Szczarba's twisting cochain is comultiplicative

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.

arXiv comment28 pages; references added, minor changes

References in corpus (8)

• Homotopy G-algebras and moduli space operad
• Combinatorial operad actions on cochains
• The spectral sequence of an equivariant chain complex and homology with local coefficients
• The loop group and the cobar construction
• A canonical enriched Adams-Hilton model for simplicial sets
• The cohomology rings of homogeneous spaces
• Szczarba's twisting cochain and the Eilenberg-Zilber maps
• Comultiplication in the Serre Spectral Sequence

Cited by in corpus (2)

• Homotopy Gerstenhaber algebras are strongly homotopy commutative
• The cohomology of biquotients via a product on the two-sided bar construction