Essential maps and manifolds

Abstract:

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