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The framed little 2-discs operad and diffeomorphisms of handlebodies / J Giansiracusa

Journal of Topology, Volume: 4, Issue: 4, Pages: 919 - 941

Swansea University Author: Giansiracusa, Jeffrey

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DOI (Published version): 10.1112/jtopol/jtr021

Abstract

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (that is, the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeo...

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Published in: Journal of Topology
ISSN: 1753-8416 1753-8424
Published: 2011
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URI: https://cronfa.swan.ac.uk/Record/cronfa13605
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Abstract: The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (that is, the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ‘open string’ modular operad made from moduli spaces of Riemann surfaces with marked intervals on the boundary. Our technique also recovers a theorem of Braun that the derived modular envelope of the cyclic operad that describes associative algebras with involution is homotopy equivalent to the modular operad made from moduli spaces of unoriented Klein surfaces with open string gluing.
Keywords: handlebodies, modular envelope, operads, moduli space, cyclic operad, modular operad, graph homology, ribbon graphs, mobius graphs, framed little discs, Batalin-Vilkovisky
College: College of Science
Issue: 4
Start Page: 919
End Page: 941