Group actions on Euclidean spaces
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{{Stub}} | {{Stub}} | ||
== Topological actions == | == Topological actions == | ||
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==== History ==== | ==== History ==== | ||
<wikitex> | <wikitex> | ||
− | The question whether contractible manifolds | + | 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 \cite{Smith1938}, \cite{Smith1939}, \cite{Smith1941}, and \cite{Smith1945}. For $G=SO(3)$, Conner and Montgomery \cite{Conner&Montgomery1962} have constructed smooth fixed point free actions of $G$ on Euclidean spaces. By generalizing their construction, Hsiang and Hsiang \cite{Hsiang&Hsiang1967} have shown that any non-abelian compact connected Lie group $G$ can admit such actions. |
− | + | Let $G=\mathbb{Z}_{pq}$ 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 smooth fixed point free actions of $G$ on Euclidean spaces (see {{cite|Bredon1972|pp. 58-61}}). In the more general case of $G$ where there exist a surjection $G\to\mathbb{Z}_p$ and an injection $\mathbb{Z}_q\to G$, smooth fixed point free actions of $G$ on Euclidean spaces have been constructed by Edmonds and Lee \cite{Edmonds&Lee1976}. | |
</wikitex> | </wikitex> | ||
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==== Results ==== | ==== Results ==== | ||
<wikitex> | <wikitex> | ||
− | The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following | + | The results of \cite{Conner&Montgomery1962}, \cite{Hsiang&Hsiang1967}, \cite{Conner&Floyd1959}, \cite{Kister1961}, and \cite{Edmonds&Lee1976} yield the following two theorems. |
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
− | A compact Lie group $G$ | + | A compact Lie group $G$ admits a smooth fixed point free action on some Euclidean space if and only if the identity connected component $G_0$ of $G$ is non-abelian or the quotient group $G/G_0$ is not of prime power order. |
− | + | {{endthm}} | |
− | + | {{beginthm|Theorem}} | |
+ | Let $G$ be a compact Lie group such that the identity connected component $G_0$ of $G$ is non-abelian or the quotient group $G/G_0$ is not of prime power order. Let $F$ be a CW complex. Then the following three statements are equivalent. | ||
+ | *$F$ is finite dimensional and countable (i.e., consists of countably many cells). | ||
+ | *There exist a finite dimensional, countable, 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}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
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==== Definitions ==== | ==== Definitions ==== | ||
<wikitex>; | <wikitex>; | ||
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− | + | For a space $X$, between the reduced real, complex, and quaternion $K$-theory groups $\widetilde{K}O(X)$, $\widetilde{K}U(X)$, and $\widetilde{K}Sp(X)$, respectively, consider | |
+ | * the induction (complexification and quaternization) homomorphisms $\widetilde{K}O(X)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(X)\xrightarrow{q_\mathbb{C}}\widetilde{X}Sp(F)$ | ||
+ | * and the forgetful (complexification and realification) homomorphisms $\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$. | ||
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{{beginrem|Definition}} | {{beginrem|Definition}} | ||
− | + | An element of an abelian group $A$ is called ''quasi-divisible'' if it belongs to the intersection of the kernels of all homomorphisms from $A$ to free abelian groups. | |
+ | {{endrem}} | ||
+ | |||
+ | The subgroup of quasi-divisible elements of $A$ is denoted by $\operatorname{qDiv}(A)$. | ||
+ | |||
+ | {{beginrem|Remark}} | ||
+ | If an abelian group $A$ is finitely generated, then $\operatorname{qDiv}(A) = \operatorname{Tor}(A)$, the group of torsion elements of $A$. | ||
{{endrem}} | {{endrem}} | ||
</wikitex> | </wikitex> | ||
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==== Results ==== | ==== Results ==== | ||
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{{beginthm|Theorem|(\cite{Oliver1996})}} | {{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] | + | Let $G$ be a finite group not of prime power order, and let $G_2$ denote a $2$-Sylow subgroup of $G$. 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 following two statements hold. |
− | + | * $F$ has a countable base of topology and $\partial F = \varnothing$. | |
− | * | + | * The class $[\tau_F]$ of $\widetilde{K}O(F)$ satisfies the following condition depending on $G$. |
− | + | ** $[\tau_F]$ is arbitrary, if $G$ is in the class $\mathcal{D}$ of finite groups with dihedral subquotient of order $2pq$ for two distinct primes $p$ and $q$. | |
− | + | ** $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$, if $G$ has a composite order element conjugate to its inverse and $G\notin\mathcal{D}$. | |
− | + | ** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$, if $G$ has a composite order element but never conjugate to its inverse and $G_2\ntrianglelefteq G$. | |
− | + | ** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex, if $G$ has a composite order element but never conjugate to its inverse and $G_2\trianglelefteq G$. | |
− | + | ** $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$, if $G$ has no composite order element and $G_2\ntrianglelefteq G$. | |
− | + | ** $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$, if $G$ has no composite order element and $G_2\trianglelefteq G$. | |
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> |
Latest revision as of 17:13, 10 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Topological actions
...
[edit] 2 Smooth actions
[edit] 2.1 Fixed point free
[edit] 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 , Conner and Montgomery [Conner&Montgomery1962] have constructed smooth fixed point free actions of on Euclidean spaces. By generalizing their construction, Hsiang and Hsiang [Hsiang&Hsiang1967] have shown that any non-abelian compact connected Lie group can admit such actions.
Let for two relatively primes integers . The construction of Conner and Floyd [Conner&Floyd1959], modified and improved by Kister [Kister1961] and [Kister1963], yields smooth fixed point free actions of on Euclidean spaces (see [Bredon1972, pp. 58-61]). In the more general case of where there exist a surjection and an injection , smooth fixed point free actions of on Euclidean spaces have been constructed by Edmonds and Lee [Edmonds&Lee1976].
[edit] 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 admits a smooth fixed point free action on some Euclidean space if and only if the identity connected component of is non-abelian or the quotient group is not of prime power order.
Theorem 2.2. Let be a compact Lie group such that the identity connected component of is non-abelian or the quotient group is not of prime power order. Let be a CW complex. Then the following three statements are equivalent.
- is finite dimensional and countable (i.e., consists of countably many cells).
- There exist a finite dimensional, countable, contractible -CW-complex with finitely many orbit types, such that the fixed point set is homeomorphic to .
- There exists a smooth action of on some Euclidean space such that the fixed point set is homotopy equivalent to .
[edit] 2.2 Fixed point sets
[edit] 2.2.1 History
[edit] 2.2.2 Definitions
For a space , between the reduced real, complex, and quaternion -theory groups , , and , respectively, consider
- the induction (complexification and quaternization) homomorphisms
- and the forgetful (complexification and realification) homomorphisms .
Definition 2.3. An element of an abelian group is called quasi-divisible if it belongs to the intersection of the kernels of all homomorphisms from to free abelian groups.
The subgroup of quasi-divisible elements of is denoted by .
Remark 2.4. If an abelian group is finitely generated, then , the group of torsion elements of .
[edit] 2.2.3 Results
Theorem 2.5 ([Oliver1996]). Let be a finite group not of prime power order, and let denote a -Sylow subgroup of . Let be a smooth manifold. Then there exists a smooth action of on some Euclidean space such that the fixed point is diffeomorphic to if and only if the following two statements hold.
- has a countable base of topology and .
- The class of satisfies the following condition depending on .
- is arbitrary, if is in the class of finite groups with dihedral subquotient of order for two distinct primes and .
- , if has a composite order element conjugate to its inverse and .
- , if has a composite order element but never conjugate to its inverse and .
- , i.e., is stably complex, if has a composite order element but never conjugate to its inverse and .
- , if has no composite order element and .
- , if has no composite order element and .
[edit] 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 , 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 without fixed points, Amer. J. Math. 85 (1963), 316–319. MR0154278 (27 #4227) Zbl 0119.18801
- [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