Group actions on Euclidean spaces
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Contents |
1 Topological actions
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2 Smooth actions
2.1 Fixed point free
2.1.1 History
The question whether contractible manifolds such as 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]. For , Conner and Montgomery [Conner&Montgomery1962] have constructed smooth fixed point free actions of on Euclidean spaces. By generalizing their construction, Hsiang and Hsiang [Hsiang&Hsiang1967] have shown that any non-abelian compact connected Lie group can admit such actions.
Let for two relatively primes integers . The construction of Conner and Floyd [Conner&Floyd1959], modified and improved by Kister [Kister1961] and [Kister1963], yields smooth fixed point free actions of on Euclidean spaces (see [Bredon1972, pp. 58-61]). In the more general case of where there exist a surjection and an injection , smooth fixed point free actions of on Euclidean spaces have been constructed 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 admits a 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 of is non-abelian.
- The quotient group is not of prime power order.
2.2 Fixed point sets
2.2.1 History
2.2.2 Definitions
Let be a finite group. Let and be two distinct primes. An element of of order is called a -element of . Moreover, has a -dihedral subquotient if contains two subgroups and such that and , the dihedral group of order . Let be a -Sylow subgroup of .
The class of finite groups not of prime power order divides into the following six mutually disjoint classes.
- has a -dihedral subquotient .
- has no -dihedral subquotient, has a -element conjugate to its inverse .
- has no -element conjugate to its inverse, has a -element, .
- has no -element conjugate to its inverse, has a -element, .
- has no -element, .
- has no -element, .
Let be a compact smooth manifold. Between the reduced real, complex, and quaternion -theory groups , , and , respectively, consider the induction (complexification and quaternization) homomorphisms and :
Definition 2.2. An element of an abelian group is called quasi-divisible if it belongs to the intersection of the kernels of all homomorphisms from to free abelian groups. The subgroup of quasi-divisible elements of is denoted by .
If an abelian group is finitely generated, then is the group of torsion elements of .
2.2.3 Results
Theorem 2.3 ([citation needed]). Let be a compact Lie group such that the identity connected component of is non-abelian, or the quotient group is not of prime power order. Let be a CW complex. Then the following three statements are equivalent.
- is finite dimensional and countable (i.e., consists of countably many cells).
- There exist a finite dimensional, countable, contractible -CW-complex with finitely many orbit types, such that the fixed point set is homeomorphic to .
- There exists a smooth action of on some Euclidean space such that the fixed point set is homotopy equivalent to .
Theorem 2.4 ([Oliver1996]). Let be a finite group not of prime power order. Let be a smooth manifold. Then there exists a smooth action of on some Euclidean space such that the fixed point is diffeomorphic to if and only if the following two statements hold.
- admits a countable atlas and .
- The class satisfies the following condition depending on :
- if , then there is no restriction on .
- if , then .
- if , then .
- if , then , i.e., is stably complex.
- if , then .
- if , then .
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 , 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 without fixed points, Amer. J. Math. 85 (1963), 316–319. MR0154278 (27 #4227) Zbl 0119.18801
- [Oliver1996] B. Oliver, Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, Topology 35 (1996), no.3, 583–615. MR1396768 (97g:57059) Zbl 0861.57047
- [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
- [citation needed] Template:Citation needed