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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
$\mathit{UV}^k$–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 (k + 1)–connected map from a topological n–manifold to a polyhedron, 2k + 3 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 5, having the disjoint disks property satisfies the linear UV (n3)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 UV k–approximation property.

Keywords
absolute neighborhood retract, homology manifolds, $UV^k$–mappings
Mathematical Subject Classification 2010
Primary: 57Q35, 57Q30, 57N99, 57P99
References
Publication
Received: 14 February 2011
Revised: 31 December 2012
Accepted: 9 January 2013
Published: 6 June 2013
Authors
John Bryant
Department of Mathematics
Florida State University
Tallahassee, FL 32306
USA
Steve Ferry
Department of Math Sciences
Rutgers University
Piscataway, NJ 08854-8019
USA
Washington Mio
Department of Mathematics
Florida State University
Tallahassee, FL 32306
USA