Volume 20, issue 1 (2016)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 Ed 1 algebra then the Hochschild cochain object of A is the universal Ed algebra acting on A. The notion of an Ed algebra acting on an Ed1 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 Ed1A–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