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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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