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.

中文翻译：

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Szczarba扭曲的共链和Eilenberg-Zilber地图

我们表明，Szczarba的扭曲笛卡尔乘积的扭曲共链与Shih构造的扭曲共链基本相同。更准确地说，如果人们使用经典Eilenberg-Mac Lane同态的“反转”版本进行Eilenberg-Zilber收缩，则可以通过基本扰动引理获得Szczarba的加捻共链。在此过程中，我们证明了涉及这些同型体的几个新身份。