| Résume | Après-midi de Topologie - jeudi 7 mars 2024
Organisée par Christian Ausoni (LAGA), Geoffroy Horel (LAGA), Muriel Livernet (IMJ-PRG), Najib Idrissi (IMJ-PRG).
Ismaïl Razack: Hochschild cohomology, Batalin-Vilkovisky algebras and operads
The Hochschild cohomology of a differential graded algebra (DGA) is naturally endowed with a Gerstenhaber algebra structure. This structure can be enhanced into a Batalin-Vilkovisky algebra (BV algebra) when the DGA verifies some form of symmetry. For instance, Luc Menichi showed that HH*(C*(M)), the Hochschild cohomology of the cochain complex of a smooth, compact, simply connected manifold M, is a BV-algebra. The goal of this talk is to give a new proof of this result using operad theory (Barratt-Eccles operad) and without assuming that M is simply connected. We'll also explain why we get a similar result if we replace a manifold by a pseudomanifold.
Ana Sopena Gilboy: Pluripotential operadic calculus
For complex manifolds, there exists a refined notion of weak equivalence related to both Dolbeault and anti-Dolbeault cohomology. This class of weak equivalences naturally defines a stronger formality notion. In particular, satisfying the ddbar-Lemma property does not imply formality in this new sense. The goal of this talk is to introduce a novel operadic framework designed to understand this homotopical situation. I will present pluripotential A-infinity algebras as well as a homotopy transfer theorem based on this strong notion of weak equivalence.
Lukas Waas: The topological stratified homotopy hypothesis
Roughly speaking, the homotopy hypothesis - due to Grothendieck - states that the homotopy theory of spaces should be the same as the homotopy theory of infinity-groupoids. Ayala, Francis and Rosenblyum conjectured a stratified topological analogue of this principle: The homotopy theory of (topological) stratified spaces should be the same as the homotopy theory of layered infinity-categories, i.e. such infinity-categories in which every endomorphism is an isomorphism. We are going to present a formal interpretation of this statement. Namely, we prove the existence of a simplicial semi-model structure for stratified spaces in which most geometrically relevant examples – such as Whitney stratified spaces and PL pseudomanifolds - are bifibrant. We then prove a Quillen equivalence (in terms of Lurie’s exit-path construction) of this model category with a left Bousfield localization of the Joyal model structure presenting layered infinity-categories. In particular, the following interpretation of the topological homotopy hypothesis follows: Lurie’s exit-path construction induces an equivalence between the localization of bifibrant stratified spaces at stratified homotopy equivalences and the homotopy theory of layered infinity-categories. |
