One fixed point actions on spheres

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This page has not been refereed. The information given here might be incomplete or provisional.

Contents

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 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 acts smoothly on a sphere S^n with exactly one fixed point. By removing from S^n an invariant open disk neighborhood around the single fixed point, one obtains a smooth fixed point free action of G on the disk D^n. It follows from the work of Oliver [Oliver1975] and [Oliver1976] that there exists such an action of G if and only if the identity connected component G_0 of G is non-abelian or the quotient group G/G_0 is an Oliver group.

2 Problem

Let G be a compact Lie group such that the identity connected component G_0 is non-abelian or the quotient group G/G_0 is an Oliver group. It is natural to ask whether G 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 G, the following three statements are equivalent.

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


3 Results so far

Smooth one fixed point actions of G on spheres have been constructed by

  • Stein [Stein1977] for G=SL_2(\mathbb{F}_5) or SL_2(\mathbb{F}_5)\times \mathbb{Z}_r with (120, r)=1.
  • Petrie [Petrie1982] for G=SL_2(\mathbb{F}_q) or PSL_2(\mathbb{F}_q), where q\geq 5 is a power of an odd prime.
  • Petrie [Petrie1982] for any finite abelian group G of odd order and with three or more non-cyclic Sylow subgroups.
  • Petrie [Petrie1982] for G=S^3 or SO(3).
  • Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawałowski1995] for any finite non-solvable group G.
  • Laitinen and Morimoto [Laitinen&Morimoto1998] for any finite Oliver group G.

4 Further discussion

The results obtained so far show that Conjecture 2.1 is true for finite groups G.

Theorem 4.1. For a finite group G, the following three statements are equivalent.

  • There exists a smooth action of G on some sphere with exactly one fixed point.
  • There exists a smooth action of G on some disk without fixed points.
  • G is an Oliver group.
Except for the two cases solved by Petrie [Petrie1982] where G = S^3 or SO(3), it is not known wheter a compact Lie group G with non-abelian identity connected component G_0 admits a smooth one fixed point action of some sphere.


5 References

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