberg's Legendrian surgery approach to wrapped Floer cohomology. The central result is Theorem 3.7, which states that for a Weinstein manifold obtained by attaching critical handles along a Legendrian link Λ, the wrapped Floer cohomology of the co-core disks C is quasi-isomorphic to the Chekanov–Eliashberg dg-algebra of Λ. The proof is presented through a Reeb chord correspondence (Lemma 3.1), an upper-triangular action-filtration statement (Lemma 3.6), and the construction of the chain map Φ^CW in Section 3.5. The remainder of the paper surveys applications: symplectic homology and a two-copy complex (Section 4.1), upside-down surgery and Hochschild complexes (Section 4.2), open-closed and closed-open maps (Section 4.3), the symplectic homology product (Section 4.4), and partially wrapped Floer cohomology with loop-space coefficients (Section 4.5). The paper is explicitly an overview and delegates many details to the prior works [7], [10], and others.","tokens_in":11687,"tokens_out":6324,"duration_ms":69247,"significance":"If the results hold as stated, the paper provides a useful conceptual roadmap to a central family of isomorphisms in symplectic topology: wrapped Floer cohomology of co-cores, symplectic homology, Hochschild homology and cohomology, and the closed-open map are all derived from one surgery picture. The main theorem is already a published theorem, and the sketched proof is consistent with the cited references, which is an important strength. The paper also clearly identifies the generator-level dictionary between Reeb chords of the co-core boundary and composable words of Reeb chords of the attaching link, and it spells out the geometric origin of the isomorphism. Its value is as a survey and a guide to the literature rather than as a new proof; the main risk is that the sketched proofs omit the limiting steps needed to pass from action-truncated comparisons to the unfiltered quasi-isomorphisms.","major_comments":[{"comment":"Lemma 3.1 supplies, for each action bound a, a handle size δ(a) such that Reeb chords of Γ of action < a correspond to composable words of Λ of total action < a. Lemma 3.6 then compares the a-truncated complexes only for handles of size < δ(a). The proof of Theorem 3.7 says 'Using a straightforward action filtration argument' but does not explain how one passes from the truncated statement to the unfiltered quasi-isomorphism: for a fixed handle size, Lemma 3.1 gives no control on chords of action ≥ a, and Lemma 3.6 does not compare their contributions. The missing step is either a uniform handle size valid for all action levels or an explicit direct-limit/invariance argument (shrink the handle and use invariance of CW^*(C) and CE^*(Λ)). Please add this step or point precisely to the place in [7, Appendix B.2] where it is carried out.","section":"§3.6, proof of Theorem 3.7"},{"comment":"The injectivity part of the proof contains the sentence 'Since CO is an isomorphism we find u...', which is circular because CO is the map whose quasi-isomorphism property is being proved; it should presumably read 'Since OC is an isomorphism' via Corollary 4.5. This typo should be corrected. More substantially, the rest of the proof is too compressed: the surjectivity argument invokes moduli spaces M(u;e), a 'dimension two' statement after gluing, and a rotation cobordism M(T) whose rigid holomorphic curves are asserted rather than derived. As written, this does not verify Theorem 4.8 without consulting the cited references [1,11]. Please expand the argument or explicitly state which parts are quoted from those sources.","section":"§4.3, proof of Theorem 4.8"},{"comment":"The statement introduces a quasi-isomorphism Φ^CW: B CW^*(C) → LCE^*(Λ) of infinity co-algebras, but the infinity co-algebra structures on the bar complex and on the linearized Chekanov–Eliashberg dg-algebra are not defined in the paper, and the augmentation needed for linearization is not specified. The proof is given as 'In direct analogy with Theorem 3.7' with no further details. Since Theorem 3.7 itself relies on the missing action-filtration limit step, Theorem 4.4 inherits that gap. Please either define the relevant structures and give a precise proof sketch with the role of Lemma 4.3, or state clearly that this result is a direct quotation of [7].","section":"§4.2, Theorem 4.4"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors that should be corrected in a revision, including 'Chekaonv-Eliashberg' (Section 2), 'W rapped' (Section 3.3 heading), 'holomorhic' (Sections 3.2 and 3.3), 'samller' and 'of of' (Lemma 3.1), 'More precisley' (Section 3.5), 'isomorhism' (Section 3.6), 'Elishberg' (Section 4.5), 'eqipped' (Lemma 4.3), and 'Gantara' (reference [11]).","section":"Throughout"},{"comment":"The entry for [7] appears to contain a duplicated title and should be cleaned up so that the journal, volume, and pages are clearly formatted.","section":"Reference [7]"},{"comment":"The phrase 'one extra puncture opposite the distinguished mixed puncture mapping to z' is unclear; it would help to specify the domain of that puncture and the role of the conformal constraint more explicitly.","section":"§4.4"},{"comment":"The isomorphism (4.1) is stated as 'straightforward to derive' but no derivation or precise reference is given; since this is a sample application, a short indication of how Theorem 3.7 and loop-space coefficients interact would make the illustration more useful.","section":"§4.5"}],"recommendation":"major_revision","confidential_remarks":"This is an overview paper by the main developer of the Legendrian surgery approach, and it overlaps substantially with his prior papers, especially [7] and [10]. I do not regard that overlap as circularity because the cited works contain detailed arguments rather than restatements of the claims. The appropriate bar for this paper is that of a survey article: it should give an accurate and sufficiently complete roadmap to the proofs. Under that bar, the main obstacles are the missing action-filtration limit step in Theorem 3.7, the circular sentence in the proof of Theorem 4.8, and the undefined infinity co-algebra structures in Theorem 4.4. These are fixable within the scope of the manuscript, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is an honest overview of Ekholm's own surgery program. The main theorem is quoted from published work, and the paper says so. What it does well is organize the material and spell out how the surgery isomorphism between wrapped Floer cohomology of co-core disks and the Chekanov-Eliashberg dg-algebra of attaching spheres drives the later relations: symplectic homology, Hochschild invariants, closed-open maps, and loop-space coefficients. The sketched proofs in Section 3 are consistent with the detailed arguments in [7], [10], [4], [5]. The Section 4.5 example is a useful illustration, even though it is asserted rather than derived.\n\nThe real soft spot is exactly what the stress-test flags: the proof of Theorem 3.7 jumps from an action-truncated statement (Lemma 3.1 picks handle size after the action bound) to the full quasi-isomorphism without explicitly passing to the limit or using invariance under handle shrinking. 'Straightforward action filtration argument' glides over that step. Since the theorem is published with detailed proofs elsewhere, this is an expositional gap, not a load-bearing flaw. A referee should ask for the missing sentence.\n\nTheorem 4.8 has a typo: 'Since CO is an isomorphism' should read 'Since OC is an isomorphism' (Corollary 4.5). The proof is sketchy but the intended logic is recoverable. The isomorphism (4.1) is also presented as straightforward with no proof; acceptable in an overview, but it should be marked as a claim to be checked rather than something the reader can verify on the spot.\n\nThe heavy self-citation is not a real vice: the cited papers are detailed, peer-reviewed, and the overview explicitly frames itself as such. The paper does not overclaim novelty.\n\nWho gets value: graduate students and researchers who want the architectural view before diving into the original sources. I would bring it to a reading group as a map. It deserves a serious referee because a good referee can fix the typos, fill the direct-limit remark, and clarify the status of the Section 4.5 example. I recommend engaging with it as a survey, not as a research advance.","headline":"Honest overview of the Legendrian surgery program: no new theorems, but a useful architectural map; the main soft spot is a real presentation gap in the action-filtration argument, not a correctness issue.","tokens_in":12253,"tokens_out":3186,"would_cite":true,"duration_ms":33664,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D40","53D42","53D35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A surgery map computes wrapped Floer cohomology from attaching spheres.","keywords":["Legendrian surgery","wrapped Floer cohomology","Chekanov-Eliashberg dg-algebra","Weinstein manifolds","symplectic homology","Hochschild homology","Reeb chords","open-closed map"],"falsifier":"Fix a Weinstein domain and a critical handle attachment; list the Reeb chords of the co-core boundary sphere with action below some level $a$ and the composable words of Reeb chords of the attaching link with total action below $a$. Any mismatch in this list, for any $a$ and any handle size, would falsify Lemma 3.1 and therefore the quasi-isomorphism of Theorem 3.7.","tokens_in":11212,"feed_emoji":"🔗","tokens_out":11216,"duration_ms":101914,"temperature":0.7,"pith_summary":"This paper lays out the Legendrian surgery approach to wrapped Floer cohomology and the connections it creates between holomorphic curve theories on Weinstein manifolds. Its central claim is that the wrapped Floer cohomology of the co-core disks of a Weinstein manifold is computed, up to quasi-isomorphism, by the Chekanov-Eliashberg dg-algebra of the Legendrian spheres along which the critical handles are attached. The argument runs through a chain map that counts anchored holomorphic disks with punctures at Reeb chords, and upgrades that map to a quasi-isomorphism by an action filtration. From this base isomorphism the paper derives relations between symplectic homology, Hochschild homology of wrapped Floer cohomology, open-closed and closed-open maps, and partially wrapped Floer cohomology with loop-space coefficients. If the main isomorphism holds, symplectic invariants of a Weinstein manifold are readable from the Legendrian geometry of its attaching data.","feed_headline":"A surgery map computes wrapped Floer cohomology from attaching spheres","feed_subtitle":"If true, all wrapped invariants of a Weinstein manifold are encoded in its Legendrian link's Reeb chords.","key_machinery":"The load-bearing mechanism is the surgery cobordism between the Weinstein domain before and after a critical handle attachment, together with an action-controlled dictionary of Reeb chords. Lemma 3.1 states that, for sufficiently small handles, Reeb chords of the co-core boundary sphere below any fixed action correspond one-to-one to composable words of Reeb chords of the attaching link with total action below that level. On top of this dictionary, the chain map $\\Phi^{\\mathrm{CW}}$ is defined by counting anchored holomorphic disks with positive punctures at Reeb chords of the co-core, negative punctures at Reeb chords of the attaching link, and two punctures asymptotic to the intersections between the core and co-core Lagrangians; the $A_\\infty$-structure on the wrapped side is handled by systems of parallel copies that prevent boundary breaking. An action filtration then turns the triangular leading term of the map into a quasi-isomorphism, and the same pattern is adapted to symplectic homology, Hochschild complexes, upside-down surgery, and partial wrapping.","core_discovery":"The paper's central discovery is Theorem 3.7: the natural $A_\\infty$-chain map $\\Phi^{\\mathrm{CW}}\\colon \\mathrm{CW}^*(C)\\to \\mathrm{CE}^*(\\Lambda)$, from the wrapped Floer cohomology of the co-core disks of a Weinstein manifold to the Chekanov-Eliashberg dg-algebra of its Legendrian attaching spheres, is a quasi-isomorphism. The map is built from holomorphic disks in the surgery cobordism with positive punctures at Reeb chords of the co-core boundary, negative punctures at Reeb chords of the attaching link, and additional punctures at the intersection points between core and co-core disks; the $A_\\infty$-relations are the splittings of one-dimensional moduli spaces. The proof uses an action-filtered induction in which the leading term of the map on each Reeb chord word is plus or minus the corresponding word, with lower-action corrections, and the required isomorphism disks are constructed by gluing. The same surgery machinery then identifies symplectic homology with cyclic words of Reeb chords, identifies the open-closed map as a quasi-isomorphism to symplectic homology, and describes partially wrapped Floer cohomology through Legendrian dg-algebras with based-loop-space coefficients.","pith_inferences":["An implication the paper leaves implicit is that the quasi-isomorphism makes wrapped Floer cohomology of co-cores insensitive to handle size and contact-form choices, so it is an invariant of the Legendrian isotopy class of the attaching link.","The finite-action dictionary in Lemma 3.1 suggests a testable low-dimensional check: compute the action spectra of Reeb chords before and after surgery and compare them with word-length spectra, which could reveal where the small-handle approximation starts to fail.","The same surgery pattern applied to partial wrapping points toward a category-level statement that partially wrapped Fukaya categories are fully encoded by Legendrian dg-algebras with local coefficients; the paper states the cohomological version rather than the full categorical one.","Upside-down surgery and cut-and-paste descriptions suggest that Legendrian surgery arguments should yield gluing formulas for symplectic cohomology along arbitrary Weinstein sectors, not only the handle-by-handle presentations shown here."],"forward_implications":["Wrapped Floer cohomology of the co-core disks of a Weinstein manifold is explicitly computable from the Chekanov-Eliashberg dg-algebra of the Legendrian attaching link.","Symplectic homology of the Weinstein manifold is quasi-isomorphic to the Hochschild complex of cyclic words of Reeb chords of the attaching link, with the subcritical part contributing only a contractible factor.","The open-closed map from the Hochschild homology of wrapped Floer cohomology to symplectic homology is a quasi-isomorphism, and the closed-open map gives the reciprocal statement for Hochschild cohomology.","Partially wrapped Floer cohomology around Legendrian stops is described by Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space, yielding geometric cut-and-paste descriptions of Floer theory.","The symplectic homology product is represented on cyclic words of Reeb chords by attaching disks, giving a concrete Legendrian picture of the Calabi-Yau structure."],"supporting_citations":[{"why":"It proves the fixed-point correspondence between Reeb chords of the co-core boundary and words of Reeb chords of the attaching link, and it gives the gluing construction of the isomorphism disks.","marker":"[10, Section 5]"},{"why":"It establishes the $A_\\infty$-structure on wrapped Floer cohomology and the action-filtration argument that promotes the chain map to a quasi-isomorphism.","marker":"[7, Appendix B.2]"},{"why":"It defines the two-copy complex and the surgery map used in Section 4.1 to identify symplectic homology with cyclic words of Reeb chords.","marker":"[4]"},{"why":"It gives the Legendrian surgery description of the symplectic homology product that Theorem 4.9 states on cyclic words of chords.","marker":"[5]"},{"why":"It proves the loop-space coefficient version of the Chekanov-Eliashberg dg-algebra for singular Legendrians used in Section 4.5.","marker":"[3]"},{"why":"It provides the cut-and-paste and sectorial descent results that Section 4.5 recovers through surgery descriptions.","marker":"[12]"}],"fun_headline_variants":["Surgery map equates wrapped Floer cohomology and Reeb chord algebra","Wrapped Floer cohomology computed from Reeb chords via surgery","Surgery reveals wrapped invariants as Legendrian dg-algebra","Quasi-isomorphism ties wrapped cohomology to Reeb chord dga"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result depends on the geometric correspondence that, for a sufficiently small handle, every Reeb chord of the co-core's boundary sphere below any fixed length bound corresponds exactly to one composable word of Reeb chords of the attaching link, with no exceptions and no missing words.","fun_headline_variants_meta":{"raw":{"variants":["Surgery map equates wrapped Floer cohomology and Reeb chord algebra","Wrapped Floer cohomology computed from Reeb chords via surgery","Surgery reveals wrapped invariants as Legendrian dg-algebra","Quasi-isomorphism ties wrapped cohomology to Reeb chord dga"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000598,"raw_usage":{"total_tokens":2748,"prompt_tokens":849,"completion_tokens":1899,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":1816}},"tokens_in":465,"tokens_out":1899,"duration_ms":15795,"temperature":1.0,"reasoning_tokens":1816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:52:02.997927+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a Weinstein domain and a critical handle attachment; list the Reeb chords of the co-core boundary sphere with action below some level $a$ and the composable words of Reeb chords of the attaching link with total action below $a$. Any mismatch in this list, for any $a$ and any handle size, would falsify Lemma 3.1 and therefore the quasi-isomorphism of Theorem 3.7.","supporting_citations":[{"cited_title":"Bourgeois, T","cited_arxiv_id":null,"evidence_quote":"It defines the two-copy complex and the surgery map used in Section 4.1 to identify symplectic homology with cyclic words of Reeb chords."},{"cited_title":"Bourgeois, T","cited_arxiv_id":null,"evidence_quote":"It gives the Legendrian surgery description of the symplectic homology product that Theorem 4.9 states on cyclic words of chords."},{"cited_title":"Asplund, T","cited_arxiv_id":null,"evidence_quote":"It proves the loop-space coefficient version of the Chekanov-Eliashberg dg-algebra for singular Legendrians used in Section 4.5."},{"cited_title":"Ganatra, J","cited_arxiv_id":null,"evidence_quote":"It provides the cut-and-paste and sectorial descent results that Section 4.5 recovers through surgery descriptions."}],"review_version":1}