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.


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
Full Text: DOI


[1] Bing, R.H., A decomposition of E3 into points and tame arcs such that the decomposition space is topologically different from E3, Ann. of math. (2), 65, 484-500, (1957) · Zbl 0079.38806
[2] Bing, R.H., The Cartesian product of a certain nonmanifold and a line is E4, Ann. of math. (2), 70, 399-412, (1959) · Zbl 0089.39501
[3] Cannon, J.W., Shrinking cell-like decompositions of manifolds. codimension three, Ann. of math. (2), 110, 83-112, (1979) · Zbl 0424.57007
[4] Cannon, J.W.; Daverman, R.J., A totally wild flow, Indiana univ. math. J., 30, 371-387, (1981) · Zbl 0432.58018
[5] Daverman, R.J., Decompositions of manifolds, (1986), Academic Press New York · Zbl 0608.57002
[6] Daverman, R.J., Detecting the disjoint disks property, Pacific J. math., 93, 277-298, (1981) · Zbl 0415.57007
[7] Daverman, R.J.; Walsh, J.J., A ghastly generalized n-manifold, Illinois J. math., 25, 555-576, (1981) · Zbl 0478.57014
[8] Halverson, D.M., Detecting codimension one manifold factors with the disjoint homotopies property, Topology appl., 117, 231-258, (2002) · Zbl 0992.57024
[9] Quinn, F., An obstruction to the resolution of homology manifolds, Michigan math. J., 34, 285-291, (1987) · Zbl 0652.57011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.