One fixed point actions on spheres
(Difference between revisions)
(→Results so far) |
(→Results so far) |
||
Line 26: | Line 26: | ||
*Stein {{cite|Stein1977}} has obtained for the first time smooth one fixed point actions on spheres. For $G=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$, he constructed a smooth action of $G$ on the sphere $S^7$ with exactly one fixed point. | *Stein {{cite|Stein1977}} has obtained for the first time smooth one fixed point actions on spheres. For $G=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$, he constructed a smooth action of $G$ on the sphere $S^7$ with exactly one fixed point. | ||
*Petrie {{cite|Petrie1982}} described smooth one fixed point actions on spheres in the case the acting group $G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for $G=S^3$ or $SO(3)$. Moreover, he announced the existence of such actions for the non-solvable groups $SL_2(\mathbb{F}_q)$ and $PSL_2(\mathbb{F}_q)$, where $q\geq 5$ is a power of an odd prime. | *Petrie {{cite|Petrie1982}} described smooth one fixed point actions on spheres in the case the acting group $G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for $G=S^3$ or $SO(3)$. Moreover, he announced the existence of such actions for the non-solvable groups $SL_2(\mathbb{F}_q)$ and $PSL_2(\mathbb{F}_q)$, where $q\geq 5$ is a power of an odd prime. | ||
− | *Laitinen, Morimoto, and Pawałowski {{cite|Laitinen&Morimoto& | + | *Laitinen, Morimoto, and Pawałowski {{cite|Laitinen&Morimoto&Pawalowski1995}} For any finite non-solvable group $G$, there exists a smooth action of $G$ on some sphere with exactly one fixed point. |
</wikitex> | </wikitex> | ||
Revision as of 18:24, 3 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
In connection with their work on fiberings with singularities, Montgomery and Samelson [Montgomery&Samelson1946] made a comment that when a compact group acts smoothly on a sphere in such a way as to have one fixed point, it is likely that there must be a second fixed point.
2 Problem
Which compact Lie groups can act smoothly on some sphere with exactly one fixed point?
3 Results so far
- Stein [Stein1977] has obtained for the first time smooth one fixed point actions on spheres. For or with , he constructed a smooth action of on the sphere with exactly one fixed point.
- Petrie [Petrie1982] described smooth one fixed point actions on spheres in the case the acting group is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for or . Moreover, he announced the existence of such actions for the non-solvable groups and , where is a power of an odd prime.
- Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawalowski1995] For any finite non-solvable group , there exists a smooth action of on some sphere with exactly one fixed point.
4 Further discussion
...
5 References
- [Laitinen&Morimoto&Pawalowski1995] E. Laitinen, M. Morimoto and K. Pawałowski, Deleting-inserting theorem for smooth actions of finite nonsolvable groups on spheres, Comment. Math. Helv. 70 (1995), no.1, 10–38. MR1314939 (96b:57043) Zbl 0843.57034
- [Montgomery&Samelson1946] D. Montgomery and H. Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51–56. MR0015794 (7,471a) Zbl 0060.41501
- [Petrie1982] T. Petrie, One fixed point actions on spheres. I, II, Adv. in Math. 46 (1982), no.1, 3–14, 15–70. MR676986 (84b:57027) Zbl 0502.57021
- [Stein1977] E. Stein, Surgery on products with finite fundamental group, Topology 16 (1977), no.4, 473–493. MR0474336 (57 #13982) Zbl 0383.57014