On manifolds with nonhomogeneous factors
Manuel Cárdenas ; Francisco Lasheras ; Antonio Quintero ; Dušan Repovš
Open Mathematics, Tome 10 (2012), p. 857-862 / Harvested from The Polish Digital Mathematics Library

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general topology concerning homogeneous spaces.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:269658
@article{bwmeta1.element.doi-10_2478_s11533-012-0026-6,
     author = {Manuel C\'ardenas and Francisco Lasheras and Antonio Quintero and Du\v san Repov\v s},
     title = {On manifolds with nonhomogeneous factors},
     journal = {Open Mathematics},
     volume = {10},
     year = {2012},
     pages = {857-862},
     zbl = {1243.57015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0026-6}
}
Manuel Cárdenas; Francisco Lasheras; Antonio Quintero; Dušan Repovš. On manifolds with nonhomogeneous factors. Open Mathematics, Tome 10 (2012) pp. 857-862. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0026-6/

[1] Ancel F.D., Duvall P.F., Singh S., Rigid 3-dimensional compacta whose squares are manifolds, Proc. Amer. Math. Soc., 1983, 88(2), 330–332 | Zbl 0509.54016

[2] Ancel F.D., Singh S., Rigid finite-dimensional compacta whose squares are manifolds, Proc. Amer. Math. Soc., 1983, 87(2), 342–346 | Zbl 0503.54024

[3] Andrews J.J., Curtis M.L., n-space modulo an arc, Ann. of Math., 1962, 75, 1–7 http://dx.doi.org/10.2307/1970414 | Zbl 0105.17403

[4] Arhangelskii A.V., Pearl E., Problems from A.V. Arhangelskii’s Structure and classification of topological spaces and cardinal invariants, Topology Atlas, Problems from Topology Proceedings, 2003, 123–134 available at http://at.yorku.ca/i/a/a/z/05.htm

[5] Bass C.D., Some products of topological spaces which are manifolds, Proc. Amer. Math. Soc., 1981, 81(4), 641–646 http://dx.doi.org/10.1090/S0002-9939-1981-0601746-0 | Zbl 0466.57006

[6] Brahana T.R., Products of generalized manifolds, Illinois J. Math., 1958, 2, 76–80 | Zbl 0080.38203

[7] Bredon G.E., Wilder manifolds are locally orientable, Proc. Nat. Acad. Sci. U.S.A., 1969, 63, 1079–1081 http://dx.doi.org/10.1073/pnas.63.4.1079 | Zbl 0186.27006

[8] Bryant J.L., Euclidean space modulo a cell, Fund. Math., 1968, 63, 43–51 | Zbl 0191.22103

[9] Bryant J.L., Reflections on the Bing-Borsuk conjecture, In: Abstracts of talks presented at the 19th Annual Workshop in Geometric Topology, Grand Rapids, June 13–15, 2002, 2–3, available at http://www.calvin.edu/~venema/workshop/proceedingspapers/bryant.pdf

[10] Daverman R.J., Decompositions of Manifolds, AMS Chelsea, Providence, 2007 | Zbl 1130.57001

[11] Daverman R.J., Venema G.A., Embeddings in Manifolds, Grad. Stud. Math., 106, American Mathematical Society, Providence, 2009 | Zbl 1209.57002

[12] Dranishnikov A.N., On a problem of P.S. Aleksandrov, Mat. Sb., 1988, 135(177)(4), 551–557 | Zbl 0643.55001

[13] Dydak J., Walsh J.J., Infinite-dimensional compacta having cohomological dimension two: an application of the Sullivan conjecture, Topology, 1993, 32(1), 93–104 http://dx.doi.org/10.1016/0040-9383(93)90040-3 | Zbl 0822.55001

[14] Fedorchuk V.V., On homogeneous Pontryagin surfaces, Dokl. Akad. Nauk, 2005, 404(5), 601–603 | Zbl 1126.54013

[15] Fox R.H., Artin E., Some wild cells and spheres in three-dimensional space, Ann. of Math., 1948, 49, 979–990 http://dx.doi.org/10.2307/1969408 | Zbl 0033.13602

[16] Fox R.H., Harrold O.G., The Wilder arcs, In: Topology of 3-Manifolds and Related Topics, The Univ. of Georgia Institute, 1961, Prentice-Hall, Englewood Cliffs, 1962, 184–187

[17] Halverson D.M., Repovš D., The Bing-Borsuk and the Busemann conjectures, Math. Commun., 2008, 13(2), 163–184 | Zbl 1163.57015

[18] Halverson D.M., Repovš D., Survey on the Generalized R.L. Moore problem, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia (in press), preprint available at http://arxiv.org/abs/1201.3897

[19] Hu S., Homotopy Theory, Pure Appl. Math., 8, Academic Press, New York, 1959

[20] Lomonaco S.J., Uncountably many mildly wild non-Wilder arcs, Proc. Amer. Math. Soc., 1968, 19(4), 895–898 http://dx.doi.org/10.1090/S0002-9939-1968-0226610-4 | Zbl 0169.26302

[21] van Mill J., A rigid space X for which X × X is homogeneous; an application of infinite-dimensional topology, Proc. Amer. Math. Soc., 1981, 83(3), 597–600 | Zbl 0484.54032

[22] Myers R., Uncountably many arcs in S 3 whose complements have non-isomorphic, indecomposable fundamental groups, J. Knot Theory Ramifications, 2000, 9(4), 505–521 http://dx.doi.org/10.1142/S021821650000027X | Zbl 1001.57032

[23] Quinn F., Problems on homology manifolds, In: Exotic Homology Manifolds, Oberwolfach, June 29–July 5, 2003, Geom. Topol. Monogr., 9, Geometry & Topology Publications, Coventry, 2006, 87–103 http://dx.doi.org/10.2140/gtm.2006.9.87 | Zbl 1108.57017

[24] Raymond F., Separation and union theorems for generalized manifolds with boundary, Michigan Math. J., 1960, 7(1), 7–21 http://dx.doi.org/10.1307/mmj/1028998337

[25] Repovš D., Detection of higher-dimensional topological manifolds among topological spaces, In: Seminari di Geometria. Giornate di Topologia e Geometria delle Varietá, Bologna, September 27–29, 1990, Universitá degli Studi di Bologna, Dipartimento di Matematica, Bologna, 1992, 113–143

[26] Rosicki W., On the uniqueness of the decomposition of continua into Cartesian products, Bull. Polish Acad. Sci. Math., 2003, 51(3), 247–250 | Zbl 1046.54010

[27] Rushing T.B., Topological Embeddings, Pure Appl. Math., 52, Academic Press, New York-London, 1973

[28] Smith B.J., Products of decompositions of E n, Trans. Amer. Math. Soc., 1973, 184, 31–41 | Zbl 0243.54004

[29] Wilder R.L., Topology of Manifolds, Amer. Math. Soc. Colloq. Publ., 32, American Mathematical Society, Providence, 1979 | Zbl 0511.57001