One fixed point actions on spheres

Revision as of 17:35, 3 December 2010

1 Introduction

In 1946, Montgomery and Samelson made a comment [Montgomery&Samelson1946] that when a compact group $<!-- 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}} == Introduction == <wikitex>; In 1946, Montgomery and Samelson made a comment [Montgomery&Samelson1946] that when a compact group GG 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=SL2(F5) or SL_2(\mathbb{F}_5)\times \mathbb{Z}_rSL2(F5)×Zr with (120, r)=1(120,r)=1, he constructed a smooth action of GG on the sphere S^7S7 with exactly one fixed point. Then Petrie [Petrie1982] described smooth one fixed point actions on spheres in the case the acting group GG is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for G=S^3G=S3 or SO(3)SO(3). Moreover, he announced the existence of such actions for the non-solvable groups SL_2(\mathbb{F}_q)SL2(Fq) and PSL_2(\mathbb{F}_q)PSL2(Fq), where q\geq 5q≥5 is a power of an odd prime. </wikitex> == Problem == <wikitex>; ... </wikitex> == Results so far == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} [[Category:Problems]]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.

2 Problem

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3 Results so far

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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