Szczarba's twisting cochain and the Eilenberg-Zilber maps

We show that Szczarba's twisting cochain for a twisted Cartesian product is essentially the same as the one constructed by Shih. More precisely, Szczarba's twisting cochain can be obtained via the basic perturbation lemma if one uses a 'reversed' version of the classical Eilenberg-Mac Lane homotopy for the Eilenberg-Zilber contraction. Along the way we prove several new identities involving these homotopies.

Automated summary

Our research demonstrates that Szczarba's twisting cochain for a twisted Cartesian product is essentially equivalent to the one created by Shih. Specifically, by utilizing a 'reversed' version of the classical Eilenberg-Mac Lane homotopy for the Eilenberg-Zilber contraction, Szczarba's twisting cochain can be derived via the basic perturbation lemma. Along the way, we establish several new identities involving these homotopies.