$\mathit{UV}^k$–mappings on homology manifolds

Abstract

We prove a strong controlled generalization of a theorem of Bestvina and Walsh, which states that a $\left(k+1\right)$–connected map from a topological $n$–manifold to a polyhedron, $2k+3\le n$, is homotopic to a ${UV}^k$–map, that is, a surjection whose point preimages are, in some sense, $k$–connected. One consequence of our main result is that a compact ENR homology $n$–manifold, $n\ge 5$, having the disjoint disks property satisfies the linear ${UV}^{\lfloor \left(n-3\right)∕2\rfloor }$–approximation property for maps to compact ANRs. The method of proof is general enough to show that any compact ENR satisfying the disjoint $\left(k+1\right)$–disks property has the linear ${UV}^k$–approximation property.