Homological embedding properties of the fibers of a map and the dimension of its image

Abstract:

A relationship is established between the homological codimension of the point inverses of a map and the dimension of its image. An infinite-dimensional version leads to the conclusion that the image of a proper map defined on Hilbert space cannot be countable dimensional. A finite-dimensional version yields: if $g:{M^n} \to Y$ is a proper map, ${M^n}$ is a $G$-orientable $n$-manifold without boundary, and $\dim Y \leqslant k$, then there is a point $y \in Y$ and an integer $i \geqslant n - k$ such that $\check {H}^i (g^{-1}(y);G) \ne 0$.