×

Covering group theory for topological groups. (English) Zbl 0982.22002

The authors develop a covering theory for a large category of “coverable” topological groups, with a generalized notion of a cover, including a number of examples and open problems. See also the review of the second part of this paper (cf. the following review).

MSC:

22D15 Group algebras of locally compact groups
22A05 Structure of general topological groups
55Q05 Homotopy groups, general; sets of homotopy classes
57T20 Homotopy groups of topological groups and homogeneous spaces
Full Text: DOI

References:

[1] Berestovskii, V.; Plaut, C., Homogeneous spaces of curvature bounded below, J. Geom. Anal., 9, 203-219 (1999) · Zbl 1009.53038
[2] Berestovskii, V.; Plaut, C., Covering group theory for locally compact groups, Topology Appl., 114, 187-199 (2001), (this issue) · Zbl 0982.22003
[3] Berestovskii, V.; Plaut, C., Covering group theory for compact groups, J. Pure Appl. Algebra, 163, 3 (2001) · Zbl 0982.22003
[4] Berestovskii, V.; Plaut, C.; Stallman, C., Geometric groups. I, Trans. Amer. Math. Soc., 351, 1403-1422 (1999) · Zbl 0909.22007
[5] Borsuk, K., Theory of Retracts. Theory of Retracts, Monografie Matematyczne, 44 (1967), PWN: PWN Warsaw · Zbl 0153.52905
[6] Bourbaki, N., Elements of Mathematics, General Topology I (1966), Addison-Wesley: Addison-Wesley London · Zbl 0145.19302
[7] J.W. Cannon, G.R. Conner, The big fundamental group, big Hawaiian earrings, and the big free groups, Preprint, 1998; J.W. Cannon, G.R. Conner, The big fundamental group, big Hawaiian earrings, and the big free groups, Preprint, 1998 · Zbl 0955.57003
[8] Chevalley, C., Theory of Lie Groups I (1946), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0063.00842
[9] Comfort, W. W.; Hofmann, K.-H.; Remus, D., Topological groups and semigroups, (Hušek, M.; van Mill, J., Recent Progress in General Topology (1992), Elsevier: Elsevier Amsterdam), 59-144 · Zbl 0798.22001
[10] G.R. Conner, J.W. Lamoreaux, On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane, Trans. Amer. Math. Soc., to appear; G.R. Conner, J.W. Lamoreaux, On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane, Trans. Amer. Math. Soc., to appear · Zbl 1092.57001
[11] Grabowski, J.; Wojtyński, W., Quotient groups of linear topological spaces, Colloq. Math., 59, 35-51 (1990) · Zbl 0727.22005
[12] Hofmann, K.; Morris, S., The Structure of Compact Groups (1998), de Gruyter: de Gruyter Berlin · Zbl 0919.22001
[13] Husain, T., Introduction to Topological Groups (1966), Saunders: Saunders Philadelphia, PA · Zbl 0136.29402
[14] Kawada, Y., On some properties of covering groups of a topological group, J. Math. Soc. Japan, 1, 203-211 (1950) · Zbl 0041.36304
[15] Mal’tsev, A., Sur les groupes topologiques locaux et complets, Comp. Rend. Acad. Sci. URSS, 32, 606-608 (1941) · Zbl 0063.03727
[16] (Mauldin, R. D., The Scottish Book (1981), Birkhäuser: Birkhäuser Boston, MA) · Zbl 0485.01013
[17] van Mill, J., A topological group having no homeomorphisms other than translations, (Sanders, T. J.; etal., Abstracts of the US Naval Academy Top. Acad., Annapolis, MD (1982)), 20 · Zbl 0573.22001
[18] Montgomery, D.; Zippin, L., Topological Transformation Groups (1955), Wiley and Sons: Wiley and Sons New York · JFM 66.0959.03
[19] Mukhin, Yu., Locally Compact Groups (1981), Sverdlovsk, (in Russian)
[20] Petersen, K., Ergodic Theory (1983), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 0507.28010
[21] Pontryagin, L., Topological Groups (1939), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0022.17104
[22] Schreier, O., Abstrakte kontinuerliche Gruppe, Hamb. Abh., 4, 15-32 (1925) · JFM 51.0112.04
[23] Stevens, T. C., Connectedness of complete metric groups, Colloq. Math., 50, 233-240 (1986) · Zbl 0599.22001
[24] Tits, J., Liesche Gruppen und Algebren (1983), Springer: Springer Berlin · Zbl 0506.22011
[25] Wilder, R. L., Evolution of the topological concept of “connected”, Amer. Math. Monthly, 85, 720-726 (1978) · Zbl 0394.54009
[26] Weil, A., Sur les Espaces à Structure Uniforme et Sur la Topologie Générale. Sur les Espaces à Structure Uniforme et Sur la Topologie Générale, Publ. Math. Univ. Strasbourg (1937), Hermann & Cie: Hermann & Cie Paris · JFM 63.0569.04
[27] A. Yelton, REU project, Summer, 1998; A. Yelton, REU project, Summer, 1998
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.