Group actions on Euclidean spaces

From Manifold Atlas
Revision as of 11:11, 27 November 2010 by Marek Kaluba (Talk | contribs)
Jump to: navigation, search


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

Contents

1 Topological actions

...


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 on Euclidean spaces, for 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} for two relatively primes integers 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 such that there exist a surjection G\to\mathbb{Z}_p and an injection \mathbb{Z}_q\to G, such actions have been obtained by Edmonds and Lee [Edmonds&Lee1976].


2.1.2 Results


The results of [Conner&Montgomery1962], [Hsiang&Hsiang1967], [Conner&Floyd1959], [Kister1961], and [Edmonds&Lee1976] yield the following theorem.

Theorem 2.1. A compact Lie group G has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.

  • The identity connected component G_0 of G is non-abelian.
  • The quotient G/G_0 is not of prime power order.

2.2 Fixed point sets

2.2.1 History

2.2.2 Results

3 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox