Homogeneous weak solenoids

Fokkink, Robbert; Oversteegen, Lex

doi:10.1090/S0002-9947-02-03017-9

Homogeneous weak solenoids
HTML articles powered by AMS MathViewer

by Robbert Fokkink and Lex Oversteegen

Trans. Amer. Math. Soc. 354 (2002), 3743-3755

DOI: https://doi.org/10.1090/S0002-9947-02-03017-9

Published electronically: April 23, 2002

PDF | Request permission

Abstract:

A (generalized) weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps. If the covering maps are regular, then we call the inverse limit space a strong solenoid. By a theorem of M.C. McCord, strong solenoids are homogeneous. We show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids. We also give an example of a weak solenoid that has simply connected path-components, but which is not homogeneous.

References

J. M. Aarts and R. J. Fokkink, The classification of solenoids, Proc. Amer. Math. Soc. 111 (1991), no. 4, 1161–1163. MR 1042260, DOI 10.1090/S0002-9939-1991-1042260-7
J. M. Aarts, C. L. Hagopian, and L. G. Oversteegen, The orientability of matchbox manifolds, Pacific J. Math. 150 (1991), no. 1, 1–12. MR 1120708
J. M. Aarts and M. Martens, Flows on one-dimensional spaces, Fund. Math. 131 (1988), no. 1, 53–67. MR 970914, DOI 10.4064/fm-131-1-53-67
Fredric D. Ancel, An alternative proof and applications of a theorem of E. G. Effros, Michigan Math. J. 34 (1987), no. 1, 39–55. MR 873018, DOI 10.1307/mmj/1029003481
R. H. Bing, A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canadian J. Math. 12 (1960), 209–230. MR 111001, DOI 10.4153/CJM-1960-018-x
Clark, A., A generalization of Hagopian’s theorem and exponents, Topology Appl. 117 (2002), no. 3, 273–283.
Clark, A., Fokkink, R.J., Bihomogeneity of solenoids, Algebraic Geom. Topol., 2 (2002), 1–8.
Dantzig, D. van, Waerden, B.L. van der, Über metrisch homogene Räume, Abh. Math. Seminar Hamburg 6, (1928), 367–376.
Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
Charles L. Hagopian, A characterization of solenoids, Pacific J. Math. 68 (1977), no. 2, 425–435. MR 458381
James Keesling, The group of homeomorphisms of a solenoid, Trans. Amer. Math. Soc. 172 (1972), 119–131. MR 315735, DOI 10.1090/S0002-9947-1972-0315735-6
Krystyna Kuperberg, On the bihomogeneity problem of Knaster, Trans. Amer. Math. Soc. 321 (1990), no. 1, 129–143. MR 989579, DOI 10.1090/S0002-9947-1990-0989579-7
Ronald de Man, On composants of solenoids, Electron. Res. Announc. Amer. Math. Soc. 1 (1995), no. 2, 87–90. MR 1350684, DOI 10.1090/S1079-6762-95-02005-1
William S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. MR 1095046
M. C. McCord, Inverse limit sequences with covering maps, Trans. Amer. Math. Soc. 114 (1965), 197–209. MR 173237, DOI 10.1090/S0002-9947-1965-0173237-0
Piotr Minc, Solenoids and bihomogeneity, Continua (Cincinnati, OH, 1994) Lecture Notes in Pure and Appl. Math., vol. 170, Dekker, New York, 1995, pp. 297–304. MR 1326852
J. Mioduszewski, Mappings of inverse limits, Colloq. Math. 10 (1963), 39–44. MR 166762, DOI 10.4064/cm-10-1-39-44
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
James T. Rogers Jr. and Jeffrey L. Tollefson, Involutions on solenoidal spaces, Fund. Math. 73 (1971/72), no. 1, 11–19. MR 296923, DOI 10.4064/fm-73-1-11-19
James T. Rogers Jr. and Jeffrey L. Tollefson, Homogeneous inverse limit spaces with nonregular covering maps as bonding maps, Proc. Amer. Math. Soc. 29 (1971), 417–420. MR 273561, DOI 10.1090/S0002-9939-1971-0273561-5
James T. Rogers Jr. and Jeffrey L. Tollefson, Homeomorphism groups of weak solenoidal spaces, Proc. Amer. Math. Soc. 28 (1971), 242–246. MR 271924, DOI 10.1090/S0002-9939-1971-0271924-5
Richard M. Schori, Inverse limits and homogeneity, Trans. Amer. Math. Soc. 124 (1966), 533–539. MR 198416, DOI 10.1090/S0002-9947-1966-0198416-9
Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
Segal, J., Homogeneity of inverse limit spaces, Notices Amer. Math. Soc. 5 (1958), 687.
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Gerald S. Ungar, On all kinds of homogeneous spaces, Trans. Amer. Math. Soc. 212 (1975), 393–400. MR 385825, DOI 10.1090/S0002-9947-1975-0385825-3
R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203. MR 348794
John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998. MR 1691054