Group actions on disks

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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$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ on a disk for $G=A_5$$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$$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$$G$ be a finite group not of prime power order. Oliver in [Oliver1975] has proven that the set

$\displaystyle \mathcal{Z}_G = \{\chi(X^G)-1 \colon \ X\textup{ is a finite contractible }G\textup{-CW-complex}\} \ \subseteq \ \mathbb{Z}$ $$\displaystyle \mathcal{Z}_G = \{\chi(X^G)-1 \colon \ X\textup{ is a finite contractible }G\textup{-CW-complex}\} \ \subseteq \ \mathbb{Z}$$

is a subgroup of the group of integers $\mathbb{Z}$$\mathbb{Z}$. Therefore, $\mathcal{Z}_G = n_G\cdot \mathbb{Z}$$\mathcal{Z}_G = n_G\cdot \mathbb{Z}$ for a unique integer $n_G\geq 0$$n_G\geq 0$, which we refer to as the Oliver number of $G$$G$.

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

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

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

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

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

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

• $n_G=n_{G/G_0}$$n_G=n_{G/G_0}$ if $G_0$$G_0$ is abelian and $G/G_0$$G/G_0$ is not of prime power order.
• $n_G=1$$n_G=1$ if $G_0$$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$$G$ not of prime power order is called an Oliver group if $n_G=1$$n_G=1$.

Examples of finite Oliver groups include:

• $\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$$\mathbb{Z}_{pqr}\times \mathbb{Z}_{pqr}$ for three distinct primes $p$$p$, $q$$q$, and $r$$r$.
• the groups $S_4\oplus \mathbb{Z}_3$$S_4\oplus \mathbb{Z}_3$ and $A_4\oplus S_3$$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$$A_n$, $S_n$$S_n$ for $n\geq 5$$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$$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$$G_0$ of $G$$G$ is non-abelian.
• The quotient $G/G_0$$G/G_0$ is not of prime power order and $n_{G/G_0}=1$$n_{G/G_0}=1$.

2.2 Fixed point sets

2.2.2 Definitions

Let $G$$G$ be a finite group. For two distinct primes $p$$p$ and $q$$q$, a $pq$$pq$-element of $G$$G$ is an element of order $pq$$pq$. One says that $G$$G$ has $pq$$pq$-dihedral subquotient if $G$$G$ contains two subgroups $H$$H$ and $K\trianglelefteq H$$K\trianglelefteq H$ such that $H/K$$H/K$ is isomorphic to the dihedral group of order $2pq$$2pq$. Denote by $G_2$$G_2$ a $2$$2$-Sylow subgroup of $G$$G$.

The class of finite groups $G$$G$ not of prime power order divides into the following six mutually disjoint classes.

• $\mathcal{A}=\{G\;\colon\; G$$\mathcal{A}=\{G\;\colon\; G$ has a $pq$$pq$-dihedral subquotient $\}$$\}$.
• $\mathcal{B}=\{G\;\colon\; G$$\mathcal{B}=\{G\;\colon\; G$ has no $pq$$pq$-dihedral subquotient, $G$$G$ has a $pq$$pq$-element conjugate to its inverse $\}$$\}$.
• $\mathcal{C}=\{G\;\colon\; G$$\mathcal{C}=\{G\;\colon\; G$ has no $pq$$pq$-element conjugate to its inverse, $G$$G$ has a $pq$$pq$-element, $G_2\ntrianglelefteq G$$G_2\ntrianglelefteq G$ $\}$$\}$.
• $\mathcal{D}=\{G\;\colon\; G$$\mathcal{D}=\{G\;\colon\; G$ has no $pq$$pq$-element conjugate to its inverse, $G$$G$ has a $pq$$pq$-element, $G_2\trianglelefteq G$$G_2\trianglelefteq G$ $\}$$\}$.
• $\mathcal{E}=\{G\;\colon\; G$$\mathcal{E}=\{G\;\colon\; G$ has no $pq$$pq$-element, $G_2\ntrianglelefteq G$$G_2\ntrianglelefteq G$ $\}$$\}$.
• $\mathcal{F}=\{G\;\colon\; G$$\mathcal{F}=\{G\;\colon\; G$ has no $pq$$pq$-element, $G_2\trianglelefteq G$$G_2\trianglelefteq G$ $\}$$\}$.

Let $F$$F$ be a compact smooth manifold. Between the reduced real, complex, and quaternion $K$$K$-theory groups $\widetilde{K}O(F)$$\widetilde{K}O(F)$, $\widetilde{K}U(F)$$\widetilde{K}U(F)$, and $\widetilde{K}Sp(F)$$\widetilde{K}Sp(F)$, respectively, consider the induction (complexification and quaternization) homomorphisms $c_\mathbb{R}$$c_\mathbb{R}$ and $q_\mathbb{C}$$q_\mathbb{C}$:

$\displaystyle \widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$ $$\displaystyle \widetilde{K}O(F)\xrightarrow{c_\mathbb{R}}\widetilde{K}U(F)\xrightarrow{q_\mathbb{C}}\widetilde{K}Sp(F)$$

and the forgetful (complexification and realification) homomorphisms $c_\mathbb{H}$$c_\mathbb{H}$ and $r_\mathbb{C}$$r_\mathbb{C}$:

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

For a finitely generated abelian group $A$$A$, denote by $\operatorname{Tor}(A)$$\operatorname{Tor}(A)$ the torsion part of $A$$A$.

2.2.3 Results

Let $G$$G$ be a compact Lie group such that the identity connected component $G_0$$G_0$ of $G$$G$ is non-abelian, or the quotient $G/G_0$$G/G_0$ is not of prime power order. Oliver [Oliver1975], [Oliver1976] has defined an integer $n_G\geq 0$$n_G\geq 0$, which we refer to as the Oliver number of $G$$G$. Recall that $n_G=n_{G/G_0}$$n_G=n_{G/G_0}$ when $G_0$$G_0$ is abelian, and otherwise $n_G=1$$n_G=1$.

Theorem 2.5 ([Oliver1975], [Oliver1976]). Let $G$$G$ be a compact Lie group such that the identity connected component $G_0$$G_0$ of $G$$G$ is non-abelian, or the quotient $G/G_0$$G/G_0$ is not of prime power order. Let $F$$F$ be a CW-complex. Then the following three statements are equivalent.

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

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

• If $G\in \mathcal{A}$$G\in \mathcal{A}$, then there is no restriction on $[\tau_F]$$[\tau_F]$.
• If $G\in \mathcal{B}$$G\in \mathcal{B}$, then $c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$$c_{\mathbb{R}}([\tau_F])\in c_{\mathbb{H}}(\widetilde{K}Sp(F))+\text{Tor}(\widetilde{K}U(F))$.
• If $G\in\mathcal{C}$$G\in\mathcal{C}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$$[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))+\text{Tor}(\widetilde{K}O(F))$.
• If $G\in\mathcal{D}$$G\in\mathcal{D}$, then $[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$$[\tau_F]\in r_{\mathbb{C}}(\widetilde{K}U(F))$, i.e., $F$$F$ is stably complex.
• If $G\in\mathcal{E}$$G\in\mathcal{E}$, then $[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$$[\tau_F]\in\text{Tor}(\widetilde{K}O(F))$.
• If $G\in\mathcal{F}$$G\in\mathcal{F}$, then $[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$$[\tau_F]\in r_{\mathbb{C}}(\text{Tor}(\widetilde{K}U(F)))$.

3 References

• [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
• [Floyd&Richardson1959] E. E. Floyd and R. W. Richardson, An action of a finite group on an $n$$n$-cell without stationary points. , Bull. Amer. Math. Soc. 65 (1959), 73–76. MR0100848 (20 #7276) Zbl 0088.15302
• [Greever1960] J. Greever, Stationary points for finite transformation groups, Duke Math. J 27 (1960), 163–170. MR0110094 (22 #977) Zbl 0113.16505
• [Laitinen&Morimoto1998] E. Laitinen and M. Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), no.4, 479–520. MR1631012 (99k:57078) Zbl 0905.57023
• [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
• [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
• [Oliver1977] R. Oliver, $G$$G$-actions on disks and permutation representations. II, Math. Z. 157 (1977), no.3, 237–263. MR0646085 (58 #31126) Zbl 0386.20002
• [Oliver1978] R. Oliver, $G$$G$-actions on disks and permutation representations, J. Algebra 50 (1978), no.1, 44–62. MR0501044 (58 #18508) Zbl 0386.20002
• [Oliver1996] B. Oliver, Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, Topology 35 (1996), no.3, 583–615. MR1396768 (97g:57059) Zbl 0861.57047