# 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 such as 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]. For $G=SO(3)$$\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=SO(3)$, Conner and Montgomery [Conner&Montgomery1962] have constructed smooth fixed point free actions of $G$$G$ on Euclidean spaces. By generalizing their construction, Hsiang and Hsiang [Hsiang&Hsiang1967] have shown that any non-abelian compact connected Lie group $G$$G$ can admit such actions.

Let $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 smooth fixed point free actions of $G$$G$ on Euclidean spaces (see [Bredon1972, pp. 58-61]). In the more general case of $G$$G$ where 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$, smooth fixed point free actions of $G$$G$ on Euclidean spaces have been constructed 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 two theorems.

Theorem 2.1. A compact Lie group $G$$G$ admits a smooth fixed point free action on some Euclidean space if and only if the identity connected component $G_0$$G_0$ of $G$$G$ is non-abelian or the quotient group $G/G_0$$G/G_0$ is not of prime power order.

Theorem 2.2. 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$$F$ is finite dimensional and countable (i.e., consists of countably many cells).
• There exist a finite dimensional, countable, contractible $G$$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$.

### 2.2 Fixed point sets

#### 2.2.2 Definitions

For a 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{X}Sp(F)$$\widetilde{K}O(X)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(X)\xrightarrow{q_\mathbb{C}}\widetilde{X}Sp(F)$
• and the forgetful (complexification and realification) homomorphisms $\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$$\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$.

Definition 2.3. An element of an abelian group $A$$A$ is called quasi-divisible if it belongs to the intersection of the kernels of all homomorphisms from $A$$A$ to free abelian groups.

The subgroup of quasi-divisible elements of $A$$A$ is denoted by $\operatorname{qDiv}(A)$$\operatorname{qDiv}(A)$.

Remark 2.4. If an abelian group $A$$A$ is finitely generated, then $\operatorname{qDiv}(A) = \operatorname{Tor}(A)$$\operatorname{qDiv}(A) = \operatorname{Tor}(A)$, the group of torsion elements of $A$$A$.

#### 2.2.3 Results

Theorem 2.5 ([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 Euclidean space $E$$E$ such that the fixed point $E^G$$E^G$ is diffeomorphic to $F$$F$ if and only if the following two statements hold.

• $F$$F$ has a countable base of topology and $\partial F = \varnothing$$\partial F = \varnothing$.
• 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$$q$.
• $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$$c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$, if $G$$G$ has a composite order element conjugate to its inverse and $G\notin\mathcal{D}$$G\notin\mathcal{D}$.
• $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$$[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$, if $G$$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$$G_2\trianglelefteq G$.
• $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$$[\tau_F]\in\text{qDiv}(\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{qDiv}(\widetilde{K}U(F)))$$[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$, if $G$$G$ has no composite order element and $G_2\trianglelefteq G$$G_2\trianglelefteq G$.