Séminaire de topologie, géométrie et algèbre (archives)

A theorem by J.D.S. Jones from 1987 identifies the cohomology of the free loop space of a simply connected space with the Hochschild homology of the singular cochain algebra of this space. There are very strong relations between the Floer homology of cotangent bundles in symplectic geometry and the homology of free loop spaces of closed manifolds. In the light of these connections, one wants to have a geometric and Morse-theoretic identification of free loop space cohomology and the Hochschild homology of Morse cochain algebras in order to establish relations between Floer homology and Hochschild homology. After describing the underlying Morse-theoretic constructions and especially the Hochschild homology of Morse cochains, I will sketch a purely Morse-theoretic version of Jones' map and discuss its most important properties.

If there is an extension of the first talk then I will outline a purely Morse-theoretic proof of Jones' theorem on free loop space homology and Hochschild homology. I will further discuss compatibility results with product structures like the Chas-Sullivan loop product and give explicit Morse-theoretic descriptions of products in Hochschild cohomology in terms of gradient flow trees.

Flat manifold bundles (i.e. manifold bundles with foliations transverse to the fibers) are classified by homotopy classes of maps to the classifying space of diffeomorphisms made discrete. In this talk, I will talk about homological stability of discrete surface diffeomorphisms and discrete symplectic diffeomorphisms which was conjectured by Morita. I will describe an infinite loop space related to the Haefliger space whose homology is the same as group homology of discrete surface diffeomorphisms in the stable range. Finally, I will discuss some interesting applications to the characteristic classes of flat surface bundles and foliated bordism groups of codimension 2 foliations.

It is possible to have a second talk on Thursday at 14:15-15:00 (Salle à déterminer)

Title : Braid groups and diffeomorphisms of the punctured disk

Abstract: Morita proved that for large enough $g$ the mapping class group of a surface of genus $g$, cannot be realized as a subgroup of the discrete surface diffeomorphism group $Diff(\Sigma_g)$, by showing that there is a homology obstruction. Surprisingly, the situation is different for the braid groups. While braid groups cannot be realized by diffeomorphism groups of punctured disks, as N.\,Salter and B.\,Tshishiku recently showed, we prove that the homology groups of the braid group are summands of the homology groups of the discrete diffeomorphisms of a disk with punctures. This situation is similar to the homeomorphism group of a surface of genus $g>5$ where the mapping class group and the homeomorphism group have the same homology but still there is no section from the mapping class group of such a surface to its homeomorphism groups. Using factorization homology, we also show that there is no homological obstruction to realize surface braid groups by diffeomorphism groups of the punctured surface. We discuss the stable homology of discrete diffeomorphisms of the punctured disk.

I will compare two combinatorial models of the Moduli space of two dimensional cobordisms. More precisely, I will construct direct connections between the space of metric admissible fat graphs due to Godin and the chain complex of black and white graphs due to Costello. Furthermore, I will construct a PROP structure on admissible fat graphs, which models the PROP of Moduli spaces of two dimensional cobordisms. I will use the connections above to give black and white graphs a PROP structure with the same property.

If there is an extension of the talk I would suggest the following.

Talk part II

Title: Other combinatorial models of the Moduli space of Riemann surfaces

Abtract: I will mention how B\"{o}digheimer's model of radial slit configurations fit into the picture of the first talk; and how this shows that the space of Sullivan diagrams, is homotopy equivalent to B\"{o}digheimer's Harmonic compactification of Moduli space. If time permits I will mention a reinterpretation of Sullivan diagrams and admissible graphs in terms of arc complexes and some new computational results.

Examples of non vanishing vector fields on closed manifolds without periodic orbits are mainly constructed by using Wilson plugs. I will introduce the definition of a plug and explain how it can be used to find examples of such vector fields. For Reeb flows plugs can not exist but I will present examples of semi contact plugs in dimension five and higher, compare their dynamics with the 3-dimensional situation and discuss relations to the Weinstein conjecture. Finally I will apply the semi contact plug construction to build Hamiltonian plugs.

Ceci est un travail en commun avec Victoria Lebed (Nantes). Un module de Yetter-Drinfeld (YD) sur une algèbre de Hopf H est un H-module qui est aussi un H-comodule tel que les deux structures soient compatibles. Dans le cas spécial d'un YD-module sur H = kG pour un groupe G, la structure de comodule donne lieu à une G-graduation compatible sur le G-module. Généralisant cette idée, un module sur un module croisé de groupes H -> G est un G-module avec une H-graduation qui est compatible (notion due à Bantay). Nous généralisons ces structures en introduisant des modules sur un système tressé ("modules de YD généralisés"). Cela donne donc en particulier une notion de module pour des modules croisés d'algèbres de Lie et de Leibniz, ainsi que pour les modules croisés de racks et de shelves. Nous étudions la possibilité d'avoir un produit tensoriel sur ces modules de YD généralisés.

I will present (in French) a generalization of the Blakers-Massey theorem which applies to a family of factorization systems (satisfying some axioms) on an arbitrary infinity topos. The classical theorem is obtained by considering the category of spaces and the n-trucated/n-connected factorization system. The proof itself is inspired by previous work on proving the Blakers-Massey theorem in Homotopy Type Theory, that is, using only the internal language of a higher topos. Time permitting, I will discuss applications to Goodwillie's Calculus of Funtors.

Le théorème de Hochschild-Kostant-Rosenberg classique identifie l'homologie de Hochschild d'une algèbre commutative lisse avec son algèbre des formes de de Rham algébriques. Connes a remarqué que l'on peut interpréter la différentielle de de Rham sur le complexe de Hochschild conduisant à la notion d'homologie cyclique vue comme une "géométrie non-commutative". Ces résultats donnent des moyens combinatoire de calculer l'homologie de Hochschild. Plus récemment, motivé par des problèmes de topologie et géométrie algébrique, Toën-Vezzosi ont démontré que l'homologie de Hochschild s'identifiait comme un espace de lacets en géométrie dérivée et l'homologie cyclique en terme d'action du cercle sur lui-même.

Le but de l'exposé est d'expliquer un théorème de type HKR pour des espaces Map(X,Y) de fonctions plus généraux et d'expliquer en particulier comment comprendre le théorème de Hochschild-Kostant-Rosenberg comme la combinaison d'un théorème de lissité et d'un théorème de formalité.

I'll define a new symplectic object called a pumpkin domain and I'll construct its Fukaya category. This simultaneously generalizes the wrapped Fukaya category of a Liouville domain and the Fukaya-Seidel category of a Lefschetz fibration. Pumpkin domains come with a natural geometric gluing operation ; at the level of Fukaya categories, it corresponds to a certain pushout. After describing this, I'll give some simple applications and a conjectural connection to Legendrian contact homology.