Séminaire de topologie, géométrie et algèbre (archives)

Talk 1: Triangulated surfaces in triangulated categories

Abstract: The symmetries inherent in the structure constants of a Frobenius algebra can be used to associate certain numerical invariants of oriented surfaces. These numbers behave nicely when chopping surfaces into pieces - in technical terms they form a 2-dimensional open topological field theory. In particular, the invariant of a given surface can be computed in terms of a chosen triangulation. In this talk, we explain how certain symmetries in the foundations of homological algebra behave like a Frobenius algebra to the extent that they define invariants of oriented surfaces.

Based on joint work with Mikhail Kapranov. "

" Talk 2: Relative Calabi-Yau structures [Salle Eole, à 14:00]

Abstract: The basic operation of oriented cobordism is to glue two oriented manifolds along a common boundary component to produce a new oriented manifold. In this talk, we discuss a generalization of this procedure to noncommutative geometry: we introduce the concept of a Calabi-Yau structure on a functor of differential graded categories which should be interpreted as an analog of an oriented manifold with boundary. As an application of the resulting theory, we show that topological Fukaya categories of surfaces give rise to a 2D TFT with values in Calabi-Yau cospans of differential graded categories.

Based on joint work in progress with Chris Brav. "

Recall that the geometric dimension $gd(G)$ of a group $G$ is the smallest dimension of a space on which $G$ acts in such a way that fixed point sets of finite subgroups are contractible. For many prominent classes of groups (e.g. for amenable groups, lattices in classical Lie groups, mapping class groups, groups of outer automorphisms of free groups...) one has equality between the geometric dimensions and the virtual cohomological dimension. On the other hand, there are some examples showing that these two notions of dimension might well differ. I will present some new examples of this phenomenon. This is joint work with Dieter Degrijse.

Les ribbon tubes sont un analogue en dimension quatre des string links. Dans ce séminaire je vais introduire une notion d'homotopie sur les ribbon tubes, qui généralise la notion de self homotopie introduite par Habegger-Lin dans le cas des string links et qui nous permet d'interpréter les (classes d'équivalence de) ribbon tubes en termes d'automorphismes de groupes libres réduits. Si le temps le permet je vais presenter aussi des autres notions d’équivalences pour les ribbon tubes, pour lesquelles nous avons trouvé une classification complète de classes d’équivalences de ribbon tubes, classifications qui généralisent en particulier des résultats de Murakami-Nakanishi et Fish-Keyman pour les string links.

Travail en commun avec B. Audoux, J-B. Meilhan et E. Wagner (arXiv: 1407.0184, 1507.00202 et 1510.04237)

The cobordism hypothesis, first formulated by Dolan an Baez in 95', asserts that if we organize the collection of all framed manifolds and framed cobordisms in dimensions 0 through n into a suitable categorical structure, they will form the free symmetric monoidal (infinity,n)-category with duals generated by a single object, namely, the point. This means, in particular, that fully extended topological field theories with value in any other (infinity,n)-category of the same nature are freely determined by the value they associate to the point. In 2009 an expository paper of Jacob Lurie paved the way to a proof of this hypothesis, but many details are still left unwritten. In this talk we will describe the hypothesis and attempt to outline the proof in the 1-dimensional case.

J'exposerai des travaux en collaboration avec Julien Marché, dans lesquels nous décrivons l'action du mapping class group sur les composantes connexes de l'espace des représenations du groupe de surface de genre 2 dans PSL(2,R).

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.