One fixed point actions on spheres

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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 $G$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}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. Contrary to this speculation, suppose that a compact Lie group $G$$G$ acts smoothly on a sphere $S^n$$S^n$ with exactly one fixed point. By removing from $S^n$$S^n$ an invariant open disk neighborhood around the single fixed point, one obtains a smooth fixed point free action of $G$$G$ on the disk $D^n$$D^n$. It follows from the work of Oliver [Oliver1975] and [Oliver1976] that the identity connected component $G_0$$G_0$ is non-abelian or the quotient group $G/G_0$$G/G_0$ is an Oliver group. So, it is natural to ask whether in fact any such a group $G$$G$ has a smooth one fixed point action on some sphere. The affirmative answer would confirm that the following conjecture is true.

2 Problem

Conjecture 2.1. For a compact Lie group $G$$G$, the following three statements are equivalent.

• There exists a smooth action of $G$$G$ on some sphere with exactly one fixed point.
• There exists a smooth action of $G$$G$ on some disk without fixed points.
• $G_0$$G_0$ is a non-abelian group or the quotient group $G/G_0$$G/G_0$ is an Oliver group.

3 Results so far

• Stein [Stein1977] has obtained for the first time smooth one fixed point actions on spheres. 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$$S^7$ with exactly one fixed point.
• Petrie [Petrie1982] described smooth one fixed point actions on spheres in the case the acting group $G$$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$$q\geq 5$ is a power of an odd prime.
• Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawalowski1995] proved that any finite non-solvable group $G$$G$ can act smoothly on some sphere with exactly one fixed point.
• Laitinen and Morimoto [Laitinen&Morimoto1998] proved that any finite Oliver group $G$$G$ can acts smoothly on some sphere with exactly one fixed point.

4 Further discussion

Except for the two cases solved by Petrie [Petrie1982] where $G = SO(3)$$G = SO(3)$ or $S^3$$S^3$, it is not known wheter any compact Lie group $G$$G$ with non-abelian $G_0$$G_0$ admits a smooth one fixed point action of some sphere.

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
• [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