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.

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)

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

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).

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.