The classical cobar construction ΩC for a coalgebra C (first introduced by Adams[1]) is an important algebraic concept motivated by the singular chain complex of a loop space ΩX. If X is a 1-reduced simplicial set with realization |X|, Adams proved that there is a natural isomorphism of homology groups

formula here

where C(X) si the coalgebra given by th chain complex on X abd tge Alexander-Whitney diagonal. Here the homology has coefficients in an abelian group A. The purpose of this paper is the extension of this result to the case of twisted coefficients given by π1Ω|X|−modules A, with π1Ω|X| = H2X.