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The framed little 2-discs operad and diffeomorphisms of handlebodies

J Giansiracusa, Jeffrey Giansiracusa Orcid Logo

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

Swansea University Author: Jeffrey Giansiracusa Orcid Logo

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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
first_indexed 2013-07-23T12:10:34Z
last_indexed 2018-02-09T04:44:25Z
id cronfa13605
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spelling 2015-07-31T17:07:25.2570716 v2 13605 2012-12-10 The framed little 2-discs operad and diffeomorphisms of handlebodies 03c4f93e1b94af60eb0c18c892b0c1d9 0000-0003-4252-0058 Jeffrey Giansiracusa Jeffrey Giansiracusa true false 2012-12-10 MACS 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. Journal Article Journal of Topology 4 4 919 941 1753-8416 1753-8424 handlebodies, modular envelope, operads, moduli space, cyclic operad, modular operad, graph homology, ribbon graphs, mobius graphs, framed little discs, Batalin-Vilkovisky 1 11 2011 2011-11-01 10.1112/jtopol/jtr021 http://jtopol.oxfordjournals.org/content/4/4/919.abstract COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2015-07-31T17:07:25.2570716 2012-12-10T14:36:00.4003761 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics J Giansiracusa 1 Jeffrey Giansiracusa 0000-0003-4252-0058 2
title The framed little 2-discs operad and diffeomorphisms of handlebodies
spellingShingle The framed little 2-discs operad and diffeomorphisms of handlebodies
Jeffrey Giansiracusa
title_short The framed little 2-discs operad and diffeomorphisms of handlebodies
title_full The framed little 2-discs operad and diffeomorphisms of handlebodies
title_fullStr The framed little 2-discs operad and diffeomorphisms of handlebodies
title_full_unstemmed The framed little 2-discs operad and diffeomorphisms of handlebodies
title_sort The framed little 2-discs operad and diffeomorphisms of handlebodies
author_id_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9
author_id_fullname_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9_***_Jeffrey Giansiracusa
author Jeffrey Giansiracusa
author2 J Giansiracusa
Jeffrey Giansiracusa
format Journal article
container_title Journal of Topology
container_volume 4
container_issue 4
container_start_page 919
publishDate 2011
institution Swansea University
issn 1753-8416
1753-8424
doi_str_mv 10.1112/jtopol/jtr021
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url http://jtopol.oxfordjournals.org/content/4/4/919.abstract
document_store_str 0
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description 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.
published_date 2011-11-01T18:25:03Z
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