  We show that there are homotopy equivalences $h:N\to M$ between closed
manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but
which are not homotopic to homeomorphisms. The phenomenon is based on
construction of cell-like maps that kill certain $\mathbb L$-classes. The image
space in these constructions is necessarily infinite-dimensional. In dimension
$>6$ we classify all such homotopy equivalences. As an application, we show
that such homotopy equivalences are realized by deformations of Riemannian
manifolds in Gromov-Hausdorff space preserving a contractibility function.
