Volume 7, issue 1 (2003)

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Arc operads and arc algebras

Ralph M Kaufmann, Muriel Livernet and R C Penner

Geometry & Topology 7 (2003) 511–568

arXiv: math.GT/0209132


Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a combinatorial compactification due to Penner of a space closely related to Riemann’s moduli space. Algebras over these operads are shown to be Batalin–Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.

moduli of Surfaces, operads, Batalin–Vilkovisky algebras
Mathematical Subject Classification 2000
Primary: 32G15
Secondary: 18D50, 17BXX, 83E30
Forward citations
Received: 6 December 2002
Accepted: 8 June 2003
Published: 18 August 2003
Proposed: Shigeyuki Morita
Seconded: Ralph Cohen, Vaughan Jones
Ralph M Kaufmann
Oklahoma State University
Max Planck Institut für Mathematik
Muriel Livernet
Université Paris 13
R C Penner
University of Southern California
Los Angeles