Bourgeois contact structures: tightness, fillability and applications

Agustin Moreno; Fabio Gironella; Jonathan Bowden

Bourgeois contact structures: tightness, fillability and applications

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Pith's one-line read The Bourgeois construction always yields a universally tight contact 5-manifold, whatever the input open book.

desk verdict A substantial paper that very likely delivers on its main claims; the step to check carefully is the cap-exclusion argument in Lemma 17. read the letter →

arxiv 1908.05749 v5 pith:VVI2T2UG submitted 2019-08-15 math.SG math.GT

Jonathan Bowden , Fabio Gironella , Agustin Moreno This is my paper · ORCID

classification math.SGmath.GT MSC 53D1053D3557R17

keywords contacttopologyBourgeoisconstructionuniversaltightnesssymplecticfillabilityopenbookdecompositionholomorphiccurvesmappingclassgroups5-manifolds

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $D^*T^n$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

This 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.

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)

[Section 5, Lemma 17] 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 6, Lemma 23] 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 7 and Introduction, Theorem J] 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 5, proof of Theorem A] 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 5, Lemma 17] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central results are derived from explicit cobordism constructions, external geometric-group-theory and hyperbolic geometry theorems, and standard holomorphic curve compactness, rather than from fitted parameters or self-referential definitions.

full rationale

The derivation chain for Theorem A is self-contained with respect to its stated hypotheses. The proof factors the monodromy via Lemma 10, which is established using Bestvina–Fujiwara quasi-homomorphisms and Thurston's hyperbolic Dehn filling theorem, and then applies Lemma 17, whose cap-exclusion argument uses the explicitly constructed pseudo-Liouville cobordism of Theorem 9 and its stated primitive data. The contradiction 0 < ∫_c Ω = ∫_{-γ} ν < 0 is a property of the constructed cobordism, not a restatement of tightness. Theorem B is proved by a capping construction and moduli-space arguments; the injectivity conclusions are derived, not assumed. Theorem J is supported by independent concurrent work [22]. The paper cites the authors' earlier work [25] and [49], but these citations are contextual or technical (e.g., for related examples and SOBD terminology) and are not load-bearing for the main results. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work to force a choice, and no known result is merely relabeled. The most delicate step, Lemma 17, is a genuine geometric argument about holomorphic caps in a pseudo-Liouville cobordism, and any concern about it is a correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper is a theorem-based contribution: there are no free parameters fitted to data and no invented entities. It rests on standard results from contact topology, mapping class groups, hyperbolic 3-manifolds, holomorphic curve theory, and polyfold theory. The most substantial external input is the Hofer-Wysocki-Zehnder polyfold framework for the general fillability obstruction, which is cited rather than redeveloped.

assumptions (8)

standard math Giroux correspondence: every closed contact manifold is supported by an open book, and contact structures correspond to stable equivalence classes of open books.
Used to define OBD(Σ,φ) and the Bourgeois construction for arbitrary contact manifolds; Section 2 and throughout.
standard math Overtwisted contact manifolds in dimension at least 5 are characterized by the existence of a Plastikstufe.
Lemma 17 assumes an overtwisted convex boundary gives an embedded Plastikstufe to run the holomorphic curve argument.
standard math Thurston's hyperbolic Dehn filling theorem: sufficiently large slopes on a hyperbolic 3-manifold yield hyperbolic Dehn fillings whose cores are geodesics.
Used in the Factorization Lemma (Section 4.3) to ensure binding components have infinite order in π1.
standard math Bestvina-Fujiwara WPD and bounded cohomology results on mapping class groups provide a pseudo-Anosov element with a homogeneous quasi-homomorphism vanishing on finite-order and reducible elements.
Needed for Corollary 13 and the factorization φ=F∘G with pseudo-Anosov factors; Section 4.1.
domain assumption Hofer-Wysocki-Zehnder polyfold Fredholm theory for closed Gromov-Witten moduli spaces gives abstract perturbations, transversality, and Stokes theorem.
The general case of Theorem B uses polyfolds [34] because semi-positivity may fail in dimensions ≥8; Remark 2 and Section 6.1.
standard math Semipositivity holds automatically for 6-dimensional symplectic manifolds, and pseudo-cycle theory applies.
Provides the dimension-5 applications and the semi-positive case of Theorem B.
standard math The s-cobordism theorem, with vanishing Whitehead group for Z^n, classifies h-cobordisms of S*T^n.
Final step of Theorem J proving diffeomorphism type; Section 7.
standard math Residual finiteness of fundamental groups of closed 3-manifolds.
Used to conclude finite-cover tightness implies universal tightness in Theorem A.

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Pith. "Pith review of Bourgeois contact structures: tightness, fillability and applications." pith.science (2026). https://pith.science/paper/VVI2T2UG

@misc{pith2026190805749,
  author       = {Pith},
  title        = {Pith review of: Bourgeois contact structures: tightness, fillability and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVI2T2UG}},
  note         = {Machine review of arXiv:1908.05749}
}

abstract

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $\mathbb S^1$-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the $n$-torus has a unique symplectically aspherical strong filling up to diffeomorphism.

Figures

Figures reproduced from arXiv: 1908.05749 by the authors.

**Figure 1.** The pseudo-Liouville cobordism (C, ωC ) given in Theorem 9. in [15, Section 5.3], we view the Bourgeois contact manifolds as S 1 -equivariant contact manifolds as in [23] (this also allows to find supporting S 1 -equivariant spinal open book decompositions or SOBDs as explained in Remark 21; cf. [49, Section 5.1]). We can do so in two different ways, one for each S 1 -factor of the Bourgeois torus T 2 = S 1 × S 1 . … view at source ↗

**Figure 2.** Topological picture of Cbot. D−,1 and D−,2, both of radius and centered at −1/2 and +1/2 respectively, removed from it. In cobordisms terms, P is seen as a smooth cobordism with concave boundary ∂D−,1 t ∂D−,2 and convex boundary ∂D+. Consider then the fiber bundle πE : E → P, with fiber the page Σ, over the pair of pants P, where the monodromies along the two negative boundary components are given by φ and ψ respe… view at source ↗

**Figure 3.** Topological picture of the cobordism (with corners) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: A schematic picture for the capping construction. [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]

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