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Detecting codimension one manifold factors with the piecewise disjoint arc-disk property and related properties. (English) Zbl 1288.57020

A space \(X\) is a codimension one manifold factor if \(X\times \mathbb{R}\) is a topological manifold. The authors introduce a new unifying general position property, called the piecewise disjoint arc-disk property and its \(1\)-complex analogue, the piecewise disjoint arc-disk property\(^*\) and show how the various position properties known to detect codimension one manifold factors are related. The main result of the paper is Theorem 1.1 If \(X\) is a resolvable generalized \(n\)-manifold that satisfies the piecewise disjoint arc-disk property, then \(X\times \mathbb{R}\) is an \((n+1)\)-manifold. The authors illustrate how the piecewise disjoint arc-disk property, the piecewise disjoint arc-disk property\(^*\) and other general position properties used in the detection of codimension one manifolds factors are related. They introduce definition for (a) a modified version of \(\delta\)- fractured maps, (b) the \(\delta\)-fractured maps property with respect to the modified definition, (c) the closed \(0\)-stitched disks property, (d) the strong fuzzy ribbons property. In the case of resolvable generalized manifolds on has (1) the modified \(\delta\)-fractured maps property implies the disjoint homotopies property, (2) the closed \(0\)-stitched disks property implies the \(\delta\)-fractured maps property, (3) the piecewise disjoint arc-disk property is equivalent to the \(\delta\)-fractured maps property, (4) the piecewise disjoint arc-disk property\(^*\) is equivalent to the strong fuzzy ribbons property.

MSC:

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57N75 General position and transversality
57P99 Generalized manifolds
53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
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