Homological embedding properties of the fibers of a map and the dimension of its image
John J. Walsh · Proceedings of the American Mathematical Society · 1982
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 → Y g:{M^n} \to Y is a proper map, M n {M^n} is a G G -orientable n n -manifold without boundary, and dim Y ⩽ k \dim Y \leqslant k , then there is a point y ∈ Y y \in Y and an integer i ⩾ n − k i \geqslant n - k such that H ˇ i ( g − 1 ( y ) ; G ) ≠ 0 \check {H}^i (g^{-1}(y);G) e 0 .