Journal article 1190 views
The framed little 2-discs operad and diffeomorphisms of handlebodies
J Giansiracusa,
Jeffrey Giansiracusa
Journal of Topology, Volume: 4, Issue: 4, Pages: 919 - 941
Swansea University Author: Jeffrey Giansiracusa
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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...
| Published in: | Journal of Topology |
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| ISSN: | 1753-8416 1753-8424 |
| Published: |
2011
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| Online Access: |
Check full text
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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. |
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| Keywords: |
handlebodies, modular envelope, operads, moduli space, cyclic operad, modular operad, graph homology, ribbon graphs, mobius graphs, framed little discs, Batalin-Vilkovisky |
| College: |
Faculty of Science and Engineering |
| Issue: |
4 |
| Start Page: |
919 |
| End Page: |
941 |