Homotopy bialgebraic structures in geometry and topology

Abstract: It is well known from the PhD thesis of Jim Stasheff that the homotopy theory of associative algebras is encoded by homotopy associative algebras, aka A_infini-algebras, since this latter notion carries infini-morphisms and satisfies a homotopy transfer theorem, for instance. A_infini-algebra structures encode the topological data of a space on the level of cochain complexes. When one wants to encode more data, like the Poincaré duality of manifolds, string topology, or non-commutative derived geometry, then one has to consider further structural operations, like symmetric bitensors or double brackets. The purpose of this talk will be to present the associated new types of homotopy bialgebras, to explain their relationship, and to show that they admit suitable homotopy properties like infini-morphisms and homotopy transfer theorem. To mention them, we will treat pre-Calabi—Yau algebras, homotopy double Poisson bialgebras, and homotopy infinitesimal balanced bialgebras. This is based on a joint work with Johan LERAY available at arXiv:2203.05062.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar