Detecting Cohomologically Stable Mappings

Abstract

Let $f$ be a cohomologically stable mapping defined from a compactum $X$ to the $(n + 1)[ - {\text {cell}}{I^{n + 1}}$, let $\pi :{I^{n + 1}} \to {I^n}$ be the projection, and let $A = {I^n} \times \{ 1\}$ and $B = {I^n} \times \{ - 1\}$ be opposite faces of ${I^{n + 1}}$. If $S$ is a separator or a continuum-wise separator of ${f^{ - 1}}(A)$ and ${f^{ - 1}}(B)$ in $X$, then $\pi f |S$ is cohomologically stable. This result is used to extend certain computations of cohomological dimension that are due to Walsh, who considered only the special case of the identity mapping on ${I^{n + 1}}$.