We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countabledimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold.If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X × [0, 1] are Q-manifolds as well. We construct a countable family x of spaces with DDP and cd-AP such that no space X ∈ xis homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ x is homeomorphic to Q. We also show that no uncountable family x with such properties exists.