Homotopy BV-algebra structure on the double cobar construction

Abstract

We show that the double cobar construction, Ω2C⁎(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 field of rational numbers Q. Indeed, the Connes–Moscovici operator defines the desired homotopy BV-algebra structure on Ω2C⁎(X) when the antipode S:ΩC⁎(X)→ΩC⁎(X) is involutive. We proceed by defining a family of obstructions On:C˜⁎(X)→C˜⁎(X)⊗n, n≥2 by computing S2−Id. When X is a suspension, the only obstruction remaining is O2:=E1,1−τE1,1 where E1,1 is the dual of the ⌣1-product. When X is a double suspension the obstructions vanish.