Group actions on disks

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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 $G$${{Stub}} == Topological actions == == Smooth actions == === Fixed point free === ==== History ==== ; 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. ==== Oliver number ==== ; 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}). ==== Oliver group ==== ; 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). ==== Results ==== ; 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}} === Fixed point sets === ==== History ==== ==== 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 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. ==== 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 \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}} == References == {{#RefList:}} [[Category:Theory]] [[Category:Group actions on 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). Next, Greever [Greever1960] has described plenty of finite solvable groups $G$$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$$G$ be a finite group not of prime power order. Oliver in [Oliver1975] proved that the set
$\displaystyle \{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}$
is a subgroup of the group of integers $\mathbb{Z}$$\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, the following lemma holds.

Lemma 2.1 (Oliver Lemma). For a finite group $G$$G$ not of prime power order, $n_G=1$$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 the same) primes $p$$p$ and $q$$q$.

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

Proposition 2.2. 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.

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

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

Definition 2.3. 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:

• $\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$).

2.1.4 Results

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

Theorem 2.4. A compact Lie group $G$$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$$G_0$ of $G$$G$ is non-abelian.
• The quotient $G/G_0$$G/G_0$ is not of prime power order and $n_{G/G_0}=1$$n_{G/G_0}=1$.

2.2 Fixed point sets

2.2.2 Definitions

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

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

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

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

$\displaystyle \widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$
and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$$c_\mathbb{H}$ and $r_\mathbb{C}$$r_\mathbb{C}$:
$\displaystyle \widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$

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

2.2.3 Results

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

Theorem 2.5 ([Oliver1975],[Oliver1976]). Let $G$$G$ be a compact Lie group such that the identity connected component $G_0$$G_0$ of $G$$G$ is non-abelian, or the quotient $G/G_0$$G/G_0$ is not of prime power order. Let $F$$F$ be a finite CW-complex. Then the following three statements are equivalent.

• The Euler-Poincaré characteristic $\chi(F)\equiv 1 \pmod{n_G}$$\chi(F)\equiv 1 \pmod{n_G}$.
• There exist a finite contractible $G$$G$-CW-complex $X$$X$ such that the fixed point set $X^G$$X^G$ is homeomorphic to $F$$F$.
• There exists a smooth action of $G$$G$ on a disk $D$$D$ such that the fixed point set $D^G$$D^G$ is homotopy equivalent to $F$$F$.

Theorem 2.6 [Oliver1996]. Let $G$$G$ be a finite group not of prime power order. Let $F$$F$ be a compact smooth manifold. Then there exists a smooth action of $G$$G$ on some disk $D$$D$ such that the fixed point $D^G$$D^G$ is diffeomorphic to $F$$F$ if and only if $\chi(F)\equiv 1\pmod{n_G}$$\chi(F)\equiv 1\pmod{n_G}$ and the class $[\tau_F]\in \widetilde{K}O(F)$$[\tau_F]\in \widetilde{K}O(F)$ satisfies one of the following condition.

• If $G\in \mathcal{A}$$G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$$[\tau_F]$.
• If $G\in \mathcal{B}$$G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$$c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.
• If $G\in\mathcal{C}$$G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$$[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.
• If $G\in\mathcal{D}$$G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$$[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$$F$ is stably complex.
• If $G\in\mathcal{E}$$G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$$[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.
• If $G\in\mathcal{F}$$G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$$[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.