#### Volume 20, issue 1 (2016)

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Kontsevich's Swiss cheese conjecture

### Justin Thomas

Geometry & Topology 20 (2016) 1–48
##### 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.

##### Keywords
operads, Hochschild cohomology, $E_n$ algebras, Swiss cheese, Deligne's conjecture
##### Mathematical Subject Classification 2010
Primary: 13D03, 18D50
Secondary: 18G55
##### Publication
Received: 5 October 2012
Revised: 22 February 2015
Accepted: 28 March 2015
Published: 29 February 2016
Proposed: Haynes Miller
Seconded: Peter Teichner, Bill Dwyer
##### Authors
 Justin Thomas 404 23rd Ave NE Norman, OK 73071 USA Department of Mathematics University of Notre Dame 255 Hurley Hall Notre Dame, IN 46556-4618 USA