# Group actions on disks

## 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$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ on a disk for $G=A_5$$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], [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 [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}$$\mathbb{Z}$. Therefore, $\mathcal{Z}_G = n_G\cdot \mathbb{Z}$$\mathcal{Z}_G = 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$.

In the papers [Oliver1975], [Oliver1977], and [Oliver1978], Oliver has computed the integer $n_G$$n_G$. 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

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

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 solvable 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.
• all non-solvable groups, e.g., $A_n$$A_n$ and $S_n$$S_n$ for $n\geq 5$$n\geq 5$.

#### 2.1.4 Results

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

Theorem 2.4. A compact Lie group $G$$G$ has a smooth fixed point free action on some disk if and only if the identity connected component $G_0$$G_0$ of $G$$G$ is non-abelian, or the quotient group $G/G_0$$G/G_0$ is not of prime power order and $n_{G/G_0}=1$$n_{G/G_0}=1$.

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

• $F$$F$ is compact and the Euler-PoincarĂ© characteristic $\chi(F)\equiv 1 \pmod{n_G}$$\chi(F)\equiv 1 \pmod{n_G}$.
• There exists 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$.

### 2.2 Fixed point sets

#### 2.2.2 Definitions

For a compact space $X$$X$, between the reduced real, complex, and quaternion $K$$K$-theory groups $\widetilde{K}O(X)$$\widetilde{K}O(X)$, $\widetilde{K}U(X)$$\widetilde{K}U(X)$, and $\widetilde{K}Sp(X)$$\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)$$\widetilde{K}O(X)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(X)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(X)$,
• and the forgetful (complexification and realification) homomorphisms $\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$$\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$.

#### 2.2.3 Results

Theorem 2.6 ([Oliver1996]). Let $G$$G$ be a finite group not of prime power order, and let $G_2$$G_2$ denote a $2$$2$-Sylow subgroup of $G$$G$. Let $F$$F$ be a smooth manifold. Then there exists a smooth action of $G$$G$ on some disk $D$$D$ such that the fixed point $D^G$$D^G$ is diffeomorphic to $F$$F$ if and only if the following two statements hold.

• $F$$F$ is compact and $\chi(F)\equiv 1\pmod{n_G}$$\chi(F)\equiv 1\pmod{n_G}$.
• The class $[\tau_F]$$[\tau_F]$ of $\widetilde{K}O(F)$$\widetilde{K}O(F)$ satisfies the following condition depending on $G$$G$.
• $[\tau_F]$$[\tau_F]$ is arbitrary, if $G$$G$ is in the class $\mathcal{D}$$\mathcal{D}$ of finite groups with dihedral subquotient of order $2pq$$2pq$ for two distinct primes $p$$p$ and $q$$q$.
• $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$$c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$, if $G$$G$ has a composite order element conjugate to its inverse and $G\notin\mathcal{D}$$G\notin\mathcal{D}$.
• $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$$[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$, if $G$$G$ has a composite order element but never conjugate to its inverse and $G_2\ntrianglelefteq G$$G_2\ntrianglelefteq G$.
• $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$$[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$$F$ is stably complex, if $G$$G$ has a composite order element but never conjugate to its inverse and $G_2\trianglelefteq G$$G_2\trianglelefteq G$.
• $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$$[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$, if $G$$G$ has no composite order element and $G_2\ntrianglelefteq G$$G_2\ntrianglelefteq G$.
• $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$$[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$, if $G$$G$ has no composite order element and $G_2\trianglelefteq G$$G_2\trianglelefteq G$.