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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The loop group and the cobar construction


Authors: Kathryn Hess and Andrew Tonks
Journal: Proc. Amer. Math. Soc. 138 (2010), 1861-1876
MSC (2010): Primary 55P35; Secondary 16T05, 18G30, 55U10, 57T05, 57T30
Published electronically: December 21, 2009
MathSciNet review: 2587471
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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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Additional Information

Kathryn Hess
Affiliation: Institut de géométrie, algèbre et topologie (IGAT), École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Email: kathryn.hess@epfl.ch

Andrew Tonks
Affiliation: Statistics, OR and Mathematics Research Centre (STORM), London Metropolitan University, 166–220 Holloway Road, London N7 8DB, United Kingdom
Email: a.tonks@londonmet.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10238-1
PII: S 0002-9939(09)10238-1
Keywords: Loop space, cobar construction, strong deformation retract, acyclic models
Received by editor(s): March 13, 2009
Published electronically: December 21, 2009
Communicated by: Brooke Shipley
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.