Kontsevich's Swiss cheese conjecture

Abstract

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.