On \(Q\)-manifolds bundles (Q2125144)

In the book [\textit{J. van Mill} (ed.) and \textit{G. M. Reed} (ed.), Open problems in topology. Amsterdam etc.: North-Holland (1990; Zbl 0718.54001)] \textit{J. West} raised the question whether there exists a Čech-cohomology version of the fibred general position theory of \textit{H. Toruńczyk} and \textit{J. West} [Fibrations and bundles with Hilbert cube manifold fibers. Providence, RI: American Mathematical Society (AMS) (1989; Zbl 0689.57013)] along the lines of \textit{R. J. Daverman} and \textit{J. J. Walsh} [Am. J. Math. 103, 411--435 (1981; Zbl 0538.57006)] that detects the \(Q\)-manifold bundles among the \(ANR\)-fibrations with compact fibers. In the present paper, the authors present a homological version of Torunczyk-West's characterization of \(Q\)-manifold bundles over \(C\)-spaces, thereby providing a partial answer to the question raised by West.