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2-ghastly spaces with the disjoint homotopies property: The method of fractured maps. (English) Zbl 1049.57015

A \(2\)-ghastly space is a resolvable generalized manifold that contains no embedded \(2\)-cells. A space \(X\) has the disjoint homotopies property if any two path homotopies \(f,g: D \times I \rightarrow X\) can be approximated by homotopies \(f',g': D \times I \rightarrow X\) such that \(f'_{t}(D) \cap g'_{t}(D) = \emptyset\) for all \(t \in T\) where \(D=I=[0,1]\). In an earlier paper [Topology Appl. 117, No. 3, 231–258 (2002; Zbl 0992.57024)] the author has shown that this latter property is sufficient to characterize resolvable generalized manifolds of dimension \(n \geq 4\) as codimension one manifold factors. R. J. Daverman and J. J. Walsh [Ill. J. Math. 25, 555–576 (1981; Zbl 0478.57014)] have shown that there exist \(2\)-ghastly spaces that are codimension one manifold factors. This paper shows that \(2\)-ghastly spaces with the disjoint homotopies property exist.

MSC:

57P05 Local properties of generalized manifolds
54B10 Product spaces in general topology
54B15 Quotient spaces, decompositions in general topology
54F65 Topological characterizations of particular spaces
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
54G15 Pathological topological spaces
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