Group actions on Euclidean spaces

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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 $G$ 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 $G$.
The class $\mathcal{G}$ of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.
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The class $\mathcal{NP}$ of finite groups not of prime power order divides into the following six mutually disjoint classes.
*$\mathcal{G}_1=\{G\in\mathcal{G}\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.
+
*$\mathcal{NP}_1=\{G\in\mathcal{NP}\;\colon\; G$ has a $pq$-dihedral subquotient $\}$.
*$\mathcal{G}_2=\{G\in\mathcal{G}\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.
+
*$\mathcal{NP}_2=\{G\in\mathcal{NP}\;\colon\; G$ has no $pq$-dihedral subquotient, $G$ has a $pq$-element conjugate to its inverse $\}$.
*$\mathcal{G}_3=\{G\in\mathcal{G}\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.
+
*$\mathcal{NP}_3=\{G\in\mathcal{NP}\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\ntrianglelefteq G$ $\}$.
*$\mathcal{G}_4=\{G\in\mathcal{G}\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.
+
*$\mathcal{NP}_4=\{G\in\mathcal{NP}\;\colon\; G$ has no $pq$-element conjugate to its inverse, $G$ has a $pq$-element, $G_2\trianglelefteq G$ $\}$.
*$\mathcal{G}_5=\{G\in\mathcal{G}\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.
+
*$\mathcal{NP}_5=\{G\in\mathcal{NP}\;\colon\; G$ has no $pq$-element, $G_2\ntrianglelefteq G$ $\}$.
*$\mathcal{G}_6=\{G\in\mathcal{G}\;\colon\; G$ has no $pq$-element, $G_2\trianglelefteq G$ $\}$.
+
*$\mathcal{NP}_6=\{G\in\mathcal{NP}\;\colon\; G$ 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 $q_\mathbb{C}$:
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 $q_\mathbb{C}$:

Revision as of 21:45, 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 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.

The class \mathcal{NP} of finite groups not of prime power order divides into the following six mutually disjoint classes.

  • \mathcal{NP}_1=\{G\in\mathcal{NP}\;\colon\; G has a pq-dihedral subquotient \}.
  • \mathcal{NP}_2=\{G\in\mathcal{NP}\;\colon\; G has no pq-dihedral subquotient, G has a pq-element conjugate to its inverse \}.
  • \mathcal{NP}_3=\{G\in\mathcal{NP}\;\colon\; G has no pq-element conjugate to its inverse, G has a pq-element, G_2\ntrianglelefteq G \}.
  • \mathcal{NP}_4=\{G\in\mathcal{NP}\;\colon\; G has no pq-element conjugate to its inverse, G has a pq-element, G_2\trianglelefteq G \}.
  • \mathcal{NP}_5=\{G\in\mathcal{NP}\;\colon\; G has no pq-element, G_2\ntrianglelefteq G \}.
  • \mathcal{NP}_6=\{G\in\mathcal{NP}\;\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}:

\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).

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

2.2.3 Results

Theorem 2.3 ([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 2.4 ([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]\in \widetilde{K}O(F) satisfies the following condition depending on G:
    • if G\in \mathcal{G}_1, then there is no restriction on [\tau_F].
    • if G\in \mathcal{G}_2, then c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{qDiv}(\widetilde{K}U(F)).
    • if G\in \mathcal{G}_3, then [\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{qDiv}(\widetilde{K}O(F)).
    • if G\in \mathcal{G}_4, then [\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F)), i.e., F is stably complex.
    • if G\in \mathcal{G}_5, then [\tau_F]\in\text{qDiv}(\widetilde{K}O(F)).
    • if G\in \mathcal{G}_6, then [\tau_F]\in r_{\mathbb{C}}(\text{qDiv}(\widetilde{K}U(F))).

3 References

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