Szczarba's twisting cochain is comultiplicative

TL;DR

This paper proves Szczarba's twisting cochain is comultiplicative, establishing a quasi-isomorphism of dg bialgebras and providing a natural model for fibre bundles, with applications to covering spaces and spectral sequences.

Contribution

It demonstrates the comultiplicativity of Szczarba's twisting cochain and constructs a natural dgc model for fibre bundles, advancing algebraic topology tools.

Findings

Szczarba's twisting cochain is proven to be comultiplicative.

The induced map is a quasi-isomorphism of dg bialgebras.

A natural dgc model for fibre bundles is established.

Abstract

We prove that Szczarba's twisting cochain is comultiplicative. In particular, the induced map from the cobar construction of the chains on a 1-reduced simplicial set X to the chains on the Kan loop group of X is a quasi-isomorphism of dg bialgebras. We also show that Szczarba's twisted shuffle map is a dgc map connecting a twisted Cartesian product with the associated twisted tensor product. This gives a natural dgc model for fibre bundles. We apply our results to finite covering spaces and to the Serre spectral sequence.