  We investigate the classical Alexandroff-Borsuk problem in the category of
non-triangulable manifolds: Given an $n$-dimensional compact non-triangulable
manifold $M^n$ and $\varepsilon > 0$, does there exist an $\varepsilon$-map of
$M^n$ onto an $n$-dimensional finite polyhedron which induces a homotopy
equivalence?
