Szczarba’s twisting cochain is comultiplicative

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.

Keywords

Szczarba’s twisting cochain, Kan loop group, extended cobar construction, homotopy Gerstenhaber coalgebra, twisted Cartesian product, twisted tensor product, $\operatorname{dgc}$ model

2010 Mathematics Subject Classification

Primary 55U10. Secondary 55R20, 55T10.

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The author was supported by an NSERC Discovery Grant.

Received 14 June 2021

Received revised 25 May 2023

Accepted 31 May 2023

Published 1 May 2024