  We investigate to what extend the density of $Z_n$-maps in the
characterization of $Q$-manifolds, and the density of maps $f\in C(\mathbb
N\times Q,X)$ having discrete images in the $l_2$-manifolds characterization
can be weakened to the density of homological $Z_n$-maps and homological
$Z$-maps, respectively. As a result, we obtain homological characterizations of
$Q$-manifolds and $l_2$-manifolds.
