The loop group and the cobar construction

Hess, Kathryn; Tonks, Andrew

doi:10.1090/S0002-9939-09-10238-1

The loop group and the cobar construction
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by Kathryn Hess and Andrew Tonks

Proc. Amer. Math. Soc. 138 (2010), 1861-1876

DOI: https://doi.org/10.1090/S0002-9939-09-10238-1

Published electronically: December 21, 2009

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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 $GX$. In order to prove this result, 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.

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