Groups, cacti and framed little discs

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Abstract

Let G be a topological group. Then the based loopspace of G is an algebra over the cacti operad, while the double loopspace 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 E-algebras, where 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 obtained by taking the homology of the loopspace of G and of the double loopspace of BG.
Original languageEnglish
Pages (from-to)2597-2636
Number of pages40
JournalTransactions of the American Mathematical Society
Volume365
Issue number5
Early online date1 Oct 2012
DOIs
Publication statusPublished - May 2013

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