Homotopy BV-algebra structure on the double cobar construction.

We show that the double cobar construction, Ω 2 C ⁎ ( 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 Ω 2 C ⁎ ( X ) when the antipode S : Ω C ⁎ ( X ) → Ω C ⁎ ( X ) is involutive. We proceed by defining a family of obstructions O n : C ˜ ⁎ ( X ) → C ˜ ⁎ ( X ) ⊗ n , n ≥ 2 by computing S 2 − Id . When X is a suspension, the only obstruction remaining is O 2 : = E 1 , 1 − τ E 1 , 1 where E 1 , 1 is the dual of the ⌣ 1 -product. When X is a double suspension the obstructions vanish. [ABSTRACT FROM AUTHOR]