Kontsevich’s Swiss cheese conjecture

Thomas, Justin

doi:10.2140/gt.2016.20.1

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Kontsevich's Swiss cheese conjecture

Justin Thomas

Geometry & Topology 20 (2016) 1–48

DOI: 10.2140/gt.2016.20.1

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

References

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

Volume 20, issue 1 (2016)