Group actions on disks

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== Topological actions ==
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== History ==
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== Smooth actions ==
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=== Fixed point free ===
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==== History ====
<wikitex>;
<wikitex>;
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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}).
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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>
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== Oliver number ==
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==== Oliver number ====
<wikitex>;
<wikitex>;
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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 $G$.
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Let $G$ be a finite group not of prime power order. Oliver \cite{Oliver1975} has proven that the set
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$$\mathcal{Z}_G := \{\chi(X^G)-1 \ | \ X\textup{ is a finite contractible }G\textup{-CW-complex}\} \ \subseteq \ \mathbb{Z}$$ 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 $G$.
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{{beginthm|Proposition|\cite{Oliver1975}}}
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In the papers \cite{Oliver1975}, \cite{Oliver1977}, and \cite{Oliver1978}, Oliver has computed the integer $n_G$. In particular, the following lemma holds.
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For a finite group $G$ not of prime power order the following two conditions are equivalent:
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*$n_G=1$
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{{beginthm|Lemma|(Oliver Lemma)}}
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**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 $q$.
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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}}
{{endthm}}
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</wikitex>
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Moreover, the work {{cite|Oliver1977|Theorem 7}} yields the following proposition.
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== Oliver group ==
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{{beginthm|Proposition}}
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<wikitex>;
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For a finite nilpotent group $G$ not of prime power order, the following conclusions hold:
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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.
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{{beginthm|Definition}}
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*$n_G=0$ if $G$ has at most one non-cyclic Sylow subgroup.
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A finite group $G$ not of prime power order is called an Oliver group if $n_G=1$.
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*$n_G=pq$ for two distinct primes $p$ and $q$, if $G$ has just one non-cyclic $p$-Sylow and $q$-Sylow subgroups.
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*$n_G=1$ if $G$ has three or more non-cyclic Sylow subgroups.
{{endthm}}
{{endthm}}
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The notion of the Oliver number $n_G$ extends to compact Lie groups $G$ as follows.
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Examples of finite Oliver groups include:
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*$n_G=n_{G/G_0}$ if $G_0$ is abelian and $G/G_0$ is not of prime power order.
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*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 $r$).
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*$n_G=1$ if $G_0$ is non-abelian (see \cite{Oliver1976}).
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*finite solvable non-nilpotent groups of order 72, such as $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$.
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*finite non-solvable (in particular, non-trivial perfect) groups (e.g. $A_n$, $S_n$ for $n\geq 5$).
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</wikitex>
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==== Oliver group ====
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<wikitex>;
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The notion of Oliver group has been introduced by Laitinen and Morimoto \cite{Laitinen&Morimoto1998}
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in connection with the work on [[Group_actions_on_spheres#Smooth_actions|smooth one fixed point actions on spheres]].
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{{beginrem|Definition}}
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A finite group $G$ not of prime power order is called an ''Oliver group'' if $n_G=1$ (cf. Oliver Lemma).
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{{endrem}}
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Examples of finite Oliver groups include:
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*$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$, $q$, and $r$.
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*the solvable groups $S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$ of order 72.
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*all non-solvable groups, e.g., $A_n$ and $S_n$ for $n\geq 5$.
</wikitex>
</wikitex>
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== Solution ==
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==== Results ====
<wikitex>;
<wikitex>;
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{{beginthm|Theorem|\cite{Oliver1975}}
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The results of Oliver \cite{Oliver1975}, \cite{Oliver1976} can be summarized as follows.
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Let $G$ be a finite group not of prime power order. Then the following three conditions are equivalent.
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There exists a smooth fixed point free action of $G$ on some disk.
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The [[Actions on disks without fixed points#Oliver number]] $n_G=1$.
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{{beginthm|Theorem}}
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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}}
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{{beginthm|Theorem}}
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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.
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*$F$ is compact and the Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$.
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*There exists a finite contractible $G$-CW-complex $X$ such that the fixed point set $X^G$ is homeomorphic to $F$.
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*There exists a smooth action of $G$ on a disk $D$ such that the fixed point set $D^G$ is homotopy equivalent to $F$.
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{{beginthm|Theorem|\cite{Oliver1976}}}
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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}}
{{endthm}}
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</wikitex>
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{{beginthm|Corollary}}
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=== Fixed point sets ===
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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 $n_{G/G_0}=1$.
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==== History ====
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{{endthm}}
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==== Definitions ====
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<wikitex>;
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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
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* the induction (complexification and quaternization) homomorphisms $\widetilde{K}O(X)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(X)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(X)$,
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* and the forgetful (complexification and realification) homomorphisms $\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$.
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</wikitex>
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==== Results ====
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<wikitex>;
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{{beginthm|Theorem|(\cite{Oliver1996})}}
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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.
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* $F$ is compact and $\chi(F)\equiv 1\pmod{n_G}$.
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* The class $[\tau_F]$ of $\widetilde{K}O(F)$ satisfies the following condition depending on $G$.
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** $[\tau_F]$ 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 $q$.
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** $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$, if $G$ has a composite order element conjugate to its inverse and $G\notin\mathcal{D}$.
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** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$, if $G$ has a composite order element but never conjugate to its inverse and $G_2\ntrianglelefteq G$.
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** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex, if $G$ has a composite order element but never conjugate to its inverse and $G_2\trianglelefteq G$.
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** $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$, if $G$ has no composite order element and $G_2\ntrianglelefteq G$.
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** $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$, if $G$ has no composite order element and $G_2\trianglelefteq G$.
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{{endthm}}
</wikitex>
</wikitex>
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[[Category:Theory]]
[[Category:Theory]]
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[[Category:Symmetries of manifolds]]
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[[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

[edit] 1 Topological actions

[edit] 2 Smooth actions

[edit] 2.1 Fixed point free

[edit] 2.1.1 History

Floyd and Richardson [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 [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], [Oliver1976] has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks.

[edit] 2.1.2 Oliver number

Let G 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}
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 [Oliver1975], [Oliver1977], and [Oliver1978], Oliver has computed the integer n_G. In particular, the following lemma holds.

Lemma 2.1 (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.

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

Proposition 2.2. 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.

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]).


[edit] 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 G not of prime power order is called an Oliver group if n_G=1 (cf. Oliver Lemma).

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.

[edit] 2.1.4 Results

The results of Oliver [Oliver1975], [Oliver1976] can be summarized as follows.

Theorem 2.4. 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.

Theorem 2.5. 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.

[edit] 2.2 Fixed point sets

[edit] 2.2.1 History

[edit] 2.2.2 Definitions

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).

[edit] 2.2.3 Results

Theorem 2.6 ([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.

[edit] 3 References

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