Group actions on disks

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Let $G$ be a finite group. For two distinct primes $p$ and $q$, ''a $pq$-element'' of $G$ is an element of order $pq$. Say that $G$ has $pq$-dihedral subquotient if $G$ contains two subgroups $H$ and $N\trianglelefteq H$ ($N$ normal in $H$) such that $H/N$ is isomorphic to the dihedral group of order $2pq$. Finally, denote by $G_2$ a $2$-Sylow subgroup of $G$.
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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 $N\trianglelefteq H$ ($N$ normal in $H$) such that $H/N$ is isomorphic to the dihedral group of order $2pq$. Finally, 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.
The class of finite groups $G$ not of prime power order divides into the following six mutually disjoint classes.

Revision as of 15:20, 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

Floyd and Richardson [Floyd&Richardson1959] have constructed for the first time a smooth fixed point free action of G on a disk for G=A_5, the alternating group on five letters (see [Bredon1972, pp. 55-58] for a transparent description of the construction). Next, Greever [Greever1960] has described plenty of finite solvable groups G, which can act smoothly on disks without fixed points. Then, Oliver [Oliver1975] and [Oliver1976], has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks.

2.1.2 Oliver number

Let G be a finite group not of prime power order. Oliver in [Oliver1975] proved that the set
\displaystyle \{\chi(X^G)-1\colon X\textup{ is a contractible }G\textup{-CW-complex}\}
is a subgroup of the group of integers \mathbb{Z}. Therefore, the set is of the form n_G\cdot \mathbb{Z} for a unique integer n_G\geq 0, which we refer to as the Oliver number of G.

Oliver has determined integer n_G in the papers [Oliver1975], [Oliver1977], and [Oliver1978]. In particular, the following lemma holds.

Lemma 2.1 (Oliver Lemma). For a finite group G not of prime power order, n_G=1 if and only if there does not exist a sequence P\trianglelefteq H\trianglelefteq G of normal subgroups such that P is a p-group, G/H is a q-group, and H/P is cyclic for two (possibly the same) primes p and q.

Moreover, the work [Oliver1977, Theorem 7] yields the following proposition.

Proposition 2.2. For a finite nilpotent group G not of prime power order, the following conclusions hold:

  • n_G=0 if G has at most one non-cyclic Sylow subgroup.
  • n_G=pq for two distinct primes p and q, if G has just one non-cyclic p-Sylow and q-Sylow subgroups.
  • n_G=1 if G has three or more non-cyclic Sylow subgroups.

The notion of the Oliver number n_G extends to compact Lie groups G as follows.

  • n_G=n_{G/G_0} if G_0 is abelian and G/G_0 is not of prime power order.
  • n_G=1 if G_0 is non-abelian (see [Oliver1976]).


2.1.3 Oliver group

In connection with the work on smooth one fixed point actions on spheres, Laitinen and Morimoto [Laitinen&Morimoto1998] have introduced the notion of Oliver group.

Definition 2.3. A finite group G not of prime power order is called an Oliver group if n_G=1.

Examples of finite Oliver groups include:

  • \mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr} for three distinct primes p, q, and 
    Tex syntax error
    .
  • the groups S_4\oplus \mathbb{Z}_3 and A_4\oplus S_3 of order 72 (which are solvable but not nilpotent).
  • finite non-solvable (in particular, non-trivial perfect) groups (e.g. A_n, S_n for n\geq 5).

2.1.4 Results

The results of Oliver [Oliver1975] and [Oliver1976], are summarized in the following theorem.

Theorem 2.4. A compact Lie group G has a smooth fixed point free action on some disk if and only if at least one of the following condition holds.

  • The identity connected component G_0 of G is non-abelian.
  • The quotient G/G_0 is not of prime power order and n_{G/G_0}=1.


2.2 Fixed point sets

2.2.1 History

2.2.2 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 2.5 ([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 CW-complex. Then the following three statements are equivalent.

  • The Euler-Poincaré characteristic \chi(F)\equiv 1 \pmod{n_G}.
  • There exist a finite contractible G-CW-complex X such that the fixed point set X^G is homeomorphic to F.
  • There exists a smooth action of G on a disk D such that the fixed point set D^G is homotopy equivalent to F.

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 N\trianglelefteq H (N normal in H) such that H/N is isomorphic to the dihedral group of order 2pq. Finally, 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 \}




Theorem 2.6 [Oliver1996]. Let G be a finite group not of prime power order. Let F be a compact smooth manifold. Then there exists a smooth action of G on some disk D such that the fixed point D^G is diffeomorphic to F if and only if \chi(F)\equiv 1\pmod{n_G} and the following conclusion about the class [\tau_F]\in \widetilde{K}O(F) holds.

  • 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}(F)).
  • If G\in\mathcal{C}, then [\tau_F]\in r_{\mathbb{C}}(\widetilde{K}(F))+\text{qDiv}(\widetilde{K}O(F)).
  • If G\in\mathcal{D}, then [\tau_F]\in r_{\mathbb{C}}(\widetilde{K}(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}O(F))).




3 References

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