One fixed point actions on spheres
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Contents |
1 Introduction
In 1946, 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
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3 Results so far
- In 1977, Stein [Stein1977] has obtained for the first time a counterexample to this speculation. For or with , he constructed a smooth action of on the sphere with exactly one fixed point.
- Then 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.
4 Further discussion
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5 References
- [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