{"id":"38b8f2e6-8fd0-496f-b5c5-55bb52a8d5e3","arxiv_id":"1908.05749","paper_version":5,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bourgeois contact 5-manifolds are universally tight, strong fillability of Bourgeois manifolds forces homological injections, and the unit cotangent bundle of the n-torus has a unique aspherical strong filling.","lead":"This mathematics paper proves that Bourgeois contact structures on five-dimensional manifolds are universally tight, no matter whether the lower-dimensional contact pieces they are built from are tight or overtwisted. It also gives strong obstructions to symplectic fillability, producing the first weakly-but-not-strongly fillable contact structures in every odd dimension.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A leans on Lemma 17's cap-exclusion Stokes argument in a pseudo-Liouville cobordism; the sign/orientation claim there is the step most worth verifying.","rationale":"The central new statement is Theorem A, and its proof has three ingredients: the cobordism Theorem 9, the Factorization Lemma 10, and the holomorphic-curve Lemma 17. I read the Factorization Lemma in detail; its use of Bestvina-Fujiwara quasimorphisms and Thurston's Dehn filling theorem is standard and the algebra checks out (φ = F τ^r τ^{-r} G = φ, slopes go to infinity, binding cores are geodesics). The sporadic cases are handled case-by-case and are also standard. Thus I cannot substantiate the reader's stated weakest assumption as an actual vulnerability. The genuinely delicate point is Lemma 17's exclusion of holomorphic caps, which the authors flag as non-automatic because the cobordism is only pseudo-Liouville. The global-primitive Stokes argument is the only thing standing between a finite-energy plane in a hypertight negative end and a cap degeneration; a sign or compactness gap there would break the non-sporadic case of Theorem A. This is a verifiable, localized concern rather than a demonstrated flaw, so the verdict stays ACCEPT (UNCHANGED). The secondary risk for Theorem B is the polyfold dependence, already acknowledged in Remark 2; it does not affect Theorem A.","tokens_in":41832,"tokens_out":34506,"duration_ms":346770,"concrete_test":"Verify Lemma 17's cap exclusion by an independent computation in the explicit negative-end model of Theorem 9. Using the global primitive ν on C (ν = λ0 + K π^*λ_std on Cbot, ν = e^t λ_B + K λ_std on Ctop), write the negative boundary as (V×T2, λ− = βΣ,φ) and take a model holomorphic cap with boundary a Reeb orbit γ in B_q. Compute ∫_{∂c} ν from the explicit coordinate identifications in Section 3.1 (including Equation (3)) and check the two signs: whether ∂c is oriented as −γ and whether ∫_γ ν > 0. Also enumerate, from the SFT compactness of the Bishop family, all possible holomorphic buildings in this pseudo-Liouville setting and confirm that the only cap-type component has exactly one boundary orbit. This finite calculation either validates the contradiction or exposes the missing term.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption field points to the Factorization Lemma, but that lemma is well supported by Bestvina-Fujiwara quasimorphisms and Thurston hyperbolic Dehn filling; I do not see a concrete gap there. The more delicate load-bearing step for Theorem A is in Lemma 17: after a Bishop family is produced from a Plastikstufe on the convex end, compactness can yield a holomorphic cap in the pseudo-Liouville cobordism (C,ω_C), and the proof rules caps out via the Stokes identity 0 < ∫_c Ω = ∫_{−γ} ν < 0. The paper itself notes this is 'not automatic from standard arguments' because the cobordism is only pseudo-Liouville. The contradiction depends on two unstated sign/data claims: (i) the cap has a single boundary component γ, a Reeb orbit of λ− in the negative end, oriented as −γ; (ii) the global primitive ν satisfies ∫_γ ν > 0 via Item 2 of Theorem 9. If the boundary orientation were +γ, or if the compactification produced a cap with additional punctures or boundary components, the argument would not go through. Since tightness of every non-sporadic case reduces to Lemma 17, this is the point where a hidden error would be most damaging. The proof is plausible but the verification is compressed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies contact structures obtained from Bourgeois' construction on V × T^2. Its main result, Theorem A, states that for every abstract open book (Σ^2, φ), the Bourgeois contact 5-manifold BO(Σ, φ) is universally tight, regardless of whether the original contact 3-manifold is tight or overtwisted. The proof combines a pseudo-Liouville cobordism between Bourgeois manifolds with the same page (Theorem 9), a factorization lemma for mapping classes of non-sporadic surfaces (Lemma 10) proved via Bestvina–Fujiwara quasi-homomorphisms and Thurston hyperbolic Dehn filling, and a holomorphic-curve argument (Lemma 17) that rules out holomorphic caps using a Stokes identity. The paper also proves a strong fillability obstruction (Theorem B): for a strong symplectic filling W of BO(Σ, φ), the page Σ injects in rational homology and the T^2 factor injects in integer homology. This obstruction is used to produce weakly but not strongly fillable contact manifolds in all odd dimensions, including planar examples (Theorem D), positive stabilizations (Corollary F), and the explicit family BO(D^*S^n, τ^k) (Theorems H and I), giving a negative answer to a question of Lisi–Marinković–Niederkrüger. Finally, Theorem J classifies symplectically aspherical strong fillings of the unit cotangent bundle S^*T^n up to diffeomorphism, via a capping construction, a moduli space of holomorphic spheres, and an s-cobordism argument.","tokens_in":41955,"tokens_out":30498,"duration_ms":293998,"significance":"If the results are correct, this is a substantial and influential paper. Theorem A is a striking 'tightening' statement: the Bourgeois construction turns any contact 3-manifold, even an overtwisted one, into a universally tight 5-manifold, with no dependence on the rigid or flexible nature of the input. Theorem B gives a very general and powerful obstruction to strong fillability, and the corollaries provide the first weakly-but-not-strongly fillable examples in all odd dimensions as well as a large new class in dimension 5. The paper is careful in separating the semi-positive case from the general polyfold case, explicitly flags the inconclusive even-even case in Theorem I, and acknowledges the independent overlap with Geiges–Kwon–Zehmisch [22]. The proofs are detailed and make use of established tools rather than ad hoc assumptions. The polyfold-dependent part of the argument is clearly identified. This paper will likely become a reference for the study of Bourgeois contact structures.","major_comments":[],"minor_comments":[{"comment":"The Stokes contradiction 0 < ∫_c Ω = ∫_{−γ} ν < 0 relies on the boundary of the holomorphic cap being a single Reeb orbit oriented as −γ and on the positivity ∫_γ ν > 0. Both assertions are correct — negative boundary components of a strong cobordism are oriented opposite to the contact orientation, and Item 2 of Theorem 9 gives ν|_{B_q} = λ_−|_{B_q} — but the orientation convention should be stated explicitly, since the proof of the non-sporadic case of Theorem A reduces to this step.","section":"Section 5, Lemma 17"},{"comment":"In the proof of Lemma 23, the exponents in the displayed term are off by one: on the manifold [0,1] × ∂X × S^2 of real dimension 2n+2, the top power of a symplectic form is n+1, not n+2, and the top power of dλ_Σ on Σ of dimension 2n−2 is n−1, not n. The sentence should read (dβ+ω_S)^{n+1} and (dλ_Σ)^{n−1} ∧ dt ∧ dx ∧ ω_S. The conclusion is unaffected, but the current indices are formally zero by dimension count.","section":"Section 6, Lemma 23"},{"comment":"Theorem J in the Introduction states n ≥ 3, but Section 7 begins with n ≥ 2 and the proof as written appears to cover n ≥ 2. Please reconcile the stated range and verify that the s-cobordism step is valid for the intended range.","section":"Section 7 and Introduction, Theorem J"},{"comment":"In the universal tightness argument, the sentence 'tightness on finite covers is equivalent to tightness on the universal cover' would benefit from a short justification: a compact overtwisted disk or Plastikstufe in the universal cover descends to some finite cover because its stabilizer has finite index, so overtwistedness in the universal cover would contradict tightness of all finite covers.","section":"Section 5, proof of Theorem A"},{"comment":"The symbol C is used both for the compact pseudo-Liouville cobordism and for its compactification obtained by adding ideal contact boundaries. Using a different notation, e.g. \\bar C, for the compactification would remove a potentially confusing ambiguity in the discussion of the holomorphic cap c.","section":"Section 5, Lemma 17"}],"recommendation":"minor_revision","confidential_remarks":"I found no novelty or citation problems; the independent overlap with [22] is openly acknowledged in Remark 3. The requested changes are local clarifications and corrections, and I do not see a load-bearing error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Bowden–Gironella–Moreno, Bourgeois contact structures. The headline: this is a substantial paper and the main theorems look right. Theorem A — every 5-dimensional Bourgeois contact structure is universally tight, independent of whether the original open book came from a tight or overtwisted 3-manifold — is genuinely new and surprising. Theorem B gives a clean homological obstruction to strong fillability, and together with [41] it yields weakly-but-not-strongly fillable examples in every odd dimension. Theorem J, the uniqueness of symplectically aspherical strong fillings of S*T^n up to diffeomorphism, overlaps with Geiges–Kwon–Zehmisch, but the paper openly says so and the proofs appear independent.\n\nThe paper does what a good paper should do. It states its technical dependencies, separates the semi-positive case from the polyfold case, and explicitly flags the even-even inconclusive case in Theorem I and the gluing sign issue in Remark 37. The Factorization Lemma is proved inside using Bestvina–Fujiwara quasi-homomorphisms and Thurston Dehn filling; I do not see a gap there. No circularity, no post-hoc selection, and the citation pattern is honest.\n\nThe softest load-bearing spot is Lemma 17, used in the proof of Theorem A. After producing a Bishop family from a Plastikstufe, the authors rule out holomorphic caps via the Stokes identity 0 < ∫_c Ω = ∫_{−γ} ν < 0. As they note, this is not automatic because the cobordism is only pseudo-Liouville, and the argument depends on the cap having exactly one boundary component γ with the orientation −γ, together with ν being positive along γ. The proof is plausible but compressed; that sign/orientation claim is exactly where a hidden error would do the most damage. I do not see one, but a referee should spend real time there. The general version of Theorem B relies on Hofer–Wysocki–Zehnder polyfold technology, which is standard at this level but limits how much of the paper can be independently verified; the semi-positive case is self-contained and covers dimension 5.\n\nThis is for contact and symplectic topologists, especially people working on high-dimensional tightness and fillings. I would bring it to a reading group. It deserves a serious referee, not a desk reject. My recommendation: send it to peer review and ask the referee to check Lemma 17 carefully.","headline":"A substantial paper that very likely delivers on its main claims; the step to check carefully is the cap-exclusion argument in Lemma 17.","tokens_in":42601,"tokens_out":1924,"would_cite":true,"duration_ms":20932,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D10","53D35","57R17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Bourgeois construction always yields a universally tight contact 5-manifold, whatever the input open book.","keywords":["contact topology","Bourgeois construction","universal tightness","symplectic fillability","open book decomposition","holomorphic curves","mapping class groups","contact 5-manifolds"],"falsifier":"One concrete way to test the central claim is to search for an abstract open book on a surface whose Bourgeois 5-manifold is overtwisted, or a strong symplectic filling of some $BO(\\Sigma,\\varphi)$ in which the page does not inject in rational homology; either example would directly refute Theorem A or Theorem B. A narrower check is to attempt to construct a monodromy on a non-sporadic surface that admits no factorization into two factors with infinite-order binding components.","tokens_in":41513,"feed_emoji":"🔒","tokens_out":9511,"duration_ms":83523,"temperature":0.7,"pith_summary":"This paper tries to establish that the Bourgeois construction is a rigidity-producing operation: in dimension five, every contact manifold built as $BO(\\Sigma,\\varphi)$ — a surface open book with monodromy $\\varphi$, crossed with a 2-torus — is universally tight, whether or not the underlying contact 3-manifold was tight or overtwisted. The same construction is then shown to be very restrictive with respect to symplectic fillings: a strong filling forces the page to inject in rational homology and the torus factor to inject in integer homology. From this the authors derive broad infinite families of weakly but not strongly fillable contact manifolds, the first such examples in all odd dimensions, and a uniqueness statement for symplectically aspherical fillings of the unit cotangent bundle of the $n$-torus. The paper also records that a natural gluing strategy for weak fillings fails because the monodromy-inversion contactomorphism flips the orientation of the torus factors, so the weak-fillable examples are not assembled by naive cobordism stacking.","feed_headline":"Bourgeois contact 5-manifolds are always universally tight","feed_subtitle":"Rigidity holds even when the input contact 3-manifold is overtwisted; strong fillings are then sharply constrained.","key_machinery":"The argument rides on three interlocking objects. The Bourgeois construction itself turns open book data $(\\Sigma,\\varphi)$ into the contact form $\\beta = \\alpha + \\Phi_1 dq_1 - \\Phi_2 dq_2$ on $OBD(\\Sigma,\\varphi) \\times T^2$. Theorem 9 builds a pseudo-Liouville symplectic cobordism from $BO(\\Sigma,\\psi) \\sqcup BO(\\Sigma,\\varphi)$ to $BO(\\Sigma,\\psi \\circ \\varphi)$, a torus-stabilized version of the standard open book composition cobordism; the cobordism is exact but its Liouville vector field is not inward pointing at the concave ends. The Factorization Lemma splits any monodromy on a non-sporadic surface into two factors whose open book bindings have infinite order in $\\pi_1$, using quasi-homomorphisms on mapping class groups and Thurston's hyperbolic Dehn filling theorem; combined with the cobordism this reduces tightness to excluding holomorphic caps via Reeb dynamics. For fillability, the central mechanism is a capping construction: attach a symplectic handle with two J-invariant co-cores to a strong filling, producing a moduli space of holomorphic spheres whose evaluation forces the homology of the page and torus to survive in the filling.","core_discovery":"The central claim, Theorem A, is that for every abstract open book $(\\Sigma^2,\\varphi)$ the Bourgeois contact 5-manifold $BO(\\Sigma,\\varphi)$ is universally tight: its universal cover is tight. This holds with no assumption on the original contact 3-manifold $OBD(\\Sigma,\\varphi)$, so even overtwisted inputs are converted into tight outputs. Theorem B states that if $BO(\\Sigma,\\varphi)$ has a strong symplectic filling $W$, then the page inclusion $\\Sigma \\to W$ is injective in rational homology and the Bourgeois torus $T^2 \\to W$ is injective in integer homology; in particular the monodromy $\\varphi$ is forced to be trivial in homology in many cases. Corollaries include: for rational homology sphere inputs only the standard $S^3$ case is strongly fillable; planar pages with nonzero same-sign Dehn twist monodromy give weakly but not strongly fillable 5-manifolds; positive stabilizations are never strongly fillable; and $S^*T^n$ has a unique symplectically aspherical strong filling up to diffeomorphism.","pith_inferences":["If the same tightness mechanism works in higher dimensions, the Bourgeois construction becomes a general source of rigid contact manifolds from flexible ones; the paper leaves this as an open question, not an established fact.","Theorem B gives a concrete test for the paper's Question 36: the likely answer is that strong fillability forces the monodromy to be smoothly trivial, and even-dimensional cotangent pages where some power of the Dehn–Seidel twist is smoothly trivial would be the sharp test case.","Because the capping argument only uses the $S^1$-invariant structure behind the Bourgeois form, a similar homology-injection obstruction should hold for other $S^1$-invariant contact manifolds built by the same double-and-glue recipe."],"forward_implications":["In dimension 5, every Bourgeois contact manifold is universally tight; in particular no Bourgeois structure can serve as an overtwisted example, even when built from an overtwisted 3-manifold.","A strong filling of $BO(\\Sigma,\\varphi)$ must rationally contain the page and integrally contain the $T^2$ factor; hence if the underlying 3-manifold is a rational homology sphere, strong fillability forces the page to be a disc and the monodromy trivial, so the manifold is $S^3$.","Bourgeois structures over planar pages with monodromy a non-trivial product of same-sign Dehn twists are weakly but not strongly fillable, giving many 5-dimensional examples of this fillability gap.","Positive stabilization of any open book produces a Bourgeois manifold that is not strongly fillable; combined with known weak fillability this yields weakly but not strongly fillable contact structures in every odd dimension.","The standard contact unit cotangent bundle $S^*T^n$ has exactly one symplectically aspherical strong filling up to diffeomorphism, namely $D^*T^n$."],"supporting_citations":[{"why":"Introduces the Bourgeois contact form on $V \\times T^2$ from an open book; the object under study.","marker":"[8]"},{"why":"Supplies prior fillability results for Bourgeois structures, including weak fillability and monodromy inversion used for sporadic cases.","marker":"[41]"},{"why":"Establishes existence and classification of overtwisted contact structures in all dimensions, so overtwistedness is detected by a Plastikstufe in the contradiction argument.","marker":"[6]"},{"why":"Defines the Plastikstufe and the Bishop family of holomorphic discs used to derive a finite-energy plane.","marker":"[51]"},{"why":"Provides the holomorphic curve machinery in symplectizations that turns a Bishop family into a holomorphic plane, contradicting hypertightness.","marker":"[31]"},{"why":"Gives unbounded quasi-homomorphisms on non-sporadic mapping class groups, used to construct the pseudo-Anosov factor in the Factorization Lemma.","marker":"[4]"},{"why":"Establishes hyperbolicity of pseudo-Anosov mapping tori, needed before Dehn filling.","marker":"[58]"},{"why":"Provides hyperbolic Dehn filling, making binding components infinite order in the fundamental group in the Factorization Lemma.","marker":"[59]"},{"why":"Supplies polyfold perturbation theory for Gromov-Witten moduli spaces, used to prove the homology injection in general non-semi-positive fillings.","marker":"[34]"}],"fun_headline_variants":["Overtwisted in, tight out: all Bourgeois 5-manifolds","Bourgeois 5-manifolds are universally tight, always","First weakly-but-not-strongly fillable in all odd dimensions","Strong fillings obstructed for Bourgeois contact manifolds","Bourgeois 5-manifolds: tightness without input assumptions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the factorization step: every monodromy of a surface that is not a disc, annulus, or pair of pants can be written as a composition of two monodromies for which every binding component of the resulting open books has infinite order in the fundamental group; if any such monodromy resisted this splitting, the proof of Theorem A for those pages would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Overtwisted in, tight out: all Bourgeois 5-manifolds","Bourgeois 5-manifolds are universally tight, always","First weakly-but-not-strongly fillable in all odd dimensions","Strong fillings obstructed for Bourgeois contact manifolds","Bourgeois 5-manifolds: tightness without input assumptions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001076,"raw_usage":{"total_tokens":4512,"prompt_tokens":963,"completion_tokens":3549,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":3455}},"tokens_in":579,"tokens_out":3549,"duration_ms":25169,"temperature":1.0,"reasoning_tokens":3455,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:06:00.677376+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete way to test the central claim is to search for an abstract open book on a surface whose Bourgeois 5-manifold is overtwisted, or a strong symplectic filling of some $BO(\\Sigma,\\varphi)$ in which the page does not inject in rational homology; either example would directly refute Theorem A or Theorem B. A narrower check is to attempt to construct a monodromy on a non-sporadic surface that admits no factorization into two factors with infinite-order binding components.","supporting_citations":[{"cited_title":"ArXiv e-prints (2018)","cited_arxiv_id":null,"evidence_quote":"Supplies prior fillability results for Bourgeois structures, including weak fillability and monodromy inversion used for sporadic cases."},{"cited_title":"Acta Math","cited_arxiv_id":null,"evidence_quote":"Establishes existence and classification of overtwisted contact structures in all dimensions, so overtwistedness is detected by a Plastikstufe in the contradiction argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Plastikstufe and the Bishop family of holomorphic discs used to derive a finite-energy plane."},{"cited_title":"ArXiv Mathematics e-prints (1998)","cited_arxiv_id":null,"evidence_quote":"Establishes hyperbolicity of pseudo-Anosov mapping tori, needed before Dehn filling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides hyperbolic Dehn filling, making binding components infinite order in the fundamental group in the Factorization Lemma."}],"review_version":1}