Groups, cacti and framed little discs

TL;DR

This paper establishes an equivalence between algebraic structures on loopspaces of topological groups and their classifying spaces, unifying cacti and framed little discs operads and recovering known BV-algebra isomorphisms.

Contribution

It proves the weak equivalence of E-algebras over cacti and framed little discs operads, linking loopspace structures and BV-algebras in a new unified framework.

Findings

Based loopspace of G is an algebra over cacti operad.

Double loopspace of BG is an algebra over framed little discs operad.

These two algebra structures are weakly equivalent E-algebras.

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.