On locally Lipschitz locally compact transformation groups of manifolds

A. A. George Michael

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 3, page 159-162
  • ISSN: 0044-8753

Abstract

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In this paper we show that a “locally Lipschitz” locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean topological manifold is a Lie group. This is a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem[1, 9] and also the Repovš-Ščepin Theorem [17] which holds only for Riemannian manifolds.

How to cite

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George Michael, A. A.. "On locally Lipschitz locally compact transformation groups of manifolds." Archivum Mathematicum 043.3 (2007): 159-162. <http://eudml.org/doc/250154>.

@article{GeorgeMichael2007,
abstract = {In this paper we show that a “locally Lipschitz” locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean topological manifold is a Lie group. This is a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem[1, 9] and also the Repovš-Ščepin Theorem [17] which holds only for Riemannian manifolds.},
author = {George Michael, A. A.},
journal = {Archivum Mathematicum},
keywords = {locally Lipschitz transformation group; Hilbert-Smith conjecture; locally Lipschitz transformation group; Hilbert-Smith conjecture},
language = {eng},
number = {3},
pages = {159-162},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On locally Lipschitz locally compact transformation groups of manifolds},
url = {http://eudml.org/doc/250154},
volume = {043},
year = {2007},
}

TY - JOUR
AU - George Michael, A. A.
TI - On locally Lipschitz locally compact transformation groups of manifolds
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 3
SP - 159
EP - 162
AB - In this paper we show that a “locally Lipschitz” locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean topological manifold is a Lie group. This is a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem[1, 9] and also the Repovš-Ščepin Theorem [17] which holds only for Riemannian manifolds.
LA - eng
KW - locally Lipschitz transformation group; Hilbert-Smith conjecture; locally Lipschitz transformation group; Hilbert-Smith conjecture
UR - http://eudml.org/doc/250154
ER -

References

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  1. Bochner S., Montgomery D., Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639–653. (1946) Zbl0061.04407MR0018187
  2. Bourbaki N., Topologie générale, Chap. 1-4, Hermann, Paris 1971. (1971) MR0358652
  3. Bredon G. E., Raymond F., Williams R. F., p -Adic transformation groups, Trans. Amer. Math. Soc. 99 (1961), 488–498. (1961) MR0142682
  4. Dieudonne J., Foundations of modern analysis, Academic Press, New York–London 1960. (1960) Zbl0100.04201MR0120319
  5. Dress A., Newman’s theorems on transformation groups, Topology, 8 (1969), 203–207. (1969) Zbl0176.53201MR0238353
  6. Federer H., Geometric measure theory, Springer-Verlag, Berlin–Heidelberg–New York, N.Y., 1969. (1969) Zbl0176.00801MR0257325
  7. Hofmann K. H., Morris S. A., The structure of compact groups, de Gruyter Stud. Math. 25 (1998). (1998) Zbl0919.22001MR1646190
  8. Karube T., Transformation groups satisfying some local metric conditions, J. Math. Soc. Japan 18, No. 1 (1966), 45–50. (1966) Zbl0136.43801MR0188342
  9. Kuranishi M., On conditions of differentiability of locally compact groups, Nagoya Math. J. 1 (1950), 71–81. (1950) Zbl0037.30502MR0038355
  10. Michael G., On the smoothing problem, Tsukuba J. Math. 25, No. 1 (2001), 13–45. Zbl0988.57014MR1846867
  11. Montgomery D., Finite dimensionality of certain transformation groups, Illinois J. Math. 1 (1957), 28–35. (1957) Zbl0077.36702MR0083680
  12. Montgomery D., Zippin L., Topological transformation groups, Interscience Publishers, New York, 1955. (1955) Zbl0068.01904MR0073104
  13. Nagami K. R., Mappings of finite order and dimension theory, Japan J. Math. 30 (1960), 25–54. (1960) Zbl0106.16002MR0142101
  14. Nagami K. R., Dimension-theoretical structure of locally compact groups, J. Math. Soc. Japan 14, No. 4 (1962), 379–396. (1962) Zbl0118.27001MR0142679
  15. Nagami K. R., Dimension theory, Academic Press, New York, 1970. (1970) Zbl0224.54060MR0271918
  16. Nagata J., Modern dimension theory, Sigma Ser. Pure Math. 2 (1983). (1983) Zbl0518.54002
  17. Repovš D., Ščepin E. V., A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Ann. 308 (1997), 361–364. (1997) Zbl0879.57025MR1464908
  18. Yang C. T., p-adic transformation groups, Michigan Math. J. 7 (1960), 201–218. (1960) Zbl0094.17502MR0120310

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