## 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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