We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note that in every example presently known to the authors of a codimension one manifold factor of dimension n ≥ 4 determined by general position properties, the piecewise disjoint arc-disk property is satisfied.
@article{bwmeta1.element.doi-10_2478_s11533-013-0291-z,
author = {Denise Halverson and Du\v san Repov\v s},
title = {Detecting codimension one manifold factors with the piecewise disjoint arc-disk property and related properties},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1932-1948},
zbl = {1288.57020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0291-z}
}
Denise Halverson; Dušan Repovš. Detecting codimension one manifold factors with the piecewise disjoint arc-disk property and related properties. Open Mathematics, Tome 11 (2013) pp. 1932-1948. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0291-z/
[1] Banakh T., Valov V., General position properties in fiberwise geometric topology, preprint available at http://arxiv.org/abs/1001.2494 | Zbl 1301.57019
[2] Borel A., Seminar on Transformation Groups, Ann. of Math. Stud., 46, Princeton University Press, Princeton, 1960 | Zbl 0091.37202
[3] Cannon J.W., Daverman R.J., A totally wild flow, Indiana Univ. Math. J., 1981, 30(3), 371–387 http://dx.doi.org/10.1512/iumj.1981.30.30029 | Zbl 0432.58018
[4] Daverman R.J., Products of cell-like decomposition, Topology Appl., 1980, 11(2), 121–139 http://dx.doi.org/10.1016/0166-8641(80)90002-4
[5] Daverman R.J., Detecting the disjoint disks property, Pacific J. Math., 1981, 93(2), 277–298 http://dx.doi.org/10.2140/pjm.1981.93.277 | Zbl 0415.57007
[6] Daverman R.J., Decompositions of Manifolds, Pure Appl. Math., 124, Academic Press, Orlando, 1986 | Zbl 0608.57002
[7] Daverman R.J., Halverson D., Path concordances as detectors of codimension-one manifold factors, Exotic Homology Manifolds, Oberwolfach, June 29–July 5, 2003, Geom. Topol. Monogr., 9, Geometry & Topology Publications, Coventry, 2006, 7–15 | Zbl 1111.57017
[8] Daverman R.J., Halverson D.M., The cell-like approximation theorem in dimension 5, Fund. Math., 2007, 197, 81–121 http://dx.doi.org/10.4064/fm197-0-5 | Zbl 1133.57013
[9] Daverman R.J., Walsh J.J., A ghastly generalized n-manifold, Illinois J. Math., 1981, 25(4), 555–576 | Zbl 0478.57014
[10] Edwards R.D., The topology of manifolds and cell-like maps, Proceedings of the International Congress of Mathematicians, Helsinki, August 15-23, 1978, Academia Scientiarum Fennica, Helsinki, 1980, 111–127
[11] Halverson D.M., Detecting codimension one manifold factors with the disjoint homotopies property, Topology Appl., 2002, 117(3), 231–258 http://dx.doi.org/10.1016/S0166-8641(01)00022-0 | Zbl 0992.57024
[12] Halverson D.M., 2-ghastly spaces with the disjoint homotopies property: the method of fractured maps, Topology Appl., 2004, 138(1–3), 277–286 http://dx.doi.org/10.1016/j.topol.2003.08.016 | Zbl 1049.57015
[13] Halverson D.M., Detecting codimension one manifold factors with 0-stitched disks, Topology Appl., 2007, 154(9), 1993–1998 http://dx.doi.org/10.1016/j.topol.2007.02.006
[14] Halverson D.M., Repovš D., Detecting codimension one manifold factors with topographical techniques, Topology Appl., 2009, 156(17), 2870–2880 http://dx.doi.org/10.1016/j.topol.2009.08.023 | Zbl 1215.57009
[15] Halverson D.M., Repovš D., A survey on the generalized R. L. Moore Problem, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 2011, 58, 175–191
[16] Halverson D.M., Repovš D., Decompositions of ℝ, n ≥ 4, into convex sets generate codimension 1 manifold factors, Mediterr. J. Math., 2013, 10(2), 1101–1106 http://dx.doi.org/10.1007/s00009-012-0197-1 | Zbl 1282.57028
[17] Mitchell W.J.R., Repovš D., Topology of cell-like mappings, Rend. Sem. Fac. Sci. Univ. Cagliari, 1988, 58(suppl.), 265–300
[18] Moore R.L., Concerning upper semi-continuous collections of continua which do not separate a given continuum, Proc. Natl. Acad. Sci. USA, 1924, 10(8), 356–360 http://dx.doi.org/10.1073/pnas.10.8.356 | Zbl 50.0130.04
[19] Moore R.L., Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc., 1925, 27(4), 416–428 http://dx.doi.org/10.1090/S0002-9947-1925-1501320-8 | Zbl 51.0464.03
[20] Rourke C.P., Sanderson B.J., Introduction to Piecewise-Linear Topology, Ergeb. Math. Grenzgeb., 69, Springer, New York-Heidelberg, 1972 http://dx.doi.org/10.1007/978-3-642-81735-9 | Zbl 0254.57010