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Path concordances as detectors of codimension-one manifold factors. (English) Zbl 1111.57017

Quinn, Frank (ed.) et al., Exotic homology manifolds. Proceedings of the mini-workshop, Oberwolfach, Germany, June 29–July 5, 2003. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 9, 7-15 (2006).
A manifold factor is a space \(X\) for which \(X\times\mathbb R^k\) is a manifold where \(k\geq 2\). This paper is concerned with whether \(X\times \mathbb R\) is necessarily a manifold. To address this question, the authors introduce a property called the disjoint path concordances property. Let \(D\) and \(I\) both denote the unit interval. Then a metric space \(X\) has the disjoint path concordances property if for any two path homotopies \(f_{i}:D\times I\rightarrow X\) and any \(\varepsilon >0\) there are path concordances \(F_{i}:D\times I\rightarrow X\times I\) with disjoint images where \(F_{i}\) is an \(\varepsilon\)-approximation to \(f_{i}.\)
For the entire collection see [Zbl 1104.57001].

MSC:

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57P05 Local properties of generalized manifolds
54B15 Quotient spaces, decompositions in general topology
57N70 Cobordism and concordance in topological manifolds
57N75 General position and transversality
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