Groups, cacti and framed little discs

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 $H_\ast (\Omega ^2 BG)$.