Group actions on disks
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− | {{Stub}}== History == | + | {{Stub}} |
+ | == Topological actions == | ||
+ | |||
+ | |||
+ | == Smooth actions == | ||
+ | === Fixed point free === | ||
+ | ==== History ==== | ||
<wikitex>; | <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). | + | 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}, \cite{Oliver1976} has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks. |
</wikitex> | </wikitex> | ||
− | == Oliver number == | + | ==== Oliver number ==== |
<wikitex>; | <wikitex>; | ||
− | Let $G$ be a finite group not of prime power order. Oliver | + | Let $G$ be a finite group not of prime power order. Oliver \cite{Oliver1975} has proven that the set |
+ | $$\mathcal{Z}_G := \{\chi(X^G)-1 \ | \ X\textup{ is a finite contractible }G\textup{-CW-complex}\} \ \subseteq \ \mathbb{Z}$$ is a subgroup of the group of integers $\mathbb{Z}$. Therefore, $\mathcal{Z}_G = n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$, which we refer to as the ''Oliver number'' of $G$. | ||
+ | In the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}, Oliver has computed the integer $n_G$. 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. | |
− | *$n_G= | + | *$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups. |
− | + | ||
{{endthm}} | {{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> | </wikitex> | ||
− | == Oliver group == | + | ==== Oliver group ==== |
<wikitex>; | <wikitex>; | ||
− | + | The notion of Oliver group has been introduced by Laitinen and Morimoto \cite{Laitinen&Morimoto1998} | |
+ | in connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]]. | ||
{{beginrem|Definition}} | {{beginrem|Definition}} | ||
− | A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$. | + | A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$ (cf. Oliver Lemma). |
{{endrem}} | {{endrem}} | ||
− | |||
Examples of finite Oliver groups include: | Examples of finite Oliver groups include: | ||
− | * | + | *$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$. |
− | * | + | *the solvable groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72. |
− | * | + | *all non-solvable groups, e.g., $A_n$ and $S_n$ for $n\geq 5$. |
− | + | ||
− | + | ||
</wikitex> | </wikitex> | ||
− | == | + | ==== Results ==== |
<wikitex>; | <wikitex>; | ||
− | + | The results of Oliver \cite{Oliver1975}, \cite{Oliver1976} can be summarized as follows. | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
+ | {{beginthm|Theorem}} | ||
+ | A compact Lie group $G$ has a smooth fixed point free action on some disk if and only if the identity connected component $G_0$ of $G$ is non-abelian, or the quotient group $G/G_0$ is not of prime power order and $n_{G/G_0}=1$. | ||
{{endthm}} | {{endthm}} | ||
+ | {{beginthm|Theorem}} | ||
+ | Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian, or the quotient group $G/G_0$ is not of prime power order. Let $F$ be a CW-complex. Then the following three statements are equivalent. | ||
+ | *$F$ is compact and the Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$. | ||
+ | *There exists 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}} | {{endthm}} | ||
+ | </wikitex> | ||
− | + | === Fixed point sets === | |
− | + | ==== History ==== | |
− | + | ||
+ | ==== Definitions ==== | ||
+ | <wikitex>; | ||
+ | For a compact space $X$, between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(X)$, $\widetilde{K}U(X)$, and $\widetilde{K}Sp(X)$, respectively, consider | ||
+ | * the induction (complexification and quaternization) homomorphisms $\widetilde{K}O(X)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(X)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(X)$, | ||
+ | * and the forgetful (complexification and realification) homomorphisms $\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$. | ||
+ | </wikitex> | ||
+ | |||
+ | ==== Results ==== | ||
+ | <wikitex>; | ||
+ | |||
+ | {{beginthm|Theorem|(\cite{Oliver1996})}} | ||
+ | Let $G$ be a finite group not of prime power order, and let $G_2$ denote a $2$-Sylow subgroup of $G$. Let $F$ be a 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 the following two statements hold. | ||
+ | * $F$ is compact and $\chi(F)\equiv 1\pmod{n_G}$. | ||
+ | * The class $[\tau_F]$ of $\widetilde{K}O(F)$ satisfies the following condition depending on $G$. | ||
+ | ** $[\tau_F]$ is arbitrary, if $G$ is in the class $\mathcal{D}$ of finite groups with dihedral subquotient of order $2pq$ for two distinct primes $p$ and $q$. | ||
+ | ** $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$, if $G$ has a composite order element conjugate to its inverse and $G\notin\mathcal{D}$. | ||
+ | ** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$, if $G$ has a composite order element but never conjugate to its inverse and $G_2\ntrianglelefteq G$. | ||
+ | ** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex, if $G$ has a composite order element but never conjugate to its inverse and $G_2\trianglelefteq G$. | ||
+ | ** $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$, if $G$ has no composite order element and $G_2\ntrianglelefteq G$. | ||
+ | ** $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$, if $G$ has no composite order element and $G_2\trianglelefteq G$. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
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[[Category:Theory]] | [[Category:Theory]] | ||
− | [[Category: | + | [[Category:Group actions on manifolds]] |
Latest revision as of 03:01, 30 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
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
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
The notion of Oliver group has been introduced by Laitinen and Morimoto [Laitinen&Morimoto1998] in connection with the work on smooth one fixed point actions on spheres.
Definition 2.3. A finite group not of prime power order is called an Oliver group if (cf. Oliver Lemma).
Examples of finite Oliver groups include:
- for three distinct primes , , and .
- the solvable groups and of order 72.
- all non-solvable groups, e.g., and for .
2.1.4 Results
The results of Oliver [Oliver1975], [Oliver1976] can be summarized as follows.
Theorem 2.4. A compact Lie group has a smooth fixed point free action on some disk if and only if the identity connected component of is non-abelian, or the quotient group is not of prime power order and .
Theorem 2.5. Let be a compact Lie group such that the identity connected component of is non-abelian, or the quotient group 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 exists 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 .
2.2 Fixed point sets
2.2.1 History
2.2.2 Definitions
For a compact space , between the reduced real, complex, and quaternion -theory groups , , and , respectively, consider
- the induction (complexification and quaternization) homomorphisms ,
- and the forgetful (complexification and realification) homomorphisms .
2.2.3 Results
Theorem 2.6 ([Oliver1996]). Let be a finite group not of prime power order, and let denote a -Sylow subgroup of . 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 the following two statements hold.
- is compact and .
- The class of satisfies the following condition depending on .
- is arbitrary, if is in the class of finite groups with dihedral subquotient of order for two distinct primes and .
- , if has a composite order element conjugate to its inverse and .
- , if has a composite order element but never conjugate to its inverse and .
- , i.e., is stably complex, if has a composite order element but never conjugate to its inverse and .
- , if has no composite order element and .
- , if has no composite order element and .
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