Group actions on spheres

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1 History

In 1946, Montgomery and Samelson made a comment [Montgomery&Samelson1946] that when a compact group $G$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == History == <wikitex>; In 1946, Montgomery and Samelson made a comment \cite{Montgomery&Samelson1946} that when a compact group $G$ 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 \cite{Stein1977} has obtained for the first time a counterexample to this speculation. 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. Then 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. </wikitex> == Solution for finite groups == <wikitex>; {{beginthm|Theorem|\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. {{endthm}} Following Laitinen and Morimoto \cite{Laitinen&Morimoto1998}, we say that a finite $G$ is an Oliver group if $G$ has a smooth fixed point free action on disk, i.e., $G$ satisfies the [[Fixed point free action on disks#Oliver groups|algebraic condition]]. {{beginthm|Theorem|\cite{Laitinen&Morimoto1998}\label{thm:one_fixed_point}}} For any finite Oliver group $G$, there exists a smooth action of $G$ on some sphere with exactly one fixed point. {{endthm}} Assume that a compact Lie group $G$ acts smoothly on a sphere with exactly one fixed point $x$. Then, the Slice Theorem allows us to remove from the sphere an invariant open ball neighbourhood of $x$. As a result, we obtain a smooth fixed point free action of $G$ on a disk, and so, $G$ is an Oliver group. {{beginthm|Corollary|[[Fixed point free actions on disks]]\label{cor:one_fixed_point}}} A finite group $G$ has a smooth one fixed point action on some sphere if and only if $G$ is an Oliver group. {{endthm}} Except for the two cases $G=S^3$ or $SO(3)$ proved by Petrie, the question whether for any compact Oliver group $G$ of positive dimension, the conclusion of Theorem \ref{thm:one_fixed_point} remains true is open. </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Symmetries of manifolds]]G$ 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=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$$SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$$(120, r)=1$, he constructed a smooth action of $G$$G$ on the sphere $S^7$$S^7$ with exactly one fixed point. Then Petrie [Petrie1982] described smooth one fixed point actions on spheres in the case the acting group $G$$G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for $G=S^3$$G=S^3$ or $SO(3)$$SO(3)$. Moreover, he announced the existence of such actions for the non-solvable groups $SL_2(\mathbb{F}_q)$$SL_2(\mathbb{F}_q)$ and $PSL_2(\mathbb{F}_q)$$PSL_2(\mathbb{F}_q)$, where $q\geq 5$$q\geq 5$ is a power of an odd prime.

2 Solution for finite groups

Theorem 2.1 [Laitinen&Morimoto&Pawalowski1995]. For any finite non-solvable group $G$$G$, there exists a smooth action of $G$$G$ on some sphere with exactly one fixed point.

Following Laitinen and Morimoto [Laitinen&Morimoto1998], we say that a finite $G$$G$ is an Oliver group if $G$$G$ has a smooth fixed point free action on disk, i.e., $G$$G$ satisfies the algebraic condition.

Theorem 2.2 [Laitinen&Morimoto1998]. For any finite Oliver group $G$$G$, there exists a smooth action of $G$$G$ on some sphere with exactly one fixed point.

Assume that a compact Lie group $G$$G$ acts smoothly on a sphere with exactly one fixed point $x$$x$. Then, the Slice Theorem allows us to remove from the sphere an invariant open ball neighbourhood of $x$$x$. As a result, we obtain a smooth fixed point free action of $G$$G$ on a disk, and so, $G$$G$ is an Oliver group.

Corollary 2.3 Fixed point free actions on disks. A finite group $G$$G$ has a smooth one fixed point action on some sphere if and only if $G$$G$ is an Oliver group.

Except for the two cases $G=S^3$$G=S^3$ or $SO(3)$$SO(3)$ proved by Petrie, the question whether for any compact Oliver group $G$$G$ of positive dimension, the conclusion of Theorem 2.2 remains true is open.

3 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
• [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
• [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