  A classical theorem of Alexandroff states that every $n$-dimensional
compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a
number of generalizations obtained by various authors. We consider
extension-dimensional and infinite dimensional analogs of strong Cantor
manifolds, Mazurkiewicz manifolds, and $V^n$-continua, and prove corresponding
versions of the above theorem. We apply our results to show that each
homogeneous metrizable continuum which is not in a given class $\mathcal C$ is
a strong Cantor manifold (or at least a Cantor manifold) with respect to
$\mathcal C$. Here, the class $\mathcal C$ is one of four classes that are
defined in terms of dimension-like invariants.
  A class of spaces having bases of neighborhoods satisfying certain special
conditions is also considered.
