One fixed point actions on spheres
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
In 1946, Montgomery and Samelson made a comment [Montgomery&Samelson1946] that when a compact group GG 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. In 1977, Stein [Stein1977] has obtained for the first time a counterexample to this speculation. For G=SL_2(\mathbb{F}_5)G=SL2(F5) or SL_2(\mathbb{F}_5)\times \mathbb{Z}_rSL2(F5)×Zr with (120, r)=1(120,r)=1, he constructed a smooth action of GG on the sphere S^7S7 with exactly one fixed point. Then Petrie [Petrie1982] described smooth one fixed point actions on spheres in the case the acting group GG is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for G=S^3G=S3 or SO(3)SO(3). Moreover, he announced the existence of such actions for the non-solvable groups SL_2(\mathbb{F}_q)SL2(Fq) and PSL_2(\mathbb{F}_q)PSL2(Fq), where q\geq 5q≥5 is a power of an odd prime.
2 Problem
...
3 Results so far
...
4 Further discussion
...