Geometry & Topology, Vol. 7 (2003) Paper no. 15, pages 511--568.

Arc Operads and Arc Algebras

Ralph M Kaufmann, Muriel Livernet and RC Penner

Abstract. 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 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, and it similarly acts on the loop space of any topological space. 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.

Keywords. Moduli of Surfaces, Operads, Bataliv-Vilkovisky algebras

AMS subject classification. Primary: 32G15. Secondary: 18D50, 17BXX, 83E30.

DOI: 10.2140/gt.2003.7.511

E-print: arXiv:math.GT/0209132

Submitted to GT on 6 December 2002. Paper accepted 8 June 2003. Paper published 18 August 2003.

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Ralph M Kaufmann, Muriel Livernet and RC Penner

RMK: Oklahoma State University, Stillwater, USA
and
Max-Planck-Institut fur Mathematik, Bonn, Germany

ML: Universite Paris 13, France

RCP: University of Southern California, Los Angeles, USA

Email: ralphk@mpim-bonn.mpg.de, livernet@math.univ-paris13.fr, rpenner@math.usc.edu

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