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], [Oliver1976] has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks.
2.1.2 Oliver number
Let be a finite group not of prime power order. Oliver [Oliver1975] has proven that the set
![\displaystyle \mathcal{Z}_G := \{\chi(X^G)-1 \ | \ X\textup{ is a finite contractible }G\textup{-CW-complex}\} \ \subseteq \ \mathbb{Z}](/images/math/0/1/0/010eee69f6f4fc8fe9740ff4e539dc4b.png)
![\mathbb{Z}](/images/math/9/3/4/9349f2f6b15e7d4ce91c301ddd64bc06.png)
![\mathcal{Z}_G = n_G\cdot \mathbb{Z}](/images/math/7/e/5/7e5f0e4231cb7a5a6a54aa1a6c4b894b.png)
![n_G\geq 0](/images/math/0/7/b/07bdd1d6c83fd40de472c7a9a33bd5d4.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
In the papers [Oliver1975], [Oliver1977], and [Oliver1978], Oliver has computed the integer . 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 solvable (but not nilpotent) groups
and
of order 72.
- finite non-solvable groups, e.g.
and
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 quaternion
-theory groups
,
, and
, respectively, consider the induction (complexification and quaternization) homomorphisms
and
:
![\displaystyle \widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)](/images/math/f/8/c/f8c48107e60da1ba9ead53924d741c6f.png)
![c_\mathbb{H}](/images/math/6/3/b/63bcfb8003a6422edb986198aa18e035.png)
![r_\mathbb{C}](/images/math/b/3/b/b3b6b99c3763a0f70143235557d41364.png)
![\displaystyle \widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)](/images/math/7/0/e/70e601e0800ad4cba9565d7d2f2a2b7e.png)
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 CW-complex. Then the following three statements are equivalent.
is compact and 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 smooth manifold. Then there exists a smooth action of
on some disk
such that the fixed point
is diffeomorphic to
if and only if
is compact,
, 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
- [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