Séminaire de topologie, géométrie et algèbre (archives)

Le posets des partitions d'ensemble a été introduit dans les années 40 par Birkhoff. D'une définition très simple (les partitions d'un ensemble, ordonnées par inclusion des parts), ce poset, ainsi que ses extensions décorées, intervient dans la résolution de plusieurs problématiques algébriques. Tout d'abord, le treillis des faces de l'arrangement de tresses est isomorphe au poset des partitions, tandis que son treillis des plats correspond au poset des compositions d'ensemble, où les parts sont ordonnées. Grâce au théorème de Zaslavsky, ces posets de partitions permettent le dénombrement des régions de cet arrangement d'hyperplans et d'arrangements associés, comme nous l'avons montré avec Guillaume Laplante-Anfossi, Vincent Pilaud et Kurt Stoeckl . De plus, Benoit Fresse (pour le cas non décoré) puis Bruno Vallette (pour le cas décoré) ont établi des liens entre la "construction bar à niveaux" d'une opérade et les complexes de chaînes (ou nerf) du poset. Cela permet notamment d'identifier, dans les cas favorables, la cohomologie d'un poset de P-partitions avec la duale de Koszul de P. Dans un travail commun avec Clément Dupont, nous avons défini et étudié une structure spécifique sur certains posets qui permet de munir leur cohomologie d'une structure d'opérade. Cela permet d'étendre à un plus grand nombre de posets les résultats obtenus sur les posets de partitions. Parmi les exemples obtenus, citons les posets de fonctions de parking, les posets d'arbres de Cayley multi-étiquetés ou encore les posets d'hyperarbres.

Je consacrerai la première partie de mon exposé au cas des posets de partitions décorées, à leur cohomologie et aux nombres de régions de k copies de l'arrangement de tresses, et présenterai dans la deuxième partie le cadre général des espèces en posets opéradiques avec de nouveaux exemples.

In this short lecture, I will explain the definition of Lefschetz fibrations on 4-manifolds and basic properties of them. In particular, I will describe a classification theorem of Kas and Matsumoto which asserts that there is one to one correspondence between the isomorphism classes of Lefschetz fibrations and the conjugacy classes of monodromy

Fine curve graphs have been introduced by Bowden, Hensel and Webb to study homeomorphism groups of closed surfaces. A main tool in their work is the fact that fine curve graphs can be approximated by curve graphs of surfaces with punctures. I will talk about joint work with Sebastian Hensel, where we study to which extent the boundary of the fine curve graph can be approximated via curve graphs of surfaces with punctures. I will also discussion of fine curve graph techniques to the study of mapping classes of infinite-type surface.

An element $g$ of an abstract group $G$ is a distortion element if there exists a finite family $S$ in $G$ such that $g\in\langle S\rangle$ and the word-length of $g^n$ (w.r.t. $S$) grows sublinearly in $n$. This is a very general group theoretic notion, but does it have a dynamical interpretation when $G$ is a group of diffeomorphisms? In this talk, we will focus on diffeomorphisms of the closed interval in different regularities. In particular, we will present some natural obstructions to distortion (such as the presence of hyperbolic fixed points in $C^1$ regularity and the positivity of the so-called asymptotic variation in $C^2$ regularity (and higher)), and we will wonder whether they are the only ones.

Le principe fondamental de la topologie algébrique est d'associer des objets algébriques à des objets de nature géométrique ou topologique. Dans cet exposé, nous commencerons par rappeler rapidement la complexité croissante de ces objets algébriques : des nombres, des groupes, des algèbres associatives, puis différents types d'algèbres de plus en plus complexes. Les opérades sont un outil pour mieux comprendre ces algèbres, et mieux les étudier. Par exemple, la construction bar des opérades et la dualité de Koszul permettent de retrouver les complexes d'homologie (Hochschild, Chevalley-Eilenberg, etc) et des constructions similaires dans des contextes plus modernes. Ceux-ci font souvent intervenir des infini-catégories ou des algèbres à homotopie près. Le but final de l'exposé est de présenter la définition des infini-opérades et de leurs algèbres, et de définir des constructions bar associées.

Ceci est un travail en commun avec Ieke Moerdijk.

Lagrangian submanifolds exhibit surprising rigidity in view of their small dimension (only half that of the ambient symplectic manifold). A famous manifestation of this rigidity is that some of them cannot be displaced from themselves by any Hamiltonian diffeomorphism or even by any diffeomorphism which preserves the symplectic form. In comparison, any submanifold of the same dimension which is not Lagrangian can be displaced from itself by a Hamiltonian diffeomorphism.

Moreover, that diffeomorphism can be chosen arbitrarily C⁰-small, so that one can wonder whether a Lagrangian L (which can be displaced from itself) admits a neighborhood of Hamiltonian (resp. symplectic) non-displacement. Such a neighborhood W is defined by the property that any Lagrangian obtained from L by a Hamiltonian (resp. symplectic) diffeomorphism, and included in W, must intersect L.

On the one hand, I will give conditions which ensure the existence of such neighborhoods for a large class of Lagrangians. On the other hand, I will construct a Lagrangian which does not admit any, in any symplectic manifold of dimension at least 6. Then, I will discuss several applications of our techniques to the topology of orbits of Lagrangians. This is based on a joint work with Marcelo Attalah, Jean-Philippe Chassé, and Egor Shelukhin.

In the late '90s, Eliashberg and Thurston established a remarkable connection between foliations and contact structures in dimension three: any co-oriented, aspherical foliation on a closed, oriented 3-manifold can be approximated by both positive and negative contact structures. Additionally, if the foliation is taut then its contact approximations are tight. In this talk, I will present a converse result on constructing taut foliations from suitable pairs of contact structures. While taut foliations are rather rigid objects, this viewpoint reveals some degree of flexibility and offers a new perspective on the L-space conjecture. A key ingredient is a generalization of a result of Burago and Ivanov on the construction of branching foliations tangent to continuous plane fields, which might be of independent interest.

We say that a link L in S^3 is negative amphichiral if there exists an orientation-reversing diffeomorphism of S3 that sends every component of L to itself with the opposite orientation. If such a map can be chosen to be an involution, then the link is said to be strongly negative amphichiral. Kawauchi proved that every strongly negative amphichiral link is rationally slice, i.e. it bounds a disjoint collection of disks in a rational homology 4-ball. In this talk, we prove that every negative amphichiral link is rationally slice, extending the aforementioned work of Kawauchi. Our proof relies on a careful analysis of the JSJ decomposition of the link complement of negative amphichiral links. This is joint work in progress with Jaewon Lee (KAIST, Daejeon) and Oğuz Şavk (CNRS, Nantes).

By a theorem of Kontsevich and Lambrechts-Volić, the little disks operad is formal in every dimension, meaning that its rational homotopy type is faithfully represented by its cohomology. This theorem has had significant consequences in both high and low-dimensional topology (e.g., Vassiliev invariants, diffeomorphisms of the disk) as well as in deformation theory (e.g., quantization deformation of Poisson manifolds). In my talk, I will review these results and explain a recent result on the equivariant formality of the little disks operad.

This is joint work with Pedro Boavida de Brito and Joana Cirici.

Spectral invariants are essential tools for studying symplectic and contact rigidity, such as non-squeezing problems and camel problems, as well as Hamiltonian dynamics. These invariants have been constructed through various approaches. In this talk, we focus on the Floer-theoretic construction of Legendrian spectral invariants in one-jet bundles.

Typically, Floer theory on contact manifolds, such as one-jet bundles, involves the use of symplectization. However, in this talk, I will introduce the concept of contact instantons, which enables the construction of Legendrian Floer cohomology and its spectral invariants without relying on symplectization.