Group actions on disks

Revision as of 13:42, 27 November 2010 (view source) Marek Kaluba (Talk | contribs) (→Results) ← Older edit		Revision as of 13:43, 27 November 2010 (view source) Marek Kaluba (Talk | contribs) (→Results) Newer edit →
Line 82:		Line 82:
	The class of finite groups not of prime power order divides into the following six mutually disjoint classes.		The class of finite groups not of prime power order divides into the following six mutually disjoint classes.
−	*$\mathcal{A}$ : $G$ has a $pq$-dihedral subquotient	+	*$\mathcal{A}=\{G\colon G$ has a $pq$-dihedral subquotient$\}$
−	*$\mathcal{B}$ : $G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse	+	*$\mathcal{B}=\{G\colon G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse$\}$
−	*$\mathcal{C}$ : $G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$	+	*$\mathcal{C}=\{G\colon G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$$\}$
−	*$\mathcal{D}$ : $G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$	+	*$\mathcal{D}=\{G\colon G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$$\}$
	*$\mathcal{E}$ : $G$ has no $pq$-element, $G_2\ntrianglelefteq G$		*$\mathcal{E}$ : $G$ has no $pq$-element, $G_2\ntrianglelefteq G$
	*$\mathcal{F}$ : $G$ has no $pq$-element, $G_2\trianglelefteq G$		*$\mathcal{F}$ : $G$ has no $pq$-element, $G_2\trianglelefteq G$

---

Revision as of 13:43, 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 2.1.2 Oliver number 2.1.3 Oliver group 2.1.4 Results 2.2 Fixed point sets 2.2.1 History 2.2.2 Results 3 References

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 ${{Stub}} == Topological actions == == Smooth actions == === Fixed point free === ==== 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). Next, Greever {{cite|Greever1960}} has described plenty of finite solvable groups $G$, which can act smoothly on disks without fixed points. Then, Oliver \cite{Oliver1975} and \cite{Oliver1976}, has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. </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, the following lemma holds. {{beginthm|Lemma|(Oliver Lemma)}} For a finite group $G$ not of prime power order, $n_G=1$ 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 the same) primes $p$ and $q$. {{endthm}} Moreover, the work {{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}} The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows. *$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order. *$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}). </wikitex> ==== Oliver group ==== <wikitex>; In connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth 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: *$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ 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> ==== Results ==== <wikitex>; The results of Oliver \cite{Oliver1975} and \cite{Oliver1976}, are summarized in the following theorem. {{beginthm|Theorem}} A compact Lie group $G$ 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 $G_0$ of $G$ is non-abelian. *The quotient $G/G_0$ is not of prime power order and $n_{G/G_0}=1$. {{endthm}} </wikitex> === Fixed point sets === ==== History ==== ==== Results ==== <wikitex>; Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver \cite{Oliver1975}, \cite{Oliver1976} has defined an integer $n_G\geq 0$, which we refer to as [[Group_actions_on_disks#Oliver_number|the Oliver number]] of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$. {{beginthm|Theorem|(\cite{Oliver1975},\cite{Oliver1976})}} Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Let $F$ be a finite CW-complex. Then the following three statements are equivalent. *The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$. *There exist a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$. *There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$. {{endthm}} The class of finite groups not of prime power order divides into the following six mutually disjoint classes. *$\mathcal{A}$ : $G$ has a $pq$-dihedral subquotient *$\mathcal{B}$ : $G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse *$\mathcal{C}$ : $G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ *$\mathcal{D}$ : $G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ *$\mathcal{E}$ : $G$ has no $pq$-element, $G_2\ntrianglelefteq G$ *$\mathcal{F}$ : $G$ has no $pq$-element, $G_2\trianglelefteq G$ {{beginthm|Theorem|\cite{Oliver1996}}} Let $G$ be a finite group not of prime power order. Let $F$ be a compact smooth manifold. Then there exists a smooth action of $G$ on some disk $D$ such that the fixed point $D^G$ is diffeomorphic to $F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$ and the following conclusion about the class $[\tau_F]\in \widetilde{K}O(F)$ holds. *If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$. *If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}(F))$. *If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}(F))+\text{qDiv}(\widetilde{K}O(F))$. *If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}(F))$, i.e., $F$ is stably complex. *If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$. *If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}O(F)))$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Group actions on manifolds]]G$ on a disk for $G=A_5$, 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 $G$, 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

Let $G$ be a finite group not of prime power order. Oliver in [Oliver1975] proved that the set 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 [Oliver1975], [Oliver1977], and [Oliver1978]. In particular, the following lemma holds.

Moreover, the work [Oliver1977, Theorem 7] yields the following proposition.

The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.

• $n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.
• $n_G=1$ if $G_0$ 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.

Examples of finite Oliver groups include:

• $\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ 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 $n\geq 5$).

2.1.4 Results

The results of Oliver [Oliver1975] and [Oliver1976], are summarized in the following theorem.

2.2 Fixed point sets

2.2.1 History

2.2.2 Results

Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient $G/G_0$ is not of prime power order. Oliver [Oliver1975], [Oliver1976] has defined an integer $n_G\geq 0$, which we refer to as the Oliver number of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.

The class of finite groups not of prime power order divides into the following six mutually disjoint classes.

• $\mathcal{A}=\{G\colon G$ has a $pq$-dihedral subquotient$\}$
• $\mathcal{B}=\{G\colon G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse$\}$
• $\mathcal{C}=\{G\colon G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G
  $$\displaystyle \}$ *$\mathcal{D}=\{G\colon G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$$
  \}$*$\mathcal{E}$:$G$has no$pq$-element,$G_2\ntrianglelefteq G$*$\mathcal{F}$:$G$has no$pq$-element,$G_2\trianglelefteq G$

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 $n$-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, $G$-actions on disks and permutation representations. II, Math. Z. 157 (1977), no.3, 237–263. MR0646085 (58 #31126) Zbl 0386.20002
• [Oliver1978] R. Oliver, $G$-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