UVk-mappings on homology manifolds

Bryant, John; Ferry, Steve; Mio, Washington

doi:10.2140/agt.2013.13.2141

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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

DOI: 10.2140/agt.2013.13.2141

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, $2 k + 3 \leq n$ , is homotopic to a ${U V}^{k}$ –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, $n \geq 5$ , having the disjoint disks property satisfies the linear ${U V}^{⌊ (n - 3) ∕ 2 ⌋}$ –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 ${U V}^{k}$ –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

Volume 13, issue 4 (2013)