November 2, 2023

Operads and graph complexes, applications to the embedding calculus, relationships with the Grothendieck-Teichmüller groups

Benoit Fresse, Lille University, France RIMS Kyoto · JP:09:00 · FR:01:00

I will survey the results of a joint research with Victor Turchin and Thomas Willwacher on applications of operads to the embedding calculus and the theory of Grothendieck-Teichmüller groups.
To be specific, the main results, which I will aim to explain in my talk, include:
- a combinatorial description, in terms of graph complexes, of the spaces of emdeddings of Euclidean spaces in the realm of the rational homotopy theory
- a description, in terms of graph complexes too, of spaces of rational homotopy automorphisms of operads, which generalize the rational Grothendieck-Teichmüller group

Our work is mainly based on a study of the rational homotopy of E_n-operads, a class of operads, defined by a reference model, the operads of little n-discs (or n-cubes), and which can be used to govern a hierarchy of homotopy commutative structures, from fully homotopy associative but non-commutative (n=1) up to fully homotopy associative and commutative (n=infinity). I will explain that certain combinatorial models of E_n-operads, defined in terms of graphs, can be used to get a graph complex description of mapping spaces associated to E_n-operads over the rationals.

The applications to the study of embeddings goes through the Goodwillie-Weiss embedding calculus, which relates the spaces of embeddings of Euclidean spaces to such operadic mapping spaces. The relationship with the Grothendieck-Teichmüller groups comes from the observation that the space of homotopy automorphisms of E_n-operads is identified with the Grothendieck-Teichmüller group in the case n=2.

Depending on time and on the interest of the audience, I will address one or the other of these connections with more details. I could also explain a generalization of the result on the rational homotopy of emdeddings of Euclidean spaces to the case of the space of embeddings of more general manifolds into an Euclidean space.

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