Groups, cacti and framed little discs
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- by Richard Hepworth PDF
- Trans. Amer. Math. Soc. 365 (2013), 2597-2636
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 $H_\ast (\Omega ^2 BG)$.References
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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
- Received by editor(s): December 15, 2010
- Received by editor(s) in revised form: September 19, 2011
- Published electronically: October 1, 2012
- © Copyright 2012 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
- MathSciNet review: 3020110