## 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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