Group actions on disks

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1 History

Floyd and Richardson [Floyd&Richardson1959] have constructed for the first time a smooth fixed point free action of $G$${{Stub}}== History == <wikitex>; Floyd and Richardson \cite{Floyd&Richardson1959} have constructed for the first time a smooth fixed point free action of $G$ on a disk for $G=A_5$, the alternating group on five letters (see {{cite|Bredon1972|pp. 55-58}} for a transparent description of the construction). Later, Greever {{cite|Greever1960}}has described plenty of finite solvable groups $G$ not of prime power order, which can act smoothly on disks without fixed points. Then Oliver has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks (see \cite{Oliver1975} and \cite{Oliver1976}). </wikitex> == Oliver number == <wikitex>; Let $G$ be a finite group not of prime power order. Oliver in \cite{Oliver1975} proved that the set $$\{\chi(X^G)-1\colon X\textup{ is a contractible }$G$\textup{-CW-complex}\}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. Oliver has determined integer $n_G$ in the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}. In particular, $n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$ is a $p$-group, $G/H$ is a $q$-group, and $H/P$ is cyclic for two (possibly equal) primes $p$ and $q$. Moreover, {{cite{Oliver1977|Theorem 7}}} yields the following proposition. {{beginthm|Proposition}} For a finite nilpotent group $G$ not of prime power order, the following conclusions hold: *$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup. *$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups. *$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups. {{endthm}} </wikitex> == Oliver group == <wikitex>; In connection with the work on smooth [[One fixed point actions on spheres|one fixed point actions on spheres]], Laitinen and Morimoto \cite{Laitinen&Morimoto1998} have introduced the notion of Oliver group. {{beginrem|Definition}} A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$. {{endrem}} Examples of finite Oliver groups include: *finite nilpotent (in particular, abelian) groups with three or more non-cyclic Sylow subgroups (e.g. $\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$). *the groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent). *finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$). </wikitex> == Solution == <wikitex>; {{beginthm|Theorem|\cite{Oliver1975}}} Let $G$ be a finite group not of prime power order. Then the following three conditions are equivalent. *There exists a smooth fixed point free action of $G$ on some disk. *The [[Actions on disks without fixed points#Oliver number]] $n_G=1$. {{endthm}} {{beginthm|Theorem|\cite{Oliver1976}}} Let $G$ be a compact Lie group with non-abelian identity connected component $G_0$. Then there exist smooth fixed point free action of $G$ on some disk. {{endthm}} {{beginthm|Corollary}} A compact Lie group $G$ has a smooth fixed point free action on soome disk if and only if $G_0$ is non-abelian or $n_{G/G_0}=1$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Symmetries of manifolds]]G$ on a disk for $G=A_5$$G=A_5$, 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 $G$$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 [Oliver1975] and [Oliver1976]).

2 Oliver number

Let $G$$G$ be a finite group not of prime power order. Oliver in [Oliver1975] proved that the set

$\displaystyle \{\chi(X^G)-1\colon X\textup{ is a contractible }$G$\textup{-CW-complex}\}$ $$\displaystyle \{\chi(X^G)-1\colon X\textup{ is a contractible }$G$\textup{-CW-complex}\}$$

is a subgroup of the group of integers $\mathbb{Z}$$\mathbb{Z}$. Therefore, the set is of the form $n_G\cdot \mathbb{Z}$$n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$$n_G\geq 0$, which we refer to as the Oliver number of $G$$G$.

Oliver has determined integer $n_G$$n_G$ in the papers [Oliver1975], [Oliver1977], and [Oliver1978]. In particular, $n_G=1$$n_G=1$ if and only if there does not exist a sequence $P\trianglelefteq H\trianglelefteq G$$P\trianglelefteq H\trianglelefteq G$ of normal subgroups such that $P$$P$ is a $p$$p$-group, $G/H$$G/H$ is a $q$$q$-group, and $H/P$$H/P$ is cyclic for two (possibly equal) primes $p$$p$ and $q$$q$. Moreover, {{cite{Oliver1977|Theorem 7}}} yields the following proposition.

Proposition 1.1. For a finite nilpotent group $G$$G$ not of prime power order, the following conclusions hold:

• $n_G=0$$n_G=0$ if $G$$G$ has at most one non-cyclic Sylow subgroup.
• $n_G=pq$$n_G=pq$ for two distinct primes $p$$p$ and $q$$q$, if $G$$G$ has just one non-cyclic $p$$p$-Sylow and $q$$q$-Sylow subgroups.
• $n_G=1$$n_G=1$ if $G$$G$ has three or more non-cyclic Sylow subgroups.

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 2.1. A finite group $G$$G$ not of prime power order is called an Oliver group if $n_G=1$$n_G=1$.

Examples of finite Oliver groups include:

• finite nilpotent (in particular, abelian) groups with three or more non-cyclic Sylow subgroups (e.g. $\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$$p$, $q$$q$, and $r$$r$).
• the groups $S_4\oplus \mathbb{Z}_3$$S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$$A_4\oplus S_3$ of order 72 (which are solvable but not nilpotent).
• finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$$A_n$, $S_n$$S_n$ for $n\geq 5$$n\geq 5$).

4 Solution

Theorem 3.1 [Oliver1975]. Let $G$$G$ be a finite group not of prime power order. Then the following three conditions are equivalent.

• There exists a smooth fixed point free action of $G$$G$ on some disk.
• The Actions on disks without fixed points#Oliver number $n_G=1$$n_G=1$.

Theorem 3.2 [Oliver1976]. Let $G$$G$ be a compact Lie group with non-abelian identity connected component $G_0$$G_0$. Then there exist smooth fixed point free action of $G$$G$ on some disk.

Corollary 3.3. A compact Lie group $G$$G$ has a smooth fixed point free action on soome disk if and only if $G_0$$G_0$ is non-abelian or $n_{G/G_0}=1$$n_{G/G_0}=1$.

5 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 $n$$n$-cell without stationary points. , Bull. Amer. Math. Soc. 65 (1959), 73–76. MR0100848 (20 #7276) Zbl 0088.15302
• [Greever1960] J. Greever, Stationary points for finite transformation groups, Duke Math. J 27 (1960), 163–170. MR0110094 (22 #977) Zbl 0113.16505
• [Laitinen&Morimoto1998] E. Laitinen and M. Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), no.4, 479–520. MR1631012 (99k:57078) Zbl 0905.57023
• [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
• [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
• [Oliver1977] R. Oliver, $G$$G$-actions on disks and permutation representations. II, Math. Z. 157 (1977), no.3, 237–263. MR0646085 (58 #31126) Zbl 0386.20002
• [Oliver1978] R. Oliver, $G$$G$-actions on disks and permutation representations, J. Algebra 50 (1978), no.1, 44–62. MR0501044 (58 #18508) Zbl 0386.20002