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Cancellation and homotopy rigidity of classical functors. (English) Zbl 1428.55005
The authors establish the existence of a unique decomposition of simply connected co-$$H$$-spaces and connected $$H$$-spaces into prime factors in the category of pointed $$p$$-local spaces of finite type.
B. Gray [Trans. Am. Math. Soc. 358, No. 8, 3305–3328 (2006; Zbl 1094.55009)] proved a Krull-Schmidt type theorem which states that each $$p$$-complete $$H$$-space can be uniquely decomposed to atomic pieces up to order and homotopy. The authors improve Gray’s result by not requiring the assumption $$p$$-complete, only $$p$$-local.
As an application, the authors show that $$\Sigma\Omega$$ and $$\Omega$$ are homotopy rigid functors on simply connected $$p$$-local co-$$H$$-spaces of finite type and $$\Omega\Sigma$$ and $$\Sigma$$ are homotopy ridgid on connected $$p$$-local $$H$$-spaces of finite type.

##### MSC:
 55P65 Homotopy functors in algebraic topology 55P45 $$H$$-spaces and duals 55P35 Loop spaces 55P40 Suspensions 55P30 Eckmann-Hilton duality
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