Group actions on Euclidean spaces

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Any smooth manifold considered here is second countable (i.e. admits a countable smooth atlas).
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It is assumed here, that any smooth manifold admits a countable smooth atlas.
{{beginthm|Theorem|(\cite{Oliver1996})}}
{{beginthm|Theorem|(\cite{Oliver1996})}}

Revision as of 16:49, 27 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 (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 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.

Theorem 2.1. 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.

2.2 Fixed point sets

2.2.1 History

2.2.2 Definitions

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
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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.

  • \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}:

\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)

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 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.

It is assumed here, that any smooth manifold admits a countable smooth atlas.

Theorem 2.3 ([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))).

3 References

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