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# Homotopy Decompositions Involving the Loops of Coassociative Co-$H$ Spaces

Published:2003-02-01
Printed: Feb 2003
• Stephen D. Theriault
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## Abstract

James gave an integral homotopy decomposition of $\Sigma\Omega\Sigma X$, Hilton-Milnor one for $\Omega (\Sigma X\vee\Sigma Y)$, and Cohen-Wu gave $p$-local decompositions of $\Omega\Sigma X$ if $X$ is a suspension. All are natural. Using idempotents and telescopes we show that the James and Hilton-Milnor decompositions have analogues when the suspensions are replaced by coassociative co-$H$ spaces, and the Cohen-Wu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative co-$H$ space.
 MSC Classifications: 55P35 - Loop spaces 55P45 - $H$-spaces and duals