Kontsevich’s Swiss cheese conjecture

Thomas, Justin¹

¹ 404 23rd Ave NE, Norman, OK 73071, USA, Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556-4618, USA

Geometry & topology, Tome 20 (2016) no. 1, pp. 1-48.

Résumé

We prove a conjecture of Kontsevich, which states that if $A$ is an $E_d-1$ algebra then the Hochschild cochain object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$ algebra acting on an $E_{d-1}$ algebra was defined by Kontsevich using the Swiss cheese operad of Voronov. The degree $0$ and $1$ pieces of the Swiss cheese operad can be used to build a cofibrant model for $A$ as an $E_{d-1}$–$A$–module. The theorem amounts to the fact that the Swiss cheese operad is generated up to homotopy by its degree $0$ and $1$ pieces.

Détail

DOI : 10.2140/gt.2016.20.1

Classification : 13D03, 18D50, 18G55
Keywords: operads, Hochschild cohomology, $E_n$ algebras, Swiss cheese, Deligne's conjecture

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