Group actions on Euclidean spaces

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==== Definitions ====
==== Definitions ====
<wikitex>;
<wikitex>;
Consider the following to classes of finite groups $G$.
* $\mathcal{C} = \{G \;\colon\; G$ has an element of order $pq$ for some distinct primes $p$ and $q$, and the element is conjugate to its inverse $\}$,
* $\mathcal{D} = \{G \;\colon\; G$ has a subquotient of the form $D_{pq}$ for two distinct primes $p$ and $q$ $\}$,
Let $G$ be a finite group. Let $p$ and $q$ be two distinct primes. An element of $G$ of order $pq$ is called a ''$pq$-element'' of $G$. Moreover, $G$ ''has a $pq$-dihedral subquotient'' if $G$ contains two subgroups $H$ and $K$ such that $K\trianglelefteq H$ and $H/K \cong D_{pq}$, the dihedral group of order $2pq$. Let $G_2$ be a $2$-Sylow subgroup of $G$.
Let $\mathcal{P}$ be the class of finite groups of prime power order. The class $\mathcal{N}$ of finite groups not of prime power order divide into six mutually disjoint classes, as follows:
*$\mathcal{G}_1 =\{G\in\mathcal{G}\;\colon\; G$ has a subquotient of the form $D_{pq}$ for two distinct primes $p$ and $q$ $\}$,
*$\mathcal{G}_2=\{G\in\mathcal{G}\;\colon\; G$ has a composite order element conjugate to its inverse, $G\notin\mathcal{G}_1$ $\}$,
*$\mathcal{G}_3=\{G\in\mathcal{G}\;\colon\; G$ has a composite order element but never conjugate to its inverse, $G_2\ntrianglelefteq G$ $\}$,
*$\mathcal{G}_4=\{G\in\mathcal{G}\;\colon\; G$ has a composite order element but never conjugate to its inverse, $G_2\trianglelefteq G$ $\}$,
*$\mathcal{G}_5=\{G\in\mathcal{G}\;\colon\; G$ has no composite order element, $G_2\ntrianglelefteq G$ $\}$,
*$\mathcal{G}_6=\{G\in\mathcal{G}\;\colon\; G$ has no composite order 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 $q_\mathbb{C}$:
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 $r_\mathbb{C}$:
$$\widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$ and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$ and $r_\mathbb{C}$:
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$
$$\widetilde{K}Sp(F)\xrightarrow{c_\mathbb{H}}\widetilde{K}U(F)\xrightarrow{r_\mathbb{C}}\widetilde{K}O(F)$$
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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.
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.
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{{endrem}}
{{endrem}}
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{{beginrem|Remark}}
If an abelian group $A$ is finitely generated, then $\operatorname{qDiv}(A)$ is the group $\operatorname{Tor}(A)$ of torsion elements of $A$.
If an abelian group $A$ is finitely generated, then $\operatorname{qDiv}(A)$ is the group $\operatorname{Tor}(A)$ of torsion elements of $A$.
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{{beginrem|Remark}}
</wikitex>
</wikitex>

Revision as of 23:38, 29 November 2010


This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Topological actions

...


2 Smooth actions

2.1 Fixed point free

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].


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 admits a 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 group G/G_0 is not of prime power order.

2.2 Fixed point sets

2.2.1 History

2.2.2 Definitions

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}:

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

Definition 2.2. 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 2.3. If an abelian group A is finitely generated, then \operatorname{qDiv}(A) is the group \operatorname{Tor}(A) of torsion elements of A.

Remark 2.4.


2.2.3 Results

Theorem 2.5 ([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 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.


Theorem 2.6 ([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 following two statements hold.

  • F admits a countable atlas 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.

3 References

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