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Relations between Non-Compact Transformation Groups and Compact Transformation Groups

Published online by Cambridge University Press:  22 January 2016

Hsin Chu
Hsin Chu*
Affiliation:
University of Maryland
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In this paper certain relations between non-compact transformation groups and compact transformation groups are studied. The notion of re-ducibility and separability of transformation groups is introduced, several necessary and sufficient conditions are established: (1) A separable transformation group to be locally weakly almost periodic, (2) A reducible and separable transformation group to be a minimal set and (3) A reducible and separable transformation group to be a fibre bundle. As applications we show, among other things, that (1) for certain reducible transformation groups its fundamental group is not trivial which is a generalization of a result in [4].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 44 , December 1971 , pp. 97 - 117
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Anderson, R.D., On raising flows and mappings, Bull. Amer. Math. Soc. 69 (1963), 259–264.CrossRefGoogle Scholar
[2] Hsin, Chu, On the existence of slices for transformation groups, Proc. of Amer. Math. Soc, 18 (1967), 513–517.Google Scholar
[3] Hsin, Chu, On universal transformation groups, 111. Jour, of Math., 6 (1962), 317–326.Google Scholar
[4] Hsin, Chu and Geraghty, M.A., The fundamental group and the first cohomology group of a minimal set, Bull, of Amer. Math. Soc, 69 (1963), 377–381.Google Scholar
[5] Ellis, R., Global sections of transformation groups, 111. Jour, of Math., 8 (1964), 380–394.Google Scholar
[6] Ellis, R. and Gottschalk, W.H., Homomorphisms of transformation groups, Trans. Amer. Math. Soc, 64 (1960), 258–271.CrossRefGoogle Scholar
[7] Gottschalk, W.H., Minimal sets, Bull. Amer. Math. Soc, 64 (1958), 336–351.CrossRefGoogle Scholar
[8] Gottschalk, W.H. and Hedlund, G.A., Topological Dynamics, Amer. Math. Soc Colloq. Publ. 36, Amer. Math. Soc, Providence, R.I., 1955.Google Scholar
[9] Hu, S.T., Homotopy Theory, Academic Press, New York, 1959.Google Scholar
[10] Iwasawa, K., On some types of topological groups, Ann. of Math., 50, (1949), 507–558.CrossRefGoogle Scholar
[11] Kahn, P.J. and Knapp, A.W., Equivariant maps onto minimal flows, 2 (1968), 319–324.Google Scholar
[12] Milnor, J., Lectures on characteristic classes, mimeographed notes, Princeton Univ.Google Scholar
[13] Schwartzman, S., Global cross sections of compact dynamical systems, Proc. Nat. Acad. Sci., U.S.A., 48 (1962) 786–791.Google Scholar
[14] Steenrod, N., The topology of fibre bundles, Princeton University Press, Princeton, New Jersey, 1951.Google Scholar