#### Volume 13, issue 4 (2013)

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

### John Bryant, Steve Ferry and Washington Mio

Algebraic & Geometric Topology 13 (2013) 2141–2170
##### 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 ${UV}^{⌊\left(n-3\right)∕2⌋}$–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.

##### Keywords
absolute neighborhood retract, homology manifolds, $UV^k$–mappings
##### Mathematical Subject Classification 2010
Primary: 57Q35, 57Q30, 57N99, 57P99