Séminaire de topologie, géométrie et algèbre (archives)

Families of groups such as symmetric groups, braid groups, general linear groups, mapping class groups of 2- or 3-dimensional manifolds, or Higman-Thompson groups share the following stability phenomenon: the homology of the nth group in the sequence is isomorphic to that of the (n+1)st group in a range of degrees increasing with n. This phenomemon is called homological stability.

In this series of talks, I will give an introduction to homological stability, showing what the above examples have in common. I'll explain through the framework of homogeneous categories how the question of stability boils down to the question of high connectivity of certain simplicial complexes and give an idea of how these connectivity results are proved in different examples.

I will give an introduction to Legendrian contact homology, which is an invariant of Legendrian submanifolds that is defined by using pseudo-holomorphic disk techniques. In particular, I will explain how one can define this homology with integer coefficients by orienting the moduli spaces of the pseudo-holomorphic disks. I will also discuss how one can make this invariant more easy to compute by replacing the pseudo-holomorphic disks with gradient flow trees, and how the moduli spaces of these trees can be oriented in a computable way.

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).