Group actions on disks
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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). Next, Greever [Greever1960] has described plenty of finite solvable groups , which can act smoothly on disks without fixed points. Then, Oliver [Oliver1975] and [Oliver1976], has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks.
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 (Oliver Lemma). 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 Results
The results of Oliver [Oliver1975] and [Oliver1976], are summarized in the following theorem.
Theorem 2.4. A compact Lie group has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.
- The identity connected component of is non-abelian.
- The quotient is not of prime power order and .
2.2 Fixed point sets
2.2.1 History
2.2.2 Definitions
Let be a finite group. For two distinct primes and , a -element of is an element of order . One says that has -dihedral subquotient if contains two subgroups and such that is isomorphic to the dihedral group of order . Denote by 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 quaternions -theory groups , , and , respectively, consider the induction (complexification and quaternization) homomorphisms and :
For a finitely generated abelian group , denote by the torsion part of .
2.2.3 Results
Let be a compact Lie group such that the identity connected component of is non-abelian, or the quotient is not of prime power order. Oliver [Oliver1975], [Oliver1976] has defined an integer , which we refer to as the Oliver number of . Recall that when is abelian, and otherwise .
Theorem 2.5 ([Oliver1975],[Oliver1976]). Let be a compact Lie group such that the identity connected component of is non-abelian, or the quotient is not of prime power order. Let be a finite CW-complex. Then the following three statements are equivalent.
- The Euler-Poincaré characteristic .
- There exist a finite contractible -CW-complex such that the fixed point set is homeomorphic to .
- There exists a smooth action of on a disk such that the fixed point set is homotopy equivalent to .
Theorem 2.6 [Oliver1996]. Let be a finite group not of prime power order. Let be a compact smooth manifold. Then there exists a smooth action of on some disk such that the fixed point is diffeomorphic to if and only if and the class satisfies one of the following condition.
- 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
- [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
- [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