Group actions on Euclidean spaces

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1 Topological actions

...

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 $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Topological actions == <wikitex>; ... </wikitex> == Smooth actions == === Fixed point free === ==== History ==== <wikitex> 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}. </wikitex> ==== Results ==== <wikitex> 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}} </wikitex> === Fixed point sets === ==== History ==== ==== Definitions ==== <wikitex>; 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 $A$. </wikitex> ==== Results ==== <wikitex>; 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}} Any smooth manifold considered here is second countable (i.e. 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}} </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Group actions on manifolds]]G$ on Euclidean spaces, for $G=SO(3)$. By generalizing their construction, Hsiang and Hsiang [Hsiang&Hsiang1967] have shown that any non-abelian compact connected Lie group admits such actions.

For $G=\mathbb{Z}_{pq}$ for two relatively primes integers $p,q\geq 2$, 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 $G$ 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 [Edmonds&Lee1976].

2.1.2 Results

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

2.2 Fixed point sets

2.2.1 History

2.2.2 Definitions

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a $pq$-element

has $pq$-dihedral subquotient

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, $G_2\trianglelefteq G$ $\}$.

Tex syntax error

$$\displaystyle \widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$

$$\displaystyle \widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$

For a finitely generated abelian group $A$, denote by $\operatorname{Tor}(A)$ the torsion part of $A$.

2.2.3 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 [Oliver1975], [Oliver1976] has defined an integer $n_G\geq 0$, which we refer to as the Oliver number of $G$. Recall that $n_G=n_{G/G_0}$ when $G_0$ is abelian, and otherwise $n_G=1$.

Theorem 2.2 ([Oliver1975], [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

Tex syntax error

$F$ 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

  Tex syntax error

  $F$.
• There exists a smooth action of $G$ on some Euclidean space

  Tex syntax error

  $E$ such that the fixed point set $E^G$ is homotopy equivalent to

  Tex syntax error

  $F$.

Any smooth manifold considered here is second countable (i.e. admits a countable smooth atlas).

Theorem 2.3 ([Oliver1996]).

Let $G$ be a finite group not of prime power order. Let

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$F$ be a smooth manifold. Then there exists a smooth action of $G$ on some Euclidean space

Tex syntax error

$E$ such that the fixed point $E^G$ is diffeomorphic to

Tex syntax error

$F$ 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.,

  Tex syntax error

  $F$ is stably complex.
• If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.
• If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.

3 References

• [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
• [Conner&Floyd1959] P. E. Conner and E. E. Floyd, On the construction of periodic maps without fixed points, Proc. Amer. Math. Soc. 10 (1959), 354–360. MR0105115 (21 #3860) Zbl 0092.39701
• [Conner&Montgomery1962] P. Conner and D. Montgomery, An example for $SO(3)$, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1918–1922. MR0148795 (26 #6300) Zbl 0107.16604
• [Edmonds&Lee1976] A. L. Edmonds and R. Lee, Compact Lie groups which act on Euclidean space without fixed points, Proc. Amer. Math. Soc. 55 (1976), no.2, 416–418. MR0420672 (54 #8684) Zbl 0326.57011
• [Hsiang&Hsiang1967] W. Hsiang and W. Hsiang, Differentiable actions of compact connected classical groups. I, Amer. J. Math. 89 (1967), 705–786. MR0217213 (36 #304) Zbl 0205.53902
• [Kister1961] J. M. Kister, Examples of periodic maps on Euclidean spaces without fixed points. , Bull. Amer. Math. Soc. 67 (1961), 471–474. MR0130929 (24 #A783) Zbl 0101.15602
• [Kister1963] J. M. Kister, Differentiable periodic actions on $E^8$ without fixed points, Amer. J. Math. 85 (1963), 316–319. MR0154278 (27 #4227) Zbl 0119.18801
• [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
• [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
• [Oliver1996] B. Oliver, Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, Topology 35 (1996), no.3, 583–615. MR1396768 (97g:57059) Zbl 0861.57047
• [Smith1938] P. A. Smith, Transformations of finite period, Ann. of Math. (2) 39 (1938), no.1, 127–164. MR1503393 Zbl 0063.07093
• [Smith1939] P. A. Smith, Transformations of finite period. II, Ann. of Math. (2) 40 (1939), 690–711. MR0000177 (1,30c) Zbl 0063.07093
• [Smith1941] P. A. Smith, Transformations of finite period. III. Newman's theorem, Ann. of Math. (2) 42 (1941), 446–458. MR0004128 (2,324c)
• [Smith1945] P. A. Smith, Transformations of finite period. IV. Dimensional parity, Ann. of Math. (2) 46 (1945), 357–364. MR0013304 (7,136e) Zbl 0063.07093