Maps into homotopy coalgebras (Q2502940)

\textit{T. Ganea} [Invent. Math. 9, 185--197 (1970; Zbl 0194.55103)] introduced the notion of homotopy coalgebra to study the conditions when a co-\(H\)-space is a suspension. For studying the classical problem when a space is a \(k\)-fold suspension, \textit{M. Arkowitz} and \textit{M. Golasiński} [Hiroshima Math. J. 27, No. 2, 209--220 (1997; Zbl 0894.55004)] introduced the more general concept of \(k\)-fold homotopy coalgebras. The main result of the paper under review gives a criterion for detecting \(k\)-fold homotopy coalgebras and homotopy coalgebra morphisms under certain connectivity conditions. As an application, the authors extend some classical results of \textit{L. Bernstein} and \textit{P. J. Hilton} [Topology 2, 73--82 (1963; Zbl 0115.40403)] and \textit{S. Saito} [Hiroshima Math. J. 6, 589--617 (1976; Zbl 0343.55012)] detecting \(k\)-fold suspensions.