Group actions on disks
m (→References) |
Marek Kaluba (Talk | contribs) (→History) |
||
Line 7: | Line 7: | ||
==== History ==== | ==== History ==== | ||
<wikitex>; | <wikitex>; | ||
− | Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Later, Greever {{cite|Greever1960}}has described plenty of finite solvable groups $G$ not of prime power order, which can act smoothly on disks without fixed points. Then Oliver has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks (see \cite{Oliver1975} and \cite{Oliver1976}). | + | Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Later, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$ not of prime power order, which can act smoothly on disks without fixed points. Then Oliver has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks (see \cite{Oliver1975} and \cite{Oliver1976}). |
</wikitex> | </wikitex> | ||
Revision as of 11:42, 27 November 2010
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
Floyd and Richardson [Floyd&Richardson1959] have constructed for the first time a smooth fixed point free action of on a disk for , the alternating group on five letters (see [Bredon1972, pp. 55-58] for a transparent description of the construction). Later, Greever [Greever1960] has described plenty of finite solvable groups not of prime power order, which can act smoothly on disks without fixed points. Then Oliver has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks (see [Oliver1975] and [Oliver1976]).
2.1.2 Oliver number
Oliver has determined integer in the papers [Oliver1975], [Oliver1977], and [Oliver1978]. In particular, the following lemma holds.
Lemma 2.1. For a finite group not of prime power order, if and only if there does not exist a sequence of normal subgroups such that is a -group, is a -group, and is cyclic for two (possibly the same) primes and .
Moreover, the work [Oliver1977, Theorem 7] yields the following proposition.
Proposition 2.2. For a finite nilpotent group not of prime power order, the following conclusions hold:
- if has at most one non-cyclic Sylow subgroup.
- for two distinct primes and , if has just one non-cyclic -Sylow and -Sylow subgroups.
- if has three or more non-cyclic Sylow subgroups.
The notion of the Oliver number extends to compact Lie groups as follows.
- if is abelian and is not of prime power order.
- if is non-abelian (see [Oliver1976]).
2.1.3 Oliver group
In connection with the work on smooth one fixed point actions on spheres, Laitinen and Morimoto [Laitinen&Morimoto1998] have introduced the notion of Oliver group.
Definition 2.3. A finite group not of prime power order is called an Oliver group if .
Examples of finite Oliver groups include:
- for three distinct primes , , and .
- the groups and of order 72 (which are solvable but not nilpotent).
- finite non-solvable (in particular, non-trivial perfect) groups (e.g. , for ).
2.1.4 Solution
Theorem 2.4 [Oliver1975]. Let be a finite group not of prime power order. Then the following two conditions are equivalent.
- There exists a smooth fixed point free action of on some disk.
- The Oliver number .
Theorem 2.5 [Oliver1976]. Let be a compact Lie group with non-abelian identity connected component . Then there exist smooth fixed point free action of on some disk.
Corollary 2.6. A compact Lie group has a smooth fixed point free action on soome disk if and only if is non-abelian or .
2.2 Fixed point sets
3 References
- [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
- [Floyd&Richardson1959] E. E. Floyd and R. W. Richardson, An action of a finite group on an -cell without stationary points. , Bull. Amer. Math. Soc. 65 (1959), 73–76. MR0100848 (20 #7276) Zbl 0088.15302
- [Greever1960] J. Greever, Stationary points for finite transformation groups, Duke Math. J 27 (1960), 163–170. MR0110094 (22 #977) Zbl 0113.16505
- [Laitinen&Morimoto1998] E. Laitinen and M. Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), no.4, 479–520. MR1631012 (99k:57078) Zbl 0905.57023
- [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
- [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
- [Oliver1977] R. Oliver, -actions on disks and permutation representations. II, Math. Z. 157 (1977), no.3, 237–263. MR0646085 (58 #31126) Zbl 0386.20002
- [Oliver1978] R. Oliver, -actions on disks and permutation representations, J. Algebra 50 (1978), no.1, 44–62. MR0501044 (58 #18508) Zbl 0386.20002