Abstract
We prove a strong controlled generalization of a theorem of Bestvina and Walsh, which states that a (k+1)–connected map from a topological n–manifold to a polyhedron, 2k+3≤n, is homotopic to a UVk–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≥5, having the disjoint disks property satisfies the linear UV⌊(n−3)∕2⌋–approximation property for maps to compact ANRs. The method of proof is general enough to show that any compact ENR satisfying the disjoint (k+1)–disks property has the linear UVk–approximation property.
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