One fixed point actions on spheres

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1 Introduction

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}} == Introduction == <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. </wikitex> == Problem == <wikitex>; ... </wikitex> == Results so far == <wikitex>; *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 [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> == 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

• 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.

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
• [Stein1977] E. Stein, Surgery on products with finite fundamental group, Topology 16 (1977), no.4, 473–493. MR0474336 (57 #13982) Zbl 0383.57014