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Szczarba’s twisting cochain and the Eilenberg–Zilber maps

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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–Mac Lane homotopy for the Eilenberg–Zilber contraction. Along the way we prove several new identities involving these homotopies.

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Notes

  1. This is a misnomer. All three maps were introduced by Eilenberg–Mac Lane [3, Eq. (5.3)], [4, Eqs. (2.8), (2.13)], with the obvious inspiration for the Alexander–Whitney map.

  2. Contrary to the claim made in [15, Thm. 6.2], Eilenberg–Mac Lane’s homotopy (3.4) does not satisfy \(d(H) = 1-\nabla AW\). The sign exponent \(n-p-q=1\) is erroneously not taken into account when the equations (65) and (70) are compared in [15, p. 85].

  3. The proof of the identity \(HH=0\) in [17, p. 26] implicitly uses an argument like Lemma 2.1 (i).

  4. Given that Gugenheim swapped the factors of the twisted tensor product compared to Shih, he might have been thinking of the homotopy \(\tilde{H}\) defined in (4.8) below, compare Propositions 4.5 and 4.6.

  5. The sign in the twisting cochain condition is given incorrectly in [17, p. 29] and [16, Def. 3.2.6], but the same sign is correct in [18, p. 198] and [9, Def. 2.3]. This difference is explained by the different orders of the factors in the twisted tensor product, compare [12, Def. II.1.4].

  6. Szczarba calls \(\varphi (x)=-(-1)^{|x|}\,t(x)\) a twisting cochain. Also note that he does not use the Koszul sign convention, so that he has \((f\otimes g)(x\otimes y)=f(x)\,g(y)\) without the sign \((-1)^{|g||x|}\). That Szczarba’s maps \(t_{{\mathrm {sz}}}\) and \(\psi \) are well-defined on normalized complexes is shown in [7, App. B].

  7. In the definition of \(\psi \) in [18, p. 201] the upper summation index should read “\(p!\)”.

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Acknowledgements

I thank Francis Sergeraert for sending me Rubio’s thesis [16].

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  1. Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada

    Matthias Franz

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The author was supported by an NSERC Discovery Grant.

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Franz, M. Szczarba’s twisting cochain and the Eilenberg–Zilber maps. Collect. Math. 72, 569–586 (2021). https://doi.org/10.1007/s13348-020-00299-x

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