Geodesible contact structures

*Geometry and Topology* 12 (2008) 1729-1776

In this paper, we study and almost completely classify contact structures on closed 3– manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case.

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth-Szabó invariants

*Mathematische Annalen* 353 (2012), n° 4, 1351-1376

Non-vanishing of the Ozsváth-Szabó contact invariant is a powerful way to prove tightness of contact structures but this invariant is known to vanish in the presence of Giroux torsion. In this note, we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth-Szabó invariant neverthe- less vanishes. Along the way, we prove a conjecture of K Honda, W Kazez and G Matić about their contact topological quantum field theory.

Tightness in contact metric 3-manifolds

avec John Etnyre and Rafał Komendarczyk

*Inventiones Mathematicae* 188 (2012), n° 3, 621-657

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S³. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.

Weak and strong fillability of higher dimensional contact manifold

avec Klaus Niederkrüger and Chris Wendl

*Inventiones Mathematicae* 192 (2013) n°3, 287-373

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

Quantitative Darboux theorems in contact geometry

avec John Etnyre and Rafał Komendarczyk

*Transactions of the AMS* 368 (2016) n°11, 7845-7881

This paper begins the study of relations between
Riemannian geometry and contact topology on (2n+1)-manifolds and
continues this study on 3-manifolds. Specifically we provide a lower
bound for the radius of a geodesic ball in a contact (2n+1)-manifold
(M,ξ) that can be embedded in the standard contact structure on
ℝ^{2n+1}, that is on the size of a Darboux ball. The bound is
established with respect to a Riemannian metric compatible with an
associated contact form α for ξ. In dimension three, it further
leads us to an estimate of the size for a standard neighborhood of a
closed Reeb orbit. The main tools are classical comparison theorems
in Riemannian geometry. In the same context, we also use holomorphic
curves techniques to provide a lower bound for the radius of a
PS-tight ball.

Examples of non-trivial contact mapping classes in all dimensions

avec Klaus Niederkrüger

*International Mathematics Research Notices (IMRN)* 2016 (15), 4784-4806

We give examples of contactomorphisms in every dimension that are smoothly isotopic to the identity but that are not contact isotopic to the identity. In fact, we prove the stronger statement that they are not even symplectically pseudo-isotopic to the identity. We also give examples of pairs of contactomorphisms which are smoothly conjugate to each other but not by contactomorphisms.

On the contact mapping class group of Legendrian circle bundles

avec Emmanuel Giroux

*Compositio Mathematica* 2017 153(2), 294-312

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. In the third section, this connectedness question is studied in more details with a number of (hopefully instructive) examples.

Contactomorphism groups and Legendrian flexibility

avec Sylvain Courte

preprint

We explain a connection between the algebraic and
geometric properties of groups of contact transformations, open
book decompositions, and flexible Legendrian embeddings. The main
result is that, if a closed contact manifold (V,ξ) has a supporting
open book whose pages are flexible Weinstein manifolds, then both
the connected component of identity in its automorphism group and
its universal cover are uniformly simple groups: for every
non-trivial element g, every other element is a product of at most
128(dimV+1) conjugates of g^{±1}. In particular any
conjugation invariant norm on these groups is bounded.

Formalising perfectoid spaces

avec Kevin Buzzard and Johan Commelin

*Certified Programs and Proofs* 2020, 299-312

Perfectoid spaces are sophisticated objects in arithmetic geometry introduced by Peter Scholze in 2012. We formalised enough definitions and theorems in topology, algebra and geometry to define perfectoid spaces in the Lean theorem prover. This experiment confirms that a proof assistant can handle complexity in that direction, which is rather different from formalising a long proof about simple objects. It also confirms that mathematicians with no computer science training can become proficient users of a proof assistant in a relatively short period of time. Finally, we observe that formalising a piece of mathematics that is a trending topic boosts the visibility of proof assistants amongst pure mathematicians.

Holonomic approximation through convex integration

avec Mélanie Theillière

to appear in *International Mathematics Research Notices (IMRN)*

Convex integration and the holonomic approximation theorem are two well-known pillars of flexibility in differential topology and geometry. They may each seem to have their own flavor and scope. The goal of this paper is to bring some new perspective on this topic. We explain how to prove the holonomic approximation theorem for first order jets using convex integration. More precisely we first prove that this theorem can easily be reduced to proving flexibility of some specific relation. Then we prove this relation is open and ample, hence its flexibility follows from off-the-shelf convex integration.

Formalising the h-principle and sphere eversion

avec Floris van Doorn and Oliver Nash

*Certified Programs and Proofs* 2023, 121-134

In differential topology and geometry, the h-principle is a property enjoyed by certain construction problems. Roughly speaking, it states that the only obstructions to the existence of a solution come from algebraic topology. We describe a formalisation in Lean of the local h-principle for first-order, open, ample partial differential relations. This is a significant result in differential topology, originally proven by Gromov in 1973 as part of his sweeping effort which greatly generalised many previous flexibility results in topology and geometry. In particular it reproves Smale's celebrated sphere eversion theorem, a visually striking and counter-intuitive construction. Our formalisation uses Theillière's implementation of convex integration from 2018. This paper is the first part of the sphere eversion project, aiming to formalise the global version of the h-principle for open and ample first order differential relations, for maps between smooth manifolds. Our current local version for vector spaces is the main ingredient of this proof, and is sufficient to prove the titular corollary of the project. From a broader perspective, the goal of this project is to show that one can formalise advanced mathematics with a strongly geometric flavour and not only algebraically-flavoured

updated on April 29 2020.