×

Covering group theory for locally compact groups. (English) Zbl 0982.22003

In the first part of this paper [ibid. 114, 141-186 (2001)] the authors introduced a concept of “coverable” topological groups, not requiring any form of local simple connectivity, such that the natural homomorphism from a certain topological group \(G\), which can be associated to any topological group \(\overline G\) in a natural way, into \(\overline G\) is an open epimorphism with central and prodiscrete kernel. In this paper coverable locally compact groups are characterized. As an application one obtains that the classical covering group theories of Poincaré and Chevalley, as well as the variants due to Tits and Hofmann-Morris, are all equivalent for locally compact groups. Furthermore an inverse sequence of locally compact groups is indicated, whose bonding homomorphisms are open surjections with discrete kernel, such that the natural projections from the inverse limit are not surjective. An earlier such example of G. Higman and A. H. Stone [J. Lond. Math. Soc. 29, 233-236 (1954; Zbl 0055.02503)] deals with infinite-dimensional vector spaces (being far from locally compact).

MSC:

22D15 Group algebras of locally compact groups
22A05 Structure of general topological groups
55Q05 Homotopy groups, general; sets of homotopy classes
08B25 Products, amalgamated products, and other kinds of limits and colimits
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berestovskii, V.; Plaut, C., Homogeneous spaces of curvature bounded below, J. Geom. Analysis, 9, 203-219 (1999) · Zbl 1009.53038
[2] Berestovskii, V.; Plaut, C., Covering group theory for topological groups, Topology Appl., 114, 141-186 (2001), (this issue) · Zbl 0982.22002
[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] Boseck, H.; Czichowski, G.; Rudolph, K., Analysis on Topological Groups—General Lie Theory (1981), Teubner: Teubner Leipzig · Zbl 0558.22012
[6] Bourbaki, N., Elements of Mathematics, General Topology, I (1966), Addison-Wesley: Addison-Wesley London · Zbl 0145.19302
[7] Chevalley, C., Theory of Lie Groups, I (1946), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0063.00842
[8] Dixmier, J., Quelques proprieties des groupes Abeliens localement compacts, Bull. Sci. Math., 81, 38-48 (1957) · Zbl 0083.02301
[9] Glushkov, V. M., Lie algebras of locally bicompact groups, Uspekhi Mat. Nauk, 12, 74, 137-142 (1957), (in Russian)
[10] Glushkov, V. M., About the structure of locally bicompact groups, Mat. Sb., 48, 90) (1, 75-92 (1959), (in Russian)
[11] Hewitt, E.; Ross, K., Abstract Harmonic Analysis, I (1963), Springer: Springer Berlin · Zbl 0115.10603
[12] Higman, G.; Stone, A., On inverse systems with trivial limits, J. London Math. Soc., 29, 233-236 (1954) · Zbl 0055.02503
[13] Hofmann, K.; Morris, S., The Structure of Compact Groups (1998), de Gruyter: de Gruyter Berlin
[14] Hurewicz, W.; Wallman, H., Dimension Theory (1948), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · JFM 67.1092.03
[15] Kawada, Y., On some properties of covering groups of a topological group, J. Math. Soc. Japan, 1, 203-211 (1950) · Zbl 0041.36304
[16] Lashof, K., Lie algebras of locally compact groups, Pacific J. Math., 108, 1145-1162 (1957) · Zbl 0081.02204
[17] Mal’tsev, A., Sur les groupes topologiques locaux et complets, Comp. Rend. Acad. Sci. URSS, 32, 606-608 (1941) · Zbl 0063.03727
[18] Mimura, M., Homotopy theory of Lie groups, (James, I. M., Handbook of Algebraic Topology (1995), Elsevier: Elsevier Amsterdam) · Zbl 0171.44101
[19] Rickert, N., Some properties of locally compact spaces, J. Austral. Math. Soc., 7, 433-454 (1967) · Zbl 0167.30103
[20] Stevens, T. C., Weakening the topology of a Lie group, Trans. Amer. Math. Soc., 276, 541-549 (1983) · Zbl 0507.22002
[21] Tits, J., Liesche Gruppen und Algebren (1983), Springer: Springer Berlin · Zbl 0506.22011
[22] 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
[23] Weil, A., L’intégration dans les groupes topologiques et ses applications (1965), Hermann: Hermann Paris
[24] Wilson, J. S., Profinite Groups (1998), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0909.20001
[25] Zelinsky, D., Rings with ideal nuclei, Duke Math. J., 18, 431-442 (1951) · Zbl 0045.31905
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.