# Models of operads and the realization problem of E_n-operads

## Topologie

Lieu:
Salle de réunion du M2
Orateur:
Benoit Fresse
Dates:
Vendredi, 4 Juin, 2021 - 14:00 - 15:00
Résumé:
The classical Sullivan model of the rational homotopy of spaces has an extension to operads. I will explain the definition of a generalization of this model for the p-complete homotopy type of operads in spaces.

The Sullivan model has been applied to E_n-operads, defined as the class of operads in spaces that have the same homotopy type as the operad of little n-discs (or n-cubes). Recall that the homology of the operad of little n-discs is identified with the n-Poisson operad, the operad that governs the category of graded Poisson algebras with a Poisson bracket of degree n-1. In the rational context, one can actually prove that every operad in spaces whose homology is equal to the n-Poisson operad has the rational homotopy type of the operad of little n-discs (provided that the operad is also equipped with an action of an involution in dimension n=4k). This statement implies that E_n-operads are rationally formal: they have the same rational homotopy type as a Sullivan realization of their cohomology cooperad.

I will explain the definition of a generalization of this Sullivan realization of the cohomology cooperad of little n-discs in the p-complete setting. The construction relies on integral versions of graph complexes. But yet the open question, equivalent to a formality conjecture for E_n-operads mod p, is whether this operads in spaces has the same p-complete homotopy type as the operad of little n-discs itself or defines another non equivalent realization of the n-Poisson operad.

The talk will mainly be based on a joint research with Lorenzo Guerra.