Szczarba's twisting cochain and the Eilenberg-Zilber maps

TL;DR

This paper demonstrates that Szczarba's twisting cochain for twisted Cartesian products aligns with Shih's construction, using the basic perturbation lemma and a reversed Eilenberg-MacLane homotopy, and introduces new identities for these homotopies.

Contribution

It reveals the equivalence of Szczarba's and Shih's twisting cochains and develops new identities involving Eilenberg-Zilber homotopies.

Findings

Szczarba's twisting cochain matches Shih's construction

The basic perturbation lemma can derive Szczarba's cochain with a reversed homotopy

New identities involving Eilenberg-MacLane homotopies are established

Abstract

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-MacLane homotopy for the Eilenberg-Zilber contraction. Along the way we prove several new identities involving these homotopies.