On the homology of the double cobar construction of a double suspension

TL;DR

This paper proves that the homology of the double cobar construction of a double suspension forms a free BV-algebra, revealing new algebraic structures and formality properties in this context.

Contribution

It establishes the formality of the double cobar construction of a double suspension and shows its homology is a free BV-algebra, including characteristic two cases.

Findings

Homology of the double cobar construction is a free BV-algebra.

Formality theorem for the double cobar construction of a double suspension.

In characteristic two, the underlying 2-restricted Gerstenhaber algebra also exhibits similar properties.

Abstract

The double cobar construction of a double suspension comes with a Connes-Moscovici structure, that is a homotopy G-algebra (or Gerstenhaber-Voronov algebra) structure together with a particular BV-operator up to a homotopy. We show that the homology of the double cobar construction of a double suspension is a free BV-algebra. In characteristic two, a similar result holds for the underlying $2$-restricted Gerstenhaber algebra. These facts rely on a formality theorem for the double cobar construction of a double suspension.