Szczarba’s twisting cochain is comultiplicative

IF 0.8 4区数学 Q2 MATHEMATICS

Homology Homotopy and Applications Pub Date : 2024-05-01 DOI:10.4310/hha.2024.v26.n1.a18

Matthias Franz

引用次数: 0

Abstract

We prove that Szczarba’s twisting cochain is comultiplicative. In particular, the induced map from the cobar construction $\Omega C(X)$ 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 $\operatorname{dg}$ bialgebras. We also show that Szczarba’s twisted shuffle map is a $\operatorname{dgc}$ map connecting a twisted Cartesian product with the associated twisted tensor product. This gives a natural $\operatorname{dgc}$ model for fibre bundles.We apply our results to finite covering spaces and to the Serre spectral sequence.

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Szczarba 的扭曲共链具有乘法性

我们证明了Szczarba的扭曲共链是乘法的。特别是，从$1$还原单纯集$X$上的链的科巴构造$\Omega C(X)$到$C(GX)$，即$X$的坎环群上的链的诱导映射，是$\operatorname{dg}$双玻的准同构。我们还证明了斯查尔巴的扭曲洗牌映射是一个连接扭曲笛卡尔积和相关扭曲张量积的\operatorname{dgc}$映射。这就为纤维束提供了一个自然的 $\operatorname{dgc}$ 模型。我们将我们的结果应用于有限覆盖空间和塞尔谱序列。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Homology Homotopy and Applications 数学-数学

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Homology, Homotopy and Applications is a refereed journal which publishes high-quality papers in the general area of homotopy theory and algebraic topology, as well as applications of the ideas and results in this area. This means applications in the broadest possible sense, i.e. applications to other parts of mathematics such as number theory and algebraic geometry, as well as to areas outside of mathematics, such as computer science, physics, and statistics. Homotopy theory is also intended to be interpreted broadly, including algebraic K-theory, model categories, homotopy theory of varieties, etc. We particularly encourage innovative papers which point the way toward new applications of the subject.