Group actions on Euclidean spaces

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1 Topological actions

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2 Smooth actions

2.1 Fixed point free

2.1.1 History

The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith [Smith1938], [Smith1939], [Smith1941], and [Smith1945]. Conner and Montgomery [Conner&Montgomery1962] have constructed smooth fixed point free actions of $G$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Topological actions == <wikitex>; ... </wikitex> == Smooth actions == === Fixed point free === ==== History ==== <wikitex> The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups $G$ such that there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, Edmonds and Lee \cite{Edmonds&Lee1976} have proven that the same conclusion holds. </wikitex> ==== Results ==== <wikitex> {{beginthm|Theorem}} A compact Lie group $G$ has smooth fixed point free action on some Euclidean space if and only if the identity connected component $G_0$ of $G$ is non-abelian or $G/G_0$ is not of prime power order. {{endthm}} </wikitex> === Fixed point sets === ==== History ==== ==== Results ==== == References == {{#RefList:}} [[Category:Theory]] [[Category:Group actions on manifolds]]G$ on Euclidean spaces, for $G=SO(3)$$G=SO(3)$. By generalizing their construction, Hsiang and Hsiang [Hsiang&Hsiang1967] have shown that any non-abelian compact connected Lie group admits such actions.

For $G=\mathbb{Z}_{pq}$$G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$$p,q\geq 2$, the construction of Conner and Floyd [Conner&Floyd1959], modified and improved by Kister [Kister1961] and [Kister1963], yields a smooth fixed point free actions on Euclidean spaces (see [Bredon1972, pp. 58-61]). For more general groups $G$$G$ such that there exist a surjection $G\to\mathbb{Z}_p$$G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$$\mathbb{Z}_q\to G$, Edmonds and Lee [Edmonds&Lee1976] have proven that the same conclusion holds.

2.1.2 Results

Theorem 2.1. A compact Lie group $G$$G$ has smooth fixed point free action on some Euclidean space if and only if the identity connected component $G_0$$G_0$ of $G$$G$ is non-abelian or $G/G_0$$G/G_0$ is not of prime power order.

3 References

• [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
• [Conner&Floyd1959] P. E. Conner and E. E. Floyd, On the construction of periodic maps without fixed points, Proc. Amer. Math. Soc. 10 (1959), 354–360. MR0105115 (21 #3860) Zbl 0092.39701
• [Conner&Montgomery1962] P. Conner and D. Montgomery, An example for $SO(3)$$SO(3)$, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1918–1922. MR0148795 (26 #6300) Zbl 0107.16604
• [Edmonds&Lee1976] A. L. Edmonds and R. Lee, Compact Lie groups which act on Euclidean space without fixed points, Proc. Amer. Math. Soc. 55 (1976), no.2, 416–418. MR0420672 (54 #8684) Zbl 0326.57011
• [Hsiang&Hsiang1967] W. Hsiang and W. Hsiang, Differentiable actions of compact connected classical groups. I, Amer. J. Math. 89 (1967), 705–786. MR0217213 (36 #304) Zbl 0205.53902
• [Kister1961] J. M. Kister, Examples of periodic maps on Euclidean spaces without fixed points. , Bull. Amer. Math. Soc. 67 (1961), 471–474. MR0130929 (24 #A783) Zbl 0101.15602
• [Kister1963] J. M. Kister, Differentiable periodic actions on $E^8$$E^8$ without fixed points, Amer. J. Math. 85 (1963), 316–319. MR0154278 (27 #4227) Zbl 0119.18801
• [Smith1938] P. A. Smith, Transformations of finite period, Ann. of Math. (2) 39 (1938), no.1, 127–164. MR1503393 Zbl 0063.07093
• [Smith1939] P. A. Smith, Transformations of finite period. II, Ann. of Math. (2) 40 (1939), 690–711. MR0000177 (1,30c) Zbl 0063.07093
• [Smith1941] P. A. Smith, Transformations of finite period. III. Newman's theorem, Ann. of Math. (2) 42 (1941), 446–458. MR0004128 (2,324c)
• [Smith1945] P. A. Smith, Transformations of finite period. IV. Dimensional parity, Ann. of Math. (2) 46 (1945), 357–364. MR0013304 (7,136e) Zbl 0063.07093