The loop group and the cobar construction

Abstract

We prove that for any 1 1 -reduced simplicial set X X , Adams’ cobar construction Ω C X \Omega CX on the normalised chain complex of X X is naturally a strong deformation retract of the normalised chains C G X CGX on the Kan loop group G X GX . In order to prove this result, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation retract for all 0 0 -reduced simplicial sets.