# Group actions on Euclidean spaces

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## 2 Smooth actions

### 2.1 Fixed point free

#### 2.1.1 History


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$.