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Groups, cacti and framed little discs


Author: Richard Hepworth
Journal: Trans. Amer. Math. Soc. 365 (2013), 2597-2636
MSC (2010): Primary 18D50, 55P48, 57T99
DOI: https://doi.org/10.1090/S0002-9947-2012-05734-5
Published electronically: October 1, 2012
MathSciNet review: 3020110
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Abstract: Let $ G$ be a topological group. Then the based loopspace $ \Omega G$ of $ G$ is an algebra over the cacti operad, while the double loopspace $ \Omega ^2 BG$ of the classifying space of $ G$ is an algebra over the framed little discs operad. This paper shows that these two algebras are equivalent, in the sense that they are weakly equivalent $ \mathcal E$-algebras, where $ \mathcal E$ is an operad weakly equivalent to both framed little discs and cacti. We recover the equivalence between cacti and framed little discs, and Menichi's isomorphism between the BV-algebras $ H_\ast (\Omega G)$ and $ H_\ast (\Omega ^2 BG)$.


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Additional Information

Richard Hepworth
Affiliation: Department of Mathematical Sciences, Copenhagen University, Universitetspark 5, 2100 Copenhagen, Denmark
Address at time of publication: Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

DOI: https://doi.org/10.1090/S0002-9947-2012-05734-5
Received by editor(s): December 15, 2010
Received by editor(s) in revised form: September 19, 2011
Published electronically: October 1, 2012
Article copyright: © Copyright 2012 Richard Hepworth

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