On the nonexistence of slices for 𝑝-adic transformation groups

Franklin, Lawrence M.

doi:10.1090/S0002-9939-1971-0267039-2

On the nonexistence of slices for $p$-adic transformation groups
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by Lawrence M. Franklin

Proc. Amer. Math. Soc. 27 (1971), 183-185

DOI: https://doi.org/10.1090/S0002-9939-1971-0267039-2

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Abstract:

If a compact Lie group acts effectively on a manifold, then local cross-sections to the orbit structure can be constructed. They are called slices. We prove that if a compact group acts effectively on a manifold, but it is not a Lie group, then a slice cannot exist at every point.

References

Hsin Chu, On the existence of slices for transformation groups, Proc. Amer. Math. Soc. 18 (1967), 513–517. MR 212126, DOI 10.1090/S0002-9939-1967-0212126-7
D. Montgomery, H. Samelson, and C. T. Yang, Groups on $E^n$ with $(n-2)$-dimensional orbits, Proc. Amer. Math. Soc. 7 (1956), 719–728. MR 78643, DOI 10.1090/S0002-9939-1956-0078643-0
D. Montgomery, H. Samelson, and L. Zippin, Singular points of a compact transformation group, Ann. of Math. (2) 63 (1956), 1–9. MR 74773, DOI 10.2307/1969986
D. Montgomery and C. T. Yang, The existence of a slice, Ann. of Math. (2) 65 (1957), 108–116. MR 87036, DOI 10.2307/1969667
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI 10.2307/1970335