Szczarba’s twisting cochain is comultiplicative

We prove that Szczarba's twisting cochain is comultiplicative.In particular, the induced map from the cobar construction C(X) of the chains on a 1-reduced simplicial set X to C(GX), the chains on the Kan loop group of X, is a quasiisomorphism of dg bialgebras.We also show that Szczarba's twisted shuffle map is a dgc map connecting a twisted Cartesian product with the associated twisted tensor product.This gives a natural dgc model for fibre bundles.We apply our results to finite covering spaces and to the Serre spectral sequence.