Finite groups of diffeomorphisms are topologically determined by a vector field

Abstract

In a previous work it is shown that every finite group G of diffeomorphisms of a connected smooth manifold M of dimension $\ge 2$ equals, up to quotient by the flow, the centralizer of the group of smooth automorphisms of a G-invariant complete vector field X (shortly X describes G). Here the foregoing result is extended to show that every finite group of diffeomorphisms of M is described, within the group of all homeomorphisms of M, by a vector field. As a consequence, it is proved that a finite group of homeomorphisms of a compact connected topological 4-manifold, whose action is free, is described by a continuous flow.

Appendix

In this section B(r), $r>0$, will be the ball in ${\mathbb {R}}^m$, endowed with coordinates $(x_1 ,\ldots ,x_m )$, of center the origin and radius r, and $\Gamma :{\mathbb {R}}^m \rightarrow {\mathbb {R}}^m$ the symmetry given by $\Gamma (x_1 ,\ldots ,x_m )=(x_1 ,\ldots ,x_{m-1},-x_m )$.

Lemma 7.1

Consider a function $\mu$ defined around the origin of ${\mathbb {R}}^m$ and $\Gamma$-invariant. Assume that the origin is a non-degenerated singularity. Then about the origin there exist coordinates $(y_1 ,\ldots ,y_m )$ such the coordinates of the origin are still $(0,\ldots ,0)$,

$$\begin{aligned} \mu =\sum _{j=1}^{k}y_{j}^{2} -\sum _{j=k+1}^{m-1}y_{j}^{2}+\varepsilon y_{m}^2 +\mu (0) \end{aligned}$$

where $\varepsilon =\pm 1$, and $\Gamma (y_1 ,\ldots ,y_m )=(y_1 ,\ldots ,y_{m-1},-y_m )$.

Proof

As $\mu$ is $\Gamma$-invariant its restriction to the hyperplane H defined by $x_m =0$ has a non-degenerated singularity at the origin. Therefore coordinates $(x_1 ,\ldots ,x_m )$ can be replaced by coordinates $(y_1 ,\ldots ,y_{m-1},x_m )$ in such a way that $\Gamma (y_1 ,\ldots ,y_{m-1},x_m )=(y_1 ,\ldots ,y_{m-1},-x_m )$ and

$$\begin{aligned} \mu (y_1 ,\ldots ,y_{m-1},0)=\sum _{j=1}^{k}y_{j}^{2} -\sum _{j=k+1}^{m-1}y_{j}^{2}+\mu (0). \end{aligned}$$

On the other hand from the Taylor expansion in variable $x_m$ transversely to H it follows

$$\begin{aligned} \mu (y_1 ,\ldots ,y_{m-1},x_m )= & {} \mu (y_1 ,\ldots ,y_{m-1},0) +x_m {\frac{\partial \mu }{\partial x_m}}(y_1 ,\ldots ,y_{m-1},0) \\&+\,x_{m}^2 f(y_1 ,\ldots ,y_{m-1},x_m ). \end{aligned}$$

By the $\Gamma$-invariance ${\frac{\partial \mu }{\partial x_m}} (y_1 ,\ldots ,y_{m-1},0)=0$ and $f(y_1 ,\ldots ,y_{m-1},-x_m ) = f(y_1 ,\ldots ,y_{m-1},x_m )$. Moreover $2f(0)={\frac{\partial ^2 \mu }{\partial x_{m}^2}}(0)\ne 0$ since the origin is a non-degenerated singularity. Therefore close to the origin $(y_1 ,\ldots ,y_{m-1},y_m )$, where $y_m =x_m |f(y_1 ,\ldots ,y_{m-1},x_m )|^{1/2}$, is a system of coordinates as required. $\square$

Proposition 7.2

For every $r>0$ there exists a Morse function $\tau :{\mathbb {R}}^m \rightarrow {\mathbb {R}}$ such that:

• (a) $\tau$ is $\Gamma$-invariant.

• (b) If p is a minimum of $\tau$ then $\Gamma (p)\ne p$, that is to say p does not belong to the hyperplane $x_m =0$.

• (c) $|\tau (x)|\le ||x||^2$ on ${\mathbb {R}}^m$ and $\tau (x)=||x||^2$ on ${\mathbb {R}}^m -B(r)$.

For proving the foregoing proposition we need:

Lemma 7.3

There exists a smooth function $\rho :{\mathbb {R}}\rightarrow {\mathbb {R}}$ such that

• (a) $\rho (t)=1$ if $t\ge 1$, $\rho (t)=-1$ if $t\le 0$, $\rho (1/2)=0$, $0<\rho <1$ on (1/2, 1) and $-1<\rho <0$ on (0, 1/2). Moreover $\rho (1-t)=-\rho (t)$, $t\in {\mathbb {R}}$, that t is to say $\rho$ is anti-symmetrical with respect to $t=1/2$.

• (b) $\rho \,'\ge 0$ on ${\mathbb {R}}$ and $\rho \,'>0$ on (0, 1). Moreover $\rho \,'(1-t)=\rho \,'(t)$, $t\in {\mathbb {R}}$.

• (c) $\rho \,''>0$ on (0, 1/2), $\rho \,''<0$ on (1/2, 1) and $\rho \,''=0$ on $({\mathbb {R}}-(0,1))\cup \{1/2\}$. Moreover $\rho \,''(1-t)=-\rho \,''(t)$, $t\in {\mathbb {R}}$.

Proof

Let $\varphi$ be a smooth function meeting the requirements of (c). Denote by $\varphi _1$ its primitive with initial condition $\varphi _1 (0)=0$ and by $\varphi _2$ the primitive of $\varphi _1$ such that $\varphi _2 (1/2)=0$.

Then $\varphi _2$ is constant and positive on $[1,\infty )$ while it is constant and negative on $(-\infty ,0]$. Moreover $\varphi _2 (0)=-\varphi _2 (1)$. The function $\rho =(\varphi _2 (1))^{-1}\varphi _2$ meets the requirements of the lemma (draw the graphics of $\varphi$, $\varphi _1$ and $\varphi _2$). $\square$

Corollary 7.4

Consider the function $\lambda$ defined by $\lambda (t)=t\rho (t)$. If $\lambda '(c)=0$ then $c\in (0,1/2)$ and $\lambda ''(c)>0$.

Proof of Proposition 7.2

First observe that if $\tau$ is like in Proposition 7.2 for $r=1$, then for any other $r>0$ it suffices to take $\tau _r (x)=r^2 \tau (r^{-1}x)$.

Consider a function $\rho$ as in Lemma 7.3 and set $\tau (x)=x^{2}_1 +\cdots +x^{2}_{m-1}+x^{2}_{m}\rho (||x||^2)$. Then $|\tau (x)|\le ||x||^2$ everywhere and $\tau (x)=||x||^2$ if $||x||\ge 1$. On the other hand an elementary computation making use of Corollary 7.4 shows that the singularities of $\tau$ are always non-degenerate and belong to the last axis, while the origin is a saddle. $\square$

Remark 7.5

As in one variable between two consecutive minima there always exists a maximum, the function $\lambda$ of Corollary 7.4 has a single singularity, which is a minimum. A more careful computation shows that function $\tau$ of the proof of Proposition 7.2 has just three singular points: a saddle and two minima.