One fixed point actions on spheres
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[edit] 1 Introduction
In connection with their work on fiberings with singularities, Montgomery and Samelson [Montgomery&Samelson1946] made a comment that when a compact Lie 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. Contrary to this speculation, suppose that a compact Lie group acts smoothly on a sphere with exactly one fixed point. By removing from an invariant open disk neighborhood around the single fixed point, one obtains a smooth fixed point free action of on the disk . It follows from the work of Oliver [Oliver1975] and [Oliver1976] that there exists such an action of if and only if the identity connected component of is non-abelian or the quotient group is an Oliver group.
[edit] 2 Problem
Let be a compact Lie group such that the identity connected component is non-abelian or the quotient group is an Oliver group. It is natural to ask whether has a smooth one fixed point action on some sphere. The affirmative answer would confirm that the following conjecture is true.
Conjecture 2.1. For a compact Lie group , the following three statements are equivalent.
- There exists a smooth action of on some sphere with exactly one fixed point.
- There exists a smooth action of on some disk without fixed points.
- is a non-abelian group or the quotient group is an Oliver group.
[edit] 3 Results so far
Smooth one fixed point actions of on spheres have been constructed by
- Stein [Stein1977] for or with .
- Petrie [Petrie1982] for or , where is a power of an odd prime.
- Petrie [Petrie1982] for any finite abelian group of odd order and with three or more non-cyclic Sylow subgroups.
- Petrie [Petrie1982] for or .
- Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawałowski1995] for any finite non-solvable group .
- Laitinen and Morimoto [Laitinen&Morimoto1998] for any finite Oliver group .
[edit] 4 Further discussion
The results obtained so far show that Conjecture 2.1 is true for finite groups .
Theorem 4.1. For a finite group , the following three statements are equivalent.
- There exists a smooth action of on some sphere with exactly one fixed point.
- There exists a smooth action of on some disk without fixed points.
- is an Oliver group.
[edit] 5 References
- [Laitinen&Morimoto&Pawałowski1995] 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
- [Laitinen&Morimoto1998] E. Laitinen and M. Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), no.4, 479–520. MR1631012 (99k:57078) Zbl 0905.57023
- [Montgomery&Samelson1946] D. Montgomery and H. Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51–56. MR0015794 (7,471a) Zbl 0060.41501
- [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
- [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
- [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