ESTIMATES OF THE COHOMOLOGICAL DIMENSION OF DECOMPOSITION SPACES

The setting is a proper surjection f:X --> Y between metrizable spaces such that each Cech cohomology group H(i)(f-1(y); Z) is finitely generated for each y is-an- element-of Y. Estimates (upper bounds) are established of the dimension of Y in terms of the dimension of X, the groups H(i)(f-1(y); Z) for y is-an-element-of Y, and the dimensions in which these groups 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 is-an-element-of Y, dim Y less-than-or-equal-to dim X + k.