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arXiv · 1009.3260

Groups, cacti and framed little discs

Richard Hepworth
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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.

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Richard Hepworth. 2010-09-16. Groups, cacti and framed little discs. https://doi.org/10.1090/s0002-9947-2012-05734-5

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