Group actions on Euclidean spaces

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== Topological actions ==
== Topological actions ==
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==== History ====
==== History ====
<wikitex>
<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.
+
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.
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}.
+
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 theorem.
+
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 $G$ has smooth fixed point free action on some Euclidean space if and only if at least one of the following conditions holds.
+
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.
*The identity connected component $G_0$ of $G$ is non-abelian.
+
{{endthm}}
*The quotient $G/G_0$ is not of prime power order.
+
{{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>;
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 $2pq$. Denote by $G_2$ a $2$-Sylow subgroup of $G$.
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.
+
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 $\widetilde{K}Sp(X)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(X)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(X)$.
*$\mathcal{A}=\{G\;\colon\; G$ 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 $r_\mathbb{C}$:
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$
{{beginrem|Definition}}
{{beginrem|Definition}}
For a abelian group $A$, denote by $\operatorname{qDiv}(A)$ subgroup of quasi divisible elements of $A$, i.e., $\operatorname{qDiv}(A)$ is the intersection of the kernels of all homomorphisms from $A$ to free abelian groups.
+
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 ====
<wikitex>;
<wikitex>;
{{beginthm|Theorem|(\cite{citation needed})}}
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 CW-complex. Then the following two statements are equivalent.
*$F$ consists of countable many cells.
*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})}}
{{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 $G$.
+
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 $\partial F = \varnothing$.
*If $G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$.
+
* The class $[\tau_F]$ of $\widetilde{K}O(F)$ satisfies the following condition depending on $G$.
*If $G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$.
+
** $[\tau_F]$ 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 $q$.
*If $G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$.
+
** $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F))$, if $G$ has a composite order element conjugate to its inverse and $G\notin\mathcal{D}$.
*If $G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex.
+
** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F))$, if $G$ has a composite order element but never conjugate to its inverse and $G_2\ntrianglelefteq G$.
*If $G\in\mathcal{E}$, then $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$.
+
** $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$ is stably complex, if $G$ has a composite order element but never conjugate to its inverse and $G_2\trianglelefteq G$.
*If $G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F)))$.
+
** $[\tau_F]\in\text{qDiv}(\widetilde{K}O(F))$, 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 G=SO(3), Conner and Montgomery [Conner&Montgomery1962] have constructed smooth fixed point free actions of G on Euclidean spaces. By generalizing their construction, Hsiang and Hsiang [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 [Conner&Floyd1959], modified and improved by Kister [Kister1961] and [Kister1963], yields smooth fixed point free actions of G on Euclidean spaces (see [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 [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 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.

Theorem 2.2. 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.

[edit] 2.2 Fixed point sets

[edit] 2.2.1 History

[edit] 2.2.2 Definitions

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).

Definition 2.3. 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.

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

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

[edit] 2.2.3 Results

Theorem 2.5 ([Oliver1996]). 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.

[edit] 3 References

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