\(UV^{k}\)-mappings on homology manifolds
This paper is about approximating maps with highly connected homotopy fiber by maps whose actual fibers are highly connected. More specifically, the authors 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 \(M\) to a polyhedron \(B\), with \(2k + 3 \leq n\), is homotopic to a \(UV^k\)-map, that is, a surjection whose point preimages are \(k\)-connected (in the sense of shape theory). It is a ``strong controlled'' generalization in the sense that the polyhedron \(B\) comes equipped with a map \(p: B\to Y\) to a metric space, a number \(\epsilon > 0\) is given, and the map \(f: M\to B\) is assumed to be \(UV^k(\epsilon)\) over \(Y\) in an appropriate sense; one then is able to conclude that \(f\) is \((C(k)\cdot\epsilon)\)-homotopicover \(Y\) to a \(UV^k\)-map, where \(C(k)\) is a constant depending only on \(k\). Moreover, one only needs to assume that \(M\) is a compact, connected ENR satisfying the disjoint \((k+1)\)-disks property and \(B\) is a compact \(ANR\). A consequence is that a compact \(ENR\) homology \(n\)-manifold, \(n \geq 5\), having the disjoint disks property satisfies the above linear \(UV^{\lfloor (n - 3)/2\rfloor}\)--approximation property for maps to compact \(ANR\)s. This is the first controlled homotopy improvement result for maps from spaces having no local linear structure. A key part of the proof is a \(UV\)-expansion theorem for compact ANRs. The methods are new, even in the case of PL or smooth manifolds, and provide an alternative proof of the Bestvina--Walsh theorem.
scientific article; zbMATH DE number 6177499
scientific article