Group actions on disks

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	*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.		*$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.
−	Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:	+	Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:
−	$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ and the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:	+	$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:
−		+	$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$
−	$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F).$$	+
	For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$.		For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$.

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Revision as of 16:24, 27 November 2010

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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 Definitions 2.2.3 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 ==== ==== Definitions ==== <wikitex>; Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. One says that $G$ ''has $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order pq$. Denote by $G_2$ a $-Sylow subgroup of $G$. The class of finite groups $G$ 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, $G_2\ntrianglelefteq G$ $\}$. *$\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$. *$\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$. *$\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$. Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$: $$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$: $$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$ For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$. </wikitex> ==== 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}} {{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 class $[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition. *If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$. *If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$. *If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$. *If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex. *If $G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$. *If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$. {{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 Definitions

Let $G$ be a finite group. For two distinct primes $p$ and $q$, a $pq$-element of $G$ is an element of order $pq$. One says that $G$ has $pq$-dihedral subquotient if $G$ contains two subgroups $H$ and $K\trianglelefteq H$ such that $H/K$ is isomorphic to the dihedral group of order $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.

The class of finite groups $G$ 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, $G_2\ntrianglelefteq G$ $\}$.
• $\mathcal{D}=\{G\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.
• $\mathcal{E}=\{G\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.
• $\mathcal{F}=\{G\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.

Let $F$ be a compact smooth manifold. Between the reduced real, complex, and quaternions $K$-theory groups $\widetilde{K}O(F)$, $\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$ and $q_\mathbb{C}$:

and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:

For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$.

2.2.3 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 $n_G=1$.

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