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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 Theilli\`ere'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, Comment: 30 pages, submitted to CPP 2023

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., Comment: 11 pages

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., Comment: Final version, with small additions requested by referees. 22 pages. See also https://leanprover-community.github.io/lean-perfectoid-spaces/

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 $(V, \xi)$ has a supporting open book whose pages are flexible Weinstein manifolds, then the connected component $G$ of the identity in its automorphism group is a uniformly simple group: for every non-trivial element $g$, every other element is a product of at most $128(\dim V + 1)$ conjugates of $g^{\pm 1}$. In particular any conjugation invariant norm on this group is bounded. We also prove the later statement still holds for the universal cover of $G$., Comment: 38 pages, 5 figures v2: corrects one silly mistake, updates references and acknowledgments. Submitted version

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., Comment: 20 pages. Extends and correct arXiv:math/0102009

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., Comment: 14 pages, 1 figure v2: Made some exposition improvements, corrected a mistake pointed out by Sylvain Courte in the proof of Observation 2.1. To appear in IMRN

These notes provide an introduction to Giroux's theory of convex surfaces in contact 3-manifolds and its simplest applications. They put a special emphasis on pictures and discussions of explicit examples. The first goal is to explain why all the information about a contact structure in a neighborhood of a generic surface is encoded by finitely many curves on the surface. Then we describe the bifurcations that happen in generic families of surfaces. As applications, we explain how Giroux used this technology to reprove Bennequin's theorem saying that the standard contact structure on S^3 is tight and Eliashberg's theorem saying that all tight contact structures on S^3 are isotopic to the standard one., Comment: 54 pages, 67 figures, based on lectures at the Contact and Symplectic Topology Summer School in Nantes in June 2011

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, 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. 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., Comment: 33 pages, corrects several inaccuracies in earlier version

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., Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor edits. v3: exposition improved using referee's comments. Published by Invent. Math

Ozsvath-Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath-Szabo invariant vanishes. We also discuss the relation between these invariants and an invariant on T^3 and construct other examples of new phenomena in Heegaard--Floer theory. Along the way, we prove two conjectures of K Honda, W Kazez and G Matic about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants., Comment: 27 pages, 12 figures

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,\xi) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure \xi is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S^3. We also describe geometric conditions in dimension three for \xi to be universally tight in the nonpositive curvature setting., Comment: 29 pages. Added the sphere theorem, removed high dimensional material and an alternate approach to the three dimensional tightness radius estimates

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., Comment: 49 pages, 6 figures. v2 includes referee's suggestions, adds some references and discussions, corrects small inaccuracies, minor exposition improvements. this is almost the published version