On Anosov–Weil problem

The setting is a proper surjection f:X → Y between metrizable spaces such that each Čech cohomology group Hi (f-1(y);Z) is finitely generated for each y ϵ Y. Estimates (upper bounds) are established of the dimension of Y in terms of the dimension of X, the groups Hi(f-1(y);Z) for y ϵ Y, and the dimensions in which these groups are nonzero. The estimates can be viewed as generalizations of the classical result that, for a proper surjection f:X → Y between metrizable spaces with cardinality of f-1(y) at most k+1 for y ϵ Y, dim Y ⩽ dim X + k.