The loop group and the cobar construction

TL;DR

This paper establishes a natural strong deformation retract between Adams' cobar construction and the chains on the Kan loop group for 1-reduced simplicial sets, enabling algebraic transfer techniques.

Contribution

It proves the existence of a strong deformation retract between the cobar construction and the chains on the Kan loop group, extending to all 0-reduced simplicial sets.

Findings

The cobar construction is a strong deformation retract of the chains on the Kan loop group for 1-reduced simplicial sets.

This retract exists for all 0-reduced simplicial sets.

The result facilitates applying homological algebra tools to transfer algebraic structures.

Abstract

We prove that for any 1-reduced simplicial set X, Adams' cobar construction, \Omega CX, on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX, opening up the possibility of applying the tools of homological algebra to transfering perturbations of algebraic structure from the latter to the former. In order to prove our theorem, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation retract for all 0-reduced simplicial sets.