Homotopy BV-algebra structure on the double cobar construction

Abstract

We show that the double cobar construction, ${\Omega }^2{C}_{\ast }(X)$, of a simplicial set $X$ is a homotopy BV-algebra if $X$ is a double suspension, or if $X$ is 2-reduced and the coefficient ring contains the ring of rational numbers $\mathbb{Q}$. Indeed, the Connes-Moscovici operator defines the desired homotopy BV-algebra structure on ${\Omega }^2{C}_{\ast }(X)$ when the antipode $S:\Omega C_{\ast }(X)\to \Omega C_{\ast }(X)$ is involutive. We proceed by defining a family of obstructions $O_n:{\tilde{C}}_{\ast }(X)\to {\tilde{C}}_{\ast }(X{)}^{\otimes n}$, $n\ge 2$ measuring the difference $S^2-Id$. When $X$ is a suspension, the only obstruction remaining is $O_2:=E^{1,1}-\tau E^{1,1}$ where $E^{1,1}$ is the dual of the ${⌣}_1$-product. When $X$ is a double suspension the obstructions vanish.