Essential Maps and Manifolds

Let (M, partial derivative M) be a compact n-manifold with boundary, orientable over a field K with characteristic q . For f : (Y, partial derivative Y) --> (M, partial derivative M) , with Y compact, and (X, partial derivative X) a compact pair, g: X --> M, let (P, partial derivative P) = {(y, x) is-an-element-of Y x (X, partial derivative X)f(y) = g(x)} denote the fibered product, with p as the projection to (X, partial derivative X) . In Cech-cohomology with coefficients K , we show that if H(n)(f) is injective then so is H* (p)-and a number of strengthenings, which point to a concept of q-essential map from one compact space to another.