# 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$$ {{Stub}} == Topological actions == ; ... == Smooth actions == === Fixed point free === ==== 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 \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of G on Euclidean spaces, for G=SO(3). By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group admits such actions. For G=\mathbb{Z}_{pq} for two relatively primes integers p,q\geq 2, the construction of Conner and Floyd \cite{Conner&Floyd1959}, modified and improved by Kister \cite{Kister1961} and \cite{Kister1963}, yields a smooth fixed point free actions on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). For more general groups G such that there exist a surjection G\to\mathbb{Z}_p and an injection \mathbb{Z}_q\to G, such actions have been obtained by Edmonds and Lee \cite{Edmonds&Lee1976}. ==== Results ==== The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following theorem. {{beginthm|Theorem}} A compact Lie group 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 of G is non-abelian. *The quotient G/G_0 is not of prime power order. {{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 quaternion 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 dimensional, with countable many cells CW-complex. Then the following two statements are equivalent. *There exist a finite dimensional, contractible G-CW-complex X with finitely many orbit types, such that the fixed point set X^G is homeomorphic to F. *There exists a smooth action of G on some Euclidean space E such that the fixed point set E^G is homotopy equivalent to F. {{endthm}} It is assumed here, that any smooth manifold admits a countable smooth atlas. {{beginthm|Theorem|(\cite{Oliver1996})}} Let G be a finite group not of prime power order. Let F be a smooth manifold. Then there exists a smooth action of G on some Euclidean space E such that the fixed point E^G is diffeomorphic to F if and only if the class [\tau_F]\in \widetilde{K}O(F) satisfies the following condition depending on G. *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{qDiv}(\widetilde{K}U(F)). *If G\in\mathcal{C}, then [\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\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{qDiv}(\widetilde{K}O(F)). *If G\in\mathcal{F}, then [\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F))). {{endthm}} == References == {{#RefList:}} [[Category:Theory]] [[Category:Group actions on manifolds]]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$$G$ contains two subgroups
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$H$ and $K\trianglelefteq H$$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

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 2.2 ([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 dimensional, with countable many cells CW-complex. Then the following two statements are equivalent.

• 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)))$.