| Title:
|
On locally Lipschitz locally compact transformation groups of manifolds (English) |
| Author:
|
George Michael, A. A. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
43 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
159-162 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
locally Lipschitz transformation group |
| Keyword:
|
Hilbert-Smith conjecture |
| MSC:
|
57S05 |
| idZBL:
|
Zbl 1164.57014 |
| idMR:
|
MR2354804 |
| . |
| Date available:
|
2008-06-06T22:50:59Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108061 |
| . |
| Reference:
|
[1] Bochner S., Montgomery D.: Locally compact groups of differentiable transformations.Ann. of Math. (2) 47 (1946), 639–653. Zbl 0061.04407, MR 0018187 |
| Reference:
|
[2] Bourbaki N.: Topologie générale, Chap. 1-4.Hermann, Paris 1971. MR 0358652 |
| Reference:
|
[3] Bredon G. E., Raymond F., Williams R. F.: $p$-Adic transformation groups.Trans. Amer. Math. Soc. 99 (1961), 488–498. MR 0142682 |
| Reference:
|
[4] Dieudonne J.: Foundations of modern analysis.Academic Press, New York–London 1960. Zbl 0100.04201, MR 0120319 |
| Reference:
|
[5] Dress A.: Newman’s theorems on transformation groups.Topology, 8 (1969), 203–207. Zbl 0176.53201, MR 0238353 |
| Reference:
|
[6] Federer H.: Geometric measure theory.Springer-Verlag, Berlin–Heidelberg–New York, N.Y., 1969. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[7] Hofmann K. H., Morris S. A.: The structure of compact groups.de Gruyter Stud. Math. 25 (1998). Zbl 0919.22001, MR 1646190 |
| Reference:
|
[8] Karube T.: Transformation groups satisfying some local metric conditions.J. Math. Soc. Japan 18, No. 1 (1966), 45–50. Zbl 0136.43801, MR 0188342 |
| Reference:
|
[9] Kuranishi M.: On conditions of differentiability of locally compact groups.Nagoya Math. J. 1 (1950), 71–81. Zbl 0037.30502, MR 0038355 |
| Reference:
|
[10] Michael G.: On the smoothing problem.Tsukuba J. Math. 25, No. 1 (2001), 13–45. Zbl 0988.57014, MR 1846867 |
| Reference:
|
[11] Montgomery D.: Finite dimensionality of certain transformation groups.Illinois J. Math. 1 (1957), 28–35. Zbl 0077.36702, MR 0083680 |
| Reference:
|
[12] Montgomery D., Zippin L.: Topological transformation groups.Interscience Publishers, New York, 1955. Zbl 0068.01904, MR 0073104 |
| Reference:
|
[13] Nagami K. R.: Mappings of finite order and dimension theory.Japan J. Math. 30 (1960), 25–54. Zbl 0106.16002, MR 0142101 |
| Reference:
|
[14] Nagami K. R.: Dimension-theoretical structure of locally compact groups.J. Math. Soc. Japan 14, No. 4 (1962), 379–396. Zbl 0118.27001, MR 0142679 |
| Reference:
|
[15] Nagami K. R.: Dimension theory.Academic Press, New York, 1970. Zbl 0224.54060, MR 0271918 |
| Reference:
|
[16] Nagata J.: Modern dimension theory.Sigma Ser. Pure Math. 2 (1983). Zbl 0518.54002 |
| Reference:
|
[17] Repovš D., Ščepin E. V.: A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps.Math. Ann. 308 (1997), 361–364. Zbl 0879.57025, MR 1464908 |
| Reference:
|
[18] Yang C. T.: p-adic transformation groups.Michigan Math. J. 7 (1960), 201–218. Zbl 0094.17502, MR 0120310 |
| . |
