# Group actions on Euclidean spaces

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## 2 Smooth actions

### 2.1 Fixed point free

#### 2.1.1 History

The question whether contractible manifolds (e.g. Euclidean spaces) admit smooth fixed point free actions of compact Lie groups has been discussed for the first time by Paul Althaus Smith [Smith1938], [Smith1939], [Smith1941], and [Smith1945]. Conner and Montgomery [Conner&Montgomery1962] have constructed smooth fixed point free actions of $G$$\newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ on Euclidean spaces, for $G=SO(3)$$G=SO(3)$. By generalizing their construction, Hsiang and Hsiang [Hsiang&Hsiang1967] have shown that any non-abelian compact connected Lie group admits such actions.

For $G=\mathbb{Z}_{pq}$$G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$$p,q\geq 2$, the construction of Conner and Floyd [Conner&Floyd1959], modified and improved by Kister [Kister1961] and [Kister1963], yields a smooth fixed point free actions on Euclidean spaces (see [Bredon1972, pp. 58-61]). For more general groups $G$$G$ such that there exist a surjection $G\to\mathbb{Z}_p$$G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$$\mathbb{Z}_q\to G$, such actions have been obtained by Edmonds and Lee [Edmonds&Lee1976].

#### 2.1.2 Results

The results of [Conner&Montgomery1962], [Hsiang&Hsiang1967], [Conner&Floyd1959], [Kister1961], and [Edmonds&Lee1976] yield the following theorem.

Theorem 2.1. A compact Lie group $G$$G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions 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.

### 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$$F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$$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

Theorem 2.2 ([citation needed]). 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 dimensional CW-complex. Then the following two statements are equivalent.

• $F$$F$ consists of countable many cells.
• There exist a finite dimensional, contractible $G$$G$-CW-complex $X$$X$ with finitely many orbit types, such that the fixed point set $X^G$$X^G$ is homeomorphic to $F$$F$.
• There exists a smooth action of $G$$G$ on some Euclidean space $E$$E$ such that the fixed point set $E^G$$E^G$ is homotopy equivalent to $F$$F$.

It is assumed here, that any smooth manifold admits a countable smooth atlas.

Theorem 2.3 ([Oliver1996]). Let $G$$G$ be a finite group not of prime power order. Let $F$$F$ be a smooth manifold. Then there exists a smooth action of $G$$G$ on some Euclidean space $E$$E$ such that the fixed point $E^G$$E^G$ is diffeomorphic to $F$$F$ if and only if the class $[\tau_F]\in \widetilde{K}O(F)$$[\tau_F]\in \widetilde{K}O(F)$ satisfies the following condition depending on $G$$G$.

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