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Formality of the framed little 2-discs operad and semidirect products

Jeff Giansiracusa, Paolo Salvatore, Jeffrey Giansiracusa

Contemporary Mathematics, Volume: 519, Pages: 115 - 121

Swansea University Author: Jeffrey Giansiracusa

Abstract

We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the oper...

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Published in: Contemporary Mathematics
Published: AMS 2010
URI: https://cronfa.swan.ac.uk/Record/cronfa8376
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spelling 2013-06-17T14:50:32.5396603 v2 8376 2012-02-22 Formality of the framed little 2-discs operad and semidirect products 03c4f93e1b94af60eb0c18c892b0c1d9 Jeffrey Giansiracusa Jeffrey Giansiracusa true false 2012-02-22 FGSEN We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the operadic semidirect product with the circle group. The idea of our proof is to show that Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions and so one can essentially take the operadic semidirect product with the homology of S^1 everywhere to obtain a chain of quasi-isomorphisms between the homology and the chains of the framed 2-discs.In Homotopy theory of function spaces and related topics, 115–121, Contemp. Math., 519, Amer. Math. Soc., Providence, RI, 2010. arXiv:0911.4428 Conference Paper/Proceeding/Abstract Contemporary Mathematics 519 115 121 AMS Framed little discs, formality, semidirect product 31 12 2010 2010-12-31 COLLEGE NANME Science and Engineering - Faculty COLLEGE CODE FGSEN Swansea University 2013-06-17T14:50:32.5396603 2012-02-22T13:37:15.0000000 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jeff Giansiracusa 1 Paolo Salvatore 2 Jeffrey Giansiracusa 3
title Formality of the framed little 2-discs operad and semidirect products
spellingShingle Formality of the framed little 2-discs operad and semidirect products
Jeffrey Giansiracusa
title_short Formality of the framed little 2-discs operad and semidirect products
title_full Formality of the framed little 2-discs operad and semidirect products
title_fullStr Formality of the framed little 2-discs operad and semidirect products
title_full_unstemmed Formality of the framed little 2-discs operad and semidirect products
title_sort Formality of the framed little 2-discs operad and semidirect products
author_id_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9
author_id_fullname_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9_***_Jeffrey Giansiracusa
author Jeffrey Giansiracusa
author2 Jeff Giansiracusa
Paolo Salvatore
Jeffrey Giansiracusa
format Conference Paper/Proceeding/Abstract
container_title Contemporary Mathematics
container_volume 519
container_start_page 115
publishDate 2010
institution Swansea University
publisher AMS
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
document_store_str 0
active_str 0
description We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the operadic semidirect product with the circle group. The idea of our proof is to show that Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions and so one can essentially take the operadic semidirect product with the homology of S^1 everywhere to obtain a chain of quasi-isomorphisms between the homology and the chains of the framed 2-discs.In Homotopy theory of function spaces and related topics, 115–121, Contemp. Math., 519, Amer. Math. Soc., Providence, RI, 2010. arXiv:0911.4428
published_date 2010-12-31T03:10:29Z
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score 11.012678

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