# On the $n$-transitivity of the group of equivariant diffeomorphisms

Marja Kankaanrinta  Tampere University<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. If $M$ is
connected and $\dim(M)\geq 2$, the group of diffeomorphisms of $M$, that are
isotopic to the identity through a compactly supported isotopy, acts
$n$-transitively on $M$, for any $n$. In this paper, we prove a version of the
$n$-transitivity result for the group of equivariant diffeomorphisms of $M$.
As a corollary we obtain a result concerning diffeomorphisms of the orbit
space $M/G$. A special case of the result for orbit spaces gives an
$n$-transitivity result for orbifold diffeomorphisms that was earlier proved
by F. Pasquotto and T. O. Rot.

##  0\. Introduction

Let $M$ be a connected smooth manifold of dimension at least two. A subgroup
$H$ of the diffeomorphism group ${\rm Diff}(M)$ is said to act
$n$-transitively on $M$, if for any two $n$-tuples of pairwise distinct points
$(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$ in $M$ there is an element
$f\in H$ such that $f(x_{i})=y_{i}$, for all $i\in\\{1,\ldots,n\\}$. In [7],
P. Michor and C. Vizman studied the $n$-transitivity of actions of various
subgroups of ${\rm Diff}(M)$ on $M$. In particular, they proved that the group
${\rm Diff}_{c}(M)$ of diffeomorphisms of $M$, that are isotopic to the
identity through a compactly supported isotopy, acts $n$-transitively on $M$,
for all $n\in{\mathbb{N}}$.

Assume a Lie group $G$ acts smoothly on $M$. It is obvious that in general the
group ${\rm Diff}^{G}(M)$ of equivariant diffeomorphisms of $M$ cannot act
transitively on $M$. For example, if $f\colon M\to M$ is a $G$-equivariant
diffeomorphism, then the isotropy groups at the points $x$ and $f(x)$ must be
the same for all $x\in M$. Also, the value of an equivariant map at $gx$, for
any $g\in G$, is already determined by the value of the map at $x$. Taking
into account these restrictions we obtain an equivariant version of the
$n$-transitivity result:

######  Theorem 0.1.

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold with orbit
space $M/G$. Let $n\in{\mathbb{N}}$, and let $(x_{1},\ldots,x_{n})$ and
$(y_{1},\ldots,y_{n})$ be two $n$-tuples of points in $M$. Assume the
following conditions are satisfied:

  1. (1)

$(Gx_{1},\ldots,Gx_{n})$ and $(Gy_{1},\ldots,Gy_{n})$ are two $n$-tuples of
pairwise distinct points in the orbit space $M/G$.

  2. (2)

For every $i$, $1\leq i\leq n$, $G_{x_{i}}=G_{y_{i}}$ and $x_{i}$ and $y_{i}$
lie in the same connected component of $M_{G_{x_{i}}}=\\{x\in M\mid
G_{x}=G_{x_{i}}\\}$.

  3. (3)

If $H$ is a compact subgroup of $G$, then any connected component $Y$ of
$M_{H}$ satisfying $\dim\bigl{(}N(H)/H\bigr{)}=\dim(Y)-1$ contains at most one
of the points $x_{i}$.

Then there exists a smooth $G$-equivariant diffeomorphism $f\colon M\to M$
isotopic to the identity through a $G$-compactly supported isotopy, such that
$f(x_{i})=y_{i}$ for all $1\leq i\leq n$.

In [10], F. Pasquotto and T. O. Rot proved a version of the $n$-transitivity
result for orbifold diffeomorphisms. As in the equivariant case, one has to
take into account the local groups at points of the orbifold. Since every
smooth reduced, i.e., effective, orbifold can be considered as a quotient of a
smooth, effective, almost free action of a compact Lie group on a smooth
manifold, it follows that the $n$-transitivity results in the equivariant case
and in the orbifold case are closely related. Indeed, by considering
diffeomorphisms of orbit spaces, we obtain an $n$-transitivity result for
diffeomorphisms of the orbit space $M/G$, see Corollary 6.1. As a special
case, we then obtain the $n$-transitivity result of Pasquotto and Rot for
orbifold diffeomorphisms:

######  Theorem 0.2.

Let $X$ be a reduced smooth orbifold. Let $n\in{\mathbb{N}}$, and let
$(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$ be two $n$-tuples of
pairwise distinct points in $X$. Assume that for every $i$, $1\leq i\leq n$,
$x_{i}$ and $y_{i}$ lie in the same connected component of the singular strata
of $X$, and that any one-dimensional connected component of the strata of $X$
contains at most one of the points $x_{i}$. Then there exists a smooth
orbifold diffeomorphism $f\colon X\to X$ isotopic to the identity through a
compactly supported isotopy, such that $f(x_{i})=y_{i}$, for all $1\leq i\leq
n$.

##  1\. Group actions

Let $G$ be a Lie group acting smoothly on a smooth (i.e., ${\rm C}^{\infty}$)
manifold $M$. The action is called proper, if the map

$\phi\colon G\times M\to M\times M,\,\,\,(g,x)\mapsto(gx,x),$

is proper, i.e., if the inverse images of compact sets are compact. In this
case we call $M$ a proper smooth $G$-manifold. Notice that $\phi$ is a closed
map, if it is proper (see [9]). In particular, every action of a compact Lie
group is proper.

Let $x\in M$. We denote the isotropy subgroup $\\{g\in G\mid gx=x\\}$ of $x$
by $G_{x}$. If $G$ acts properly on $M$, then the isotropy subgroup of every
$x\in M$ is compact. The action is called free, if $G_{x}=\\{e\\}$, for every
$x\in M$, and almost free, if $G_{x}$ is finite for every $x\in M$. If the
identity element $e$ of $G$ is the only element of $G$ that fixes every point
of $M$, we call the action effective.

For a compact subgroup $H$ of $G$ we define

$M_{H}=\\{x\in M\mid G_{x}=H\\},\quad M_{(H)}=\\{x\in M\mid G_{x}\sim H\\}$

and

$M^{H}=\\{x\in M\mid H\subset G_{x}\\},$

where $G_{x}\sim H$ means that $G_{x}=gHg^{-1}$, for some $g\in G$. If $M$ is
a proper smooth $G$-manifold, then the sets $M_{H}$, $M_{(H)}$ and $M^{H}$ are
$\Sigma$-submanifolds of $M$. In other words, each connected component of
$M_{H}$, $M_{(H)}$ and $M^{H}$ is a submanifold of $M$. The dimension of the
connected components may vary. The sets satisfy the equality

$M_{H}=M^{H}\cap M_{(H)},$

see Corollary 4.2.8 in [11]. The set $M^{H}$, being a fixed point set, is
closed in $M$.

The orbit of $x\in M$ is $Gx=\\{gx\mid g\in G\\},$ and the orbit space is
denoted by $G/M$. A subset $A\subset M$ is called $G$-compact, if $\pi(A)$ is
compact, where $\pi\colon M\to M/G$, $x\mapsto Gx$, denotes the natural
projection.

Let $H$ be a closed subgroup of a Lie group $G$, and assume $M$ is a proper
smooth $H$-manifold. We define the twisted product $G\times_{H}M$ to be the
orbit space of the smooth $H$-manifold $G\times M$, where $H$ acts on $G\times
M$ by $h(g,x)=(gh^{-1},hx)$. Let $p\colon G\times M\to G\times_{H}M$ be the
natural projection, and denote $p(g,x)=[g,x]$. Then $G$ acts on $G\times_{H}M$
by $g^{\prime}[g,x]=[g^{\prime}g,x]$ and $G\times_{H}M$ is a proper smooth
$G$-manifold. Let $N$ be a smooth $G$-manifold. Then any smooth
$H$-equivariant map $f\colon M\to N$ induces a smooth $G$-equivariant map

$\tilde{f}\colon G\times_{H}M\to N,\,\,\,[g,x]\mapsto gf(x).$

For basic properties of twisted products, see [4] or [5].

We will be using the differentiable slice theorem (Proposition 2.2.2. in [8])
for proper actions: Let $x\in M$, and let $Gx$ be the orbit of $x$ in $M$.
Equip a $G$-invariant neighborhood of $x$ with a smooth $G$-invariant
Riemannian metric. Then $M$ has a smooth $G_{x}$-invariant submanifold ${\rm
N}_{x}$ containing $x$ that is $G_{x}$-equivariantly diffeomorphic to an open
$G_{x}$-invariant neighborhood of the origin in the normal space ${\rm
T}_{x}M/{\rm T}_{x}Gx$ to $Gx$ at $x$. The manifold ${\rm N}_{x}$ is called a
linear slice (or just a slice) at $x$. The exponential map takes an open
$G$-invariant neighborhood of the zero section of the normal bundle of $Gx$
diffeomorphically to the open neighborhood $G{\rm N}_{x}$ of $Gx$. The map

$G\times_{G_{x}}{\rm N}_{x}\to G{\rm N}_{x},\,\,\,[g,y]\mapsto gy,$

is a smooth $G$-equivariant diffeomorphism. Therefore, we may identify $G{\rm
N}_{x}$ with the twisted product $G\times_{G_{x}}{\rm N}_{x}$. The map

$G{\rm N}_{x}\to G/G_{x},\,\,\,gy\mapsto gG_{x},$

is smooth and the inverse image of $eG_{x}$ equals ${\rm N}_{x}$. The map

$G\times{\rm N}_{x}\to G{\rm N}_{x},\,\,\,(g,y)\mapsto gy,$

is open.

######  Lemma 1.1.

Let $G$ be a Lie group and let $H$ be a compact subgroup of $G$. Let $M$ be a
proper smooth $G$-manifold. Assume $x\in M$ and $G_{x}=H$. Let ${\rm N}$ be a
slice at $x$. Then ${\rm N}_{H}={\rm N}_{(H)}={\rm N}^{H}$ and $(G{\rm
N})_{H}=(G{\rm N})^{H}$.

###### Proof.

By Lemma 4.2.9 in [11], the only closed subgroup of $H$ that is conjugate to
$H$ is $H$ itself. The claims follow, since for every $y\in{\rm N}$,
$G_{y}\subset G_{x}=H$ and since $G_{gy}=gG_{y}g^{-1}$, for all $y\in{\rm N}$
and for all $g\in G$. ∎

######  Lemma 1.2.

Let $G$ be a Lie group and let $H$ be a compact subgroup of $G$. Let $M$ be a
proper smooth $G$-manifold. Assume $G_{x}=H$ for all $x\in M$. Let $\phi\colon
G\times M\to M\times M$, $(g,x)\mapsto(gx,x)$. Let $G$ act diagonally on
$M\times M$. Then $\phi(G\times M)$ is a closed, smooth $G$-invariant
submanifold of $M\times M$, and the map

$f\colon G/H\times M\to M\times M,\,\,\,(gH,x)\mapsto(gx,x),$

is a smooth embedding with image $\phi(G\times M)$.

###### Proof.

The map $\phi$ is a proper map, since $G$ acts properly on $M$. Thus $\phi$ is
a closed map, which implies that $\phi(G\times M)$ is closed in $M\times M$.
Clearly, $\phi(G\times M)$ is $G$-invariant.

The map $f$ is well-defined, because $G_{x}=H$, for every $x\in M$. Clearly,
the image of $f$ is $\phi(G\times M)$. The map $f$ is injective, the inverse
map, defined in the image of $f$ is

$f^{-1}\colon\phi(G\times M)\to G/H\times M,\,\,\,(gx,x)\mapsto(gH,x).$

Let $(gx,x)\in M\times M$, and let ${\rm N}$ be a linear slice at $x$. Then
$G{\rm N}$ is open in $M$, and the map

$\nu\colon G{\rm N}\times G{\rm N}\to G/H\times G{\rm
N},\,\,\,(gy,z)\mapsto(gH,z),$

where $g\in G$ and $y\in{\rm N}$, is smooth. The maps $f^{-1}$ and $\nu$ agree
in the intersection of their domains. Thus the restriction of $f^{-1}$ to this
intersection is continuous. Since $x$ was chosen arbitrarily, it follows that
$f^{-1}$ is continuous.

Let $g_{0}H\in G/H$. Then $g_{0}H$ has an open neighborhood $U$ such that the
quotient map $G\to G/H$, $g\mapsto gH$, has a smooth local section
$\gamma\colon U\to G$. Then $f(gH,x)=(\gamma(gH)x,x)$, for every $(gH,x)\in
U\times M$. It follows that the restriction of $f$ to $U\times M$ is smooth
and furthermore that $f$ is smooth. To complete the proof, it remains to show
that $f$ is an immersion. Let $(gH,x)\in G/H\times M$. The restriction of $f$
to $G/H\times\\{x\\}$ is a smooth embedding with the image $Gx\times\\{x\\}$.
Thus the restriction of $df$ to ${\rm T}_{gH}(G/H)$ is injective. Moreover,
the restriction of $f$ to $\\{gH\\}\times M$ is a smooth embedding, and thus
also the restriction of $df$ to ${\rm T}_{x}M$ is injective. Since

$df\bigl{(}{\rm T}_{gH}(G/H)\bigr{)}\cap df({\rm T}_{x}M)=\\{0\\},$

it follows that $df$ is injective. Thus $f$ is an immersion. ∎

######  Lemma 1.3.

Let $X$ and $Y$ be topological spaces, and let $f\colon X\to Y$ be an open
map. Assume $Z\subset Y$ satisfies $f\bigl{(}f^{-1}(Z)\bigr{)}=Z$. Then the
induced map

$\tilde{f}\colon f^{-1}(Z)\to Z,\,\,\,x\mapsto f(x),$

is open.

###### Proof.

Let $U$ be an open subset of $f^{-1}(Z)$. Then $U=O\cap f^{-1}(Z)$ for some
open subset $O$ of $X$, and $\tilde{f}(U)=f\bigl{(}O\cap f^{-1}(Z)\bigl{)}$.
We will show that $f\bigl{(}O\cap f^{-1}(Z)\bigr{)}=f(O)\cap Z$, which will
prove the claim. Clearly,

$f\bigl{(}O\cap f^{-1}(Z)\bigr{)}\subset f(O)\cap
f\bigl{(}f^{-1}(Z)\bigr{)}=f(O)\cap Z.$

Assume then $y\in f(O)\cap Z$. Then $y=f(x)$, for some $x\in O$. But then
$x\in f^{-1}(y)\subset f^{-1}(Z)$, which implies that $x\in O\cap f^{-1}(Z)$.
Therefore, $y=f(x)\in f\bigl{(}O\cap f^{-1}(Z)\bigr{)}$. Consequently,
$f(O)\cap Z\subset f\bigl{(}O\cap f^{-1}(Z)\bigr{)}$. ∎

Let $H$ be a compact subgroup of a Lie group $G$. We denote the normalizer of
$H$ in $G$ by $N(H)$ and the connected component of $N(H)$ containing the
identity element by $N(H)_{0}$. Then $N(H)$ and $N(H)_{0}$ are closed
subgroups of $G$.

######  Lemma 1.4.

Let $G$ be a Lie group and let $H$ be a compact subgroup of $G$. Let $M$ be a
proper smooth $G$-manifold, and let $\pi\colon M\to M/G$ be the natural
projection. Then the connected components of $\pi(M_{H})$ are the sets
$\pi(Y_{i})$, where $Y_{i}$, $i\in J$, are the connected components of
$M_{H}$.

###### Proof.

Since the sets $Y_{i}$ are connected, also the images $\pi(Y_{i})$ are
connected. Assume $\pi(Y_{i})\cap\pi(Y_{j})\not=\emptyset$, for some $i,j\in
J$. Then there are $g_{i},g_{j}\in G$, $y_{i}\in Y_{i}$ and $y_{j}\in Y_{j}$,
with $g_{i}y_{i}=g_{j}y_{j}$. Hence $g_{j}^{-1}g_{i}y_{i}=y_{j}\in M_{H}$, and
it follows that $g_{j}^{-1}g_{i}\in N(H)$. As a diffeomorphism $M_{H}\to
M_{H}$, $g_{j}^{-1}g_{i}$ takes connected components onto connected
components. Thus $g_{j}^{-1}g_{i}Y_{i}=Y_{j}$, and $\pi(Y_{i})=\pi(Y_{j})$. By
Lemma 1.3, the restriction $\pi|\colon M_{(H)}\to\pi(M_{(H)})$ is open. It
follows from Theorem 4.3.10 in [11], that for any open subset $A$ in $M_{H}$,
the subset $GA$ is open in $M_{(H)}$. Thus also the restriction $\pi|\colon
M_{H}\to\pi(M_{H})$ is open. Since $M_{H}$ is a $\Sigma$-manifold, each
$Y_{i}$ is open in $M_{H}$. Thus each $\pi(Y_{i})$ is open in $\pi(M_{H})$,
and the claim follows. ∎

##  2\. Equivariant isotopies

Let $G$ be a Lie group and let $M$ and $N$ be proper smooth $G$-manifolds. Let
$I=[0,1]$. A map

$F\colon M\times I\to N$

is a $G$-equivariant isotopy, if it is smooth and if every map

$F_{t}\colon M\to N,\,\,\,x\mapsto F(x,t),$

is a smooth $G$-equivariant diffeomorphism. The support ${\rm supp}(f)$ of a
$G$-equivariant diffeomorphism $f\colon M\to M$ is the closure of the set

$\\{x\in M\mid f(x)\not=x\\},$

and the support of a $G$-equivariant isotopy $F\colon M\times I\to M$ is the
closure of the set

$\\{x\in M\mid F(x,t)\not=x\,\,{\rm for\,\,some\,\,}t\in I\\}.$

The support of a map $f\colon M\to{\mathbb{R}}$ is the closure of the set
$\\{x\in M\mid f(x)\not=0\\}$. An equivariant diffeomorphism, isotopy or a
real valued map is called $G$-compactly supported, if it’s support is
$G$-compact. We denote by ${\rm Diff}^{G}_{c}(M)$ the group of $G$-equivariant
diffeomorphisms of $M$ isotopic to the identity through a $G$-equivariant
$G$-compactly supported isotopy.

######  Theorem 2.1.

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. Let $N$ be
a closed smooth $G$-invariant submanifold of $M$. Let $F\colon N\times I\to M$
be a $G$-equivariant, $G$-compactly supported isotopy of embeddings such that
$F_{0}$ is the canonical inclusion. Then $F$ extends to a $G$-equivariant,
$G$-compactly supported isotopy of $M$ starting at the identity.

###### Proof.

The proof is similar to the proof of Theorem 8.6 in [6]. When the assumption
of the isotopy having bounded velocity is replaced by the assumption of the
isotopy being $G$-compactly supported, also the obtained isotopy can be made
$G$-compactly supported. Notice that what we call an isotopy here is called a
diffeotopy in [6] and what we call an isotopy of embeddings is called an
isotopy in [6]. ∎

######  Lemma 2.2.

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. Let $x\in
M$ and let ${\rm N}$ be a linear slice at $x$. Denote the isotropy subgroup
$G_{x}$ of $x$ by $H$. Assume $y\in N(H)_{0}{\rm N}_{H}$. Then there exists a
smooth $G$-equivariant isotopy

$F\colon G{\rm N}_{H}\times I\to G{\rm N}_{H},$

with $G$-compact support, starting at the identity and such that $F_{1}(x)=y$.

###### Proof.

Assume first that $y\in{\rm N}_{H}$. Since ${\rm N}$ is a linear slice at $x$,
we may identify it with an orthogonal $H$-space. Then ${\rm N}_{H}$ is the
fixed point set of $H$ in ${\rm N}$, and hence connected as a linear subspace
of ${\rm N}$. Thus the group of compactly supported diffeomorphisms isotopic
to the identity of ${\rm N}_{H}$ acts transitively on ${\rm N}_{H}$ (see [7]).
Notice that this holds also if $\dim({\rm N}_{H})=1$. Let $f\colon{\rm
N}_{H}\to{\rm N}_{H}$ be such a diffeomorphism taking $x$ to $y$, and let $F$
be a compactly supported isotopy from ${\rm id}_{{\rm N}_{H}}$ to $f$. Since
$H$ acts trivially on ${\rm N}_{H}$, it follows that the diffeomorphisms
$F_{t}$, $t\in I$, are $H$-equivariant. Then

$\tilde{F}\colon G{\rm N}_{H}\times I\to G{\rm N}_{H},\,\,\,(gz,t)\mapsto
gF(z,t),$

is a $G$-equivariant, $G$-compactly supported isotopy starting at the identity
and such that $\tilde{F}_{1}(x)=y$.

Let then $k\in N(H)_{0}$. Let $u\colon I\to N(H)_{0}$ be a smooth path from
the identity element $e$ of $G$ to $k$. Let $\psi\colon{\rm N}_{H}\to I$ be a
smooth function having compact support and taking $y$ to $1$. Let

$\varphi\colon{\rm N}_{H}\times I\to G{\rm N}_{H},\,\,\,(z,t)\mapsto
u\bigl{(}\psi(z)t\bigr{)}z.$

Then $\varphi_{0}$ is the inclusion ${\rm N}_{H}\hookrightarrow G{\rm N}_{H}$,
$\varphi_{1}(y)=ky$, and each $\varphi_{t}$ is a smooth $H$-equivariant map.
It is easy to check that each $\varphi_{t}$ is injective. We check that each
$\varphi_{t}$ is an immersion: Let

$f_{0}\colon G{\rm N}\to G/H,\,\,\,gz\mapsto gH,$

for all $g\in G$ and for all $z\in{\rm N}$. Since ${\rm N}$ is a linear slice
at $x$, there is a smooth cross-section $\sigma\colon O\to G$ of the map $G\to
G/H$, $g\mapsto gH$, defined in some $H$-invariant open neighborhood $O$ of
$eH$ in $G/H$, and a diffeomorphism

$q\colon O\times{\rm N}\to W,\,\,\,(o,z)\mapsto\sigma(o)z,$

onto an open subset $W$ of $M$. The inverse map of $q$ is

$q^{-1}\colon W\to O\times{\rm
N},\,\,\,z\mapsto\Bigl{(}f_{0}(z),\sigma\bigl{(}f_{0}(z)\bigr{)}^{-1}z\Bigr{)}.$

Similarly, for every $g_{0}\in G$, there is a diffeomorphism

$q_{g_{0}}^{-1}\colon g_{0}W\to g_{0}O\times{\rm
N},\,\,\,z\mapsto\Bigl{(}g_{0}f_{0}(g_{0}^{-1}z),\sigma\bigl{(}f_{0}(g_{0}^{-1}z)\bigr{)}^{-1}g_{0}^{-1}z\Bigr{)}.$

Let $z\in{\rm N}_{H}$. Then $\varphi_{t}(z)\in g_{0}W$, for some $g_{0}\in G$.
Let

${\rm pr}\colon g_{0}O\times{\rm N}\to{\rm N}$

be the projection. Then

$\displaystyle({\rm pr}\circ q_{g_{0}}^{-1}\circ\varphi_{t})(z)$
$\displaystyle=({\rm pr}\circ
q_{g_{0}}^{-1})\Bigl{(}u\bigl{(}\psi(z)t\bigr{)}z\Bigr{)}$
$\displaystyle=\sigma\Bigl{(}f_{0}\bigl{(}g_{0}^{-1}u\bigl{(}\psi(z)t\bigr{)}z\bigr{)}\Bigr{)}^{-1}g_{0}^{-1}u\bigl{(}\psi(z)t\bigr{)}z$
$\displaystyle=hz$ $\displaystyle=z,$

for some $h\in H$. Thus the restriction

${\rm pr}\circ
q_{g_{0}}^{-1}\circ\varphi_{t}|\colon\varphi^{-1}_{t}(g_{0}W)\to{\rm N}$

is the inclusion, which implies that $\varphi_{t}$ is immersive at $z$.

We next show that each $\varphi_{t}$ is a closed map: Let $A$ be a closed
subset of ${\rm N}_{H}$. Assume $w$ is a point in the closure of the image
$\varphi_{t}(A)$. Then there is a sequence of points
$\varphi_{t}(z_{n})=u\bigl{(}\psi(z_{n})t\bigr{)}z_{n}$ converging to $w$,
where $z_{n}\in A$, for every $n$. The set

$K=\bigl{\\{}u\bigl{(}\psi(z)t\bigr{)}\mid z\in{\rm N}_{H}\bigr{\\}}$

is compact, since $\psi({\rm N}_{H})=I$ and $u$ is a path. By passing to a
subsequence, if necessary, we may assume that the sequence of points
$u\bigl{(}\psi(z_{n})t\bigr{)}$ converges to a point
$u\bigl{(}\psi(z)t\bigr{)}$, for some $z\in{\rm N}_{H}$. It follows that the
sequence $(z_{n})$ converges to $u\bigl{(}\psi(z)t\bigr{)}^{-1}w=\tilde{z}$.
Since $A$ is closed, it follows that $\tilde{z}\in A$. The sequence
$\Bigl{(}u\bigl{(}\psi(z_{n})t\bigr{)}z_{n}\Bigr{)}$ converges to
$u(\psi(z)t)\tilde{z}$. Now, $z_{n}\to\tilde{z}$ implies
$u\bigl{(}\psi(z_{n})t\bigr{)}\to u\bigl{(}\psi(\tilde{z})t\bigr{)}$. Thus
$u\bigl{(}\psi(\tilde{z})t)\bigr{)}=u\bigl{(}\psi(z)t\bigr{)}$ and hence
$w=u\bigl{(}\psi(z)t\bigr{)}\tilde{z}=u\bigl{(}\psi(\tilde{z})t\bigr{)}\tilde{z}=\varphi_{t}(\tilde{z})\in\varphi_{t}(A)$.
It follows that $\varphi_{t}(A)$ is closed, and hence that $\varphi_{t}$ is a
closed map.

We proved that $\varphi$ is a compactly supported $H$-equivariant isotopy of
embeddings. Let

$\tilde{\varphi}\colon G{\rm N}_{H}\times I\to G{\rm
N}_{H},\,\,\,(gz,t)\mapsto g\varphi(z,t).$

Clearly, $\tilde{\varphi}_{1}(y)=ky$. To show that $\tilde{\varphi}$ is a
$G$-compactly supported $G$-equivariant isotopy starting at the identity, it
suffices to show that each $\tilde{\varphi}_{t}$ is bijective, and that the
inverse map $\tilde{\varphi}_{t}^{-1}$ is smooth. Let $g_{0}\in G$ and
$z_{0}\in{\rm N}_{H}$, so that $g_{0}z_{0}\in G{\rm N}_{H}$. Then
$g_{0}u\bigl{(}\psi(z_{0})t\bigr{)}^{-1}\in G$ and

$\tilde{\varphi}_{t}\Bigl{(}g_{0}u\bigl{(}\psi(z_{0})t\bigr{)}^{-1}z_{0}\Bigr{)}=g_{0}u\bigl{(}\psi(z_{0})t\bigr{)}^{-1}u\bigl{(}\psi(z_{0})t\bigr{)}z_{0}=g_{0}z_{0}.$

Thus $\tilde{\varphi}_{t}$ is surjective.

Assume then $\tilde{\varphi}_{t}(g_{1}z_{1})=\tilde{\varphi}_{t}(g_{2}z_{2})$.
Then
$g_{1}u\bigl{(}\psi(z_{1})t\bigr{)}z_{1}=g_{2}u\bigl{(}\psi(z_{2})t\bigr{)}z_{2}$.
Thus $g_{1}u\bigl{(}\psi(z_{1})t\bigr{)}=g_{2}u\bigl{(}\psi(z_{2})t\bigr{)}h$,
for some $h\in H$. It follows that $z_{1}=z_{2}$, and moreover that
$u\bigl{(}\psi(z_{1})t)\bigr{)}=u\bigl{(}\psi(z_{2})t\bigr{)}$. Since
$u\bigl{(}\psi(z_{2})t\bigr{)}^{-1}g_{1}^{-1}g_{2}u\bigl{(}\psi(z_{2})t\bigr{)}\in
H$, $g_{1}^{-1}g_{2}\in
u\bigl{(}\psi(z_{2})t\bigr{)}Hu\bigl{(}\psi(z_{2})t\bigr{)}^{-1}=H$. Thus
$g_{1}=g_{2}h^{\prime}$, for some $h^{\prime}\in H$, and it follows that
$g_{1}z_{1}=g_{2}z_{2}$. Hence $\tilde{\varphi}_{t}$ is injective.

It remains to check that $\tilde{\varphi}_{t}^{-1}\colon G{\rm N}_{H}\to G{\rm
N}_{H}$ is smooth. An arbitrary element in $G{\rm N}_{H}$ can be written as
$g_{0}u\bigl{(}\psi(z_{0}t)\bigr{)}z_{0}$, where $g_{0}\in G$ and
$z_{0}\in{\rm N}_{H}$. Then $g_{0}u\bigl{(}\psi(z_{0})t\bigr{)}\in G$. The
restriction

$f_{0}|\colon G{\rm N}_{H}\to G/H,\,\,\,gz\mapsto gH,$

is smooth. The map $G\to G/H$, $g\mapsto gH$, has a smooth local cross-section
$\sigma\colon U\to G$, where $U$ is an open $H$-invariant neighborhood of
$g_{0}u\bigl{(}\psi(z_{0})t\bigr{)}H$ in $G/H$. The set $f_{0}^{-1}(U)$ is
open in $G{\rm N}_{H}$ and $g_{0}u\bigl{(}\psi(z_{0})t\bigr{)}z_{0}\in
f_{0}^{-1}(U)$. The map

$\displaystyle(\sigma^{-1}\circ f_{0}|,{\rm id})\colon f_{0}^{-1}(U)$
$\displaystyle\to G\times f_{0}^{-1}(U),$ $\displaystyle
gu\bigl{(}\psi(z)t\bigr{)}z$
$\displaystyle\mapsto\Bigl{(}\sigma\bigl{(}gu\bigl{(}\psi(z)t\bigr{)}H\bigr{)}^{-1},gu\bigl{(}\psi(z)t\bigr{)}z\Bigr{)},$

is smooth. Since $G$ acts smoothly on $M$, it follows that the map

$f_{1}\colon f_{0}^{-1}(U)\to G{\rm
N}_{H},\,\,\,gu\bigl{(}\psi(z)t\bigr{)}z\mapsto\sigma\bigl{(}gu\bigl{(}\psi(z)t\bigr{)}H\bigr{)}^{-1}gu\bigl{(}\psi(z)t\bigr{)}z=z,$

is smooth. Finally, the map

$p\colon{\rm N}_{H}\to G{\rm N}_{H},\,\,\,z\mapsto
u\bigl{(}\psi(z)t\bigr{)}^{-1}z,$

is smooth. Thus the map

$\displaystyle(\sigma\circ f_{0}|,p\circ f_{1})\colon f_{0}^{-1}(U)$
$\displaystyle\to G\times G{\rm N}_{H},$ $\displaystyle
gu\bigl{(}\psi(z)t\bigr{)}z$
$\displaystyle\mapsto\Bigl{(}\sigma\bigl{(}gu\bigl{(}\psi(z)t\bigr{)}H\bigr{)},u\bigl{(}\psi(z)t\bigr{)}^{-1}z\Bigr{)},$

is smooth. Consequently,

$\displaystyle\tilde{\varphi}_{t}^{-1}|\colon f_{0}^{-1}(U)$ $\displaystyle\to
G{\rm N}_{H},$ $\displaystyle gu\bigl{(}\psi(z)t\bigr{)}z$
$\displaystyle\mapsto\sigma\bigl{(}gu\bigl{(}\psi(z)t)H\bigr{)}u\bigl{(}\psi(z)t\bigr{)}^{-1}z=gz,$

is smooth.

An isotopy, as in the claim of the lemma, is obtained as a composition of
isotopies of the form $\tilde{F}$ and $\tilde{\varphi}$ of the proof. ∎

The following proposition will be used to prove Theorem 0.1.

######  Proposition 2.3.

Let $G$ be a Lie group and let $H$ be a compact subgroup of $G$. Let $M$ be a
proper smooth $G$-manifold and let $x\in M$ with $G_{x}=H$. Let $U$ be an open
$G$-invariant neighborhood of $x$. Then there exists an open neighborhood $V$
of $x$ in $M_{H}$ with the property that $GV\subset U$, and for each $y\in V$
there exists $f\in{\rm Diff}^{G}_{c}(M)$ such that $f(x)=y$ and $f$ is
isotopic to the identity through a $G$-equivariant isotopy with $G$-compact
support contained in $U$.

###### Proof.

We may assume that $U=G{\rm N}$, for some linear slice ${\rm N}$ at $x$. Let
$V=N(H)_{0}{\rm N}_{H}$. Since $G{\rm N}$ is open in $M$, it follows that
$N(H){\rm N}_{H}=G{\rm N}\cap M_{H}$ is open in $M_{H}$. The map

$k\colon N(H){\rm N}_{H}\to N(H)/H,\quad gz\mapsto gH,$

is continuous as a restriction of the continuous map $G{\rm N}\to G/H$,
$gz\mapsto gH$. Thus $V=N(H)_{0}{\rm N}_{H}=k^{-1}\bigl{(}N(H)_{0}H/H\bigr{)}$
is open in $N(H){\rm N}_{H}$ and, consequently, also in $M_{H}$. Assume $y\in
V$. By Lemma 2.2, there is a smooth $G$-equivariant isotopy

$F\colon G{\rm N}_{H}\times I\to G{\rm N}_{H},$

with $G$-compact support, starting at the identity and such that $F_{1}(x)=y$.
Now, ${\rm N}_{H}$ is closed in ${\rm N}$. Thus $G{\rm N}\setminus G{\rm
N}_{H}=G({\rm N}\setminus{\rm N}_{H})$ is open in $G{\rm N}$, and hence $G{\rm
N}_{H}$ is closed in $G{\rm N}$. By Theorem 2.1, the isotopy $F$ can be
extended to a smooth $G$-equivariant isotopy

$\tilde{F}\colon G{\rm N}\times I\to G{\rm N},$

with $G$-compact support and starting at the identity. Since $G{\rm N}$ is
open in $M$, and since the support of $\tilde{F}$ is $G$-compact, it follows
that extending each $\tilde{F}_{t}$, $t\in I$, to be the identity on
$M\setminus G{\rm N}$, yields an isotopy $M\times I\to M$ having the
properties in the claim. ∎

##  3\. Orbifolds

In this section we recall the definition and some basic properties of
orbifolds.

######  Definition 3.1.

Let $X$ be a topological space and let $n>0$.

  1. (1)

An $n$-dimensional orbifold chart for an open subset $V$ of $X$ is a triple
$(\tilde{V},G,\varphi)$ satisfying the following conditions:

    1. (a)

$\tilde{V}$ is a connected open subset of ${\mathbb{R}}^{n}$,

    2. (b)

$G$ is a finite group of homeomorphisms acting on $\tilde{V}$, ${\rm ker}(G)$
denotes the subgroup of $G$ acting trivially on $\tilde{V}$,

    3. (c)

$\varphi\colon\tilde{V}\to V$ is a $G$-invariant map that induces a
homeomorphism from $\tilde{V}/G$ onto $V$.

  2. (2)

If $V_{i}\subset V_{j}$, an embedding
$(\lambda_{ij},h_{ij})\colon(\tilde{V}_{i},G_{i},\varphi_{i})\to(\tilde{V}_{j},G_{j},\varphi_{j})$
means

    1. (a)

an injective homomorphism $h_{ij}\colon G_{i}\to G_{j}$, such that $h_{ij}$ is
an isomorphism from ${\rm ker}(G_{i})$ to ${\rm ker}(G_{j})$, and

    2. (b)

an equivariant embedding $\lambda_{ij}\colon\tilde{V}_{i}\to\tilde{V}_{j}$
such that $\varphi_{j}\circ\lambda_{ij}=\varphi_{i}$. (Thus
$\lambda_{ij}(gx)=h_{ij}(g)\lambda_{ij}(x)$ for every $g\in G_{i}$ and every
$x\in\tilde{V}_{i}$.)

  3. (3)

An orbifold atlas on $X$ is a family
${\mathcal{V}}=\\{(\tilde{V_{i}},G_{i},\varphi_{i})\\}_{i\in J}$ of orbifold
charts satisfying the following conditions:

    1. (a)

$\\{V_{i}\\}_{i\in J}$ is a covering of $X$,

    2. (b)

given two charts $(\tilde{V}_{i},G_{i},\varphi_{i})$ and
$(\tilde{V}_{j},G_{j},\varphi_{j})$ and a point $x\in V_{i}\cap V_{j}$, there
exists an open neighborhood $V_{k}\subset V_{i}\cap V_{j}$ of $x$ and a chart
$(\tilde{V}_{k},G_{k},\varphi_{k})$ such that there are embeddings
$(\lambda_{ki},h_{ki})\colon(\tilde{V}_{k},G_{k},\varphi_{k})\to(\tilde{V}_{i},G_{i},\varphi_{i})$
and
$(\lambda_{kj},h_{kj})\colon(\tilde{V}_{k},G_{k},\varphi_{k})\to(\tilde{V}_{j},G_{j},\varphi_{j})$.

  4. (4)

An atlas ${\mathcal{U}}$ is called a refinement of an atlas ${\mathcal{W}}$ if
every chart in ${\mathcal{U}}$ admits an embedding into some chart of
${\mathcal{W}}$. Two orbifold atlases having a common refinement are called
equivalent.

######  Definition 3.2.

An $n$-dimensional orbifold is a paracompact Hausdorff space $X$ equipped with
an equivalence class of $n$-dimensional orbifold atlases.

If the actions of the finite groups on the orbifold charts are effective, the
orbifold is called reduced. Reduced orbifolds are also called effective.

An orbifold is called smooth if each $G_{i}$ acts by smooth diffeomorphisms on
$\tilde{V}_{i}$ and if each embedding
$\lambda_{ij}\colon{\tilde{V}}_{i}\to\tilde{V}_{j}$ is smooth. By the
differentiable slice theorem we may always choose orbifold charts
$(\tilde{V_{i}},G_{i},\varphi_{i})$ of a smooth orbifold in such a way that
$\tilde{V}_{i}$ is diffeomorphic to a euclidean space ${\mathbb{R}}^{n}$,
where $n$ is the dimension of the orbifold, and $G_{i}$ acts linearly on
${\mathbb{R}}^{n}$. In this paper we only consider smooth orbifolds.

Let $G$ be a compact Lie group acting smoothly, effectively and almost freely
on a smooth manifold $M$. Then the quotient space $M/G$ is a reduced smooth
orbifold. The orbifold charts of $M/G$ are the triples $({\rm
N}_{x},G_{x},\pi_{x})$, where ${\rm N}_{x}$ is a linear slice at $x\in M$ and
$\pi_{x}\colon{\rm N}_{x}\to{\rm N}_{x}/G_{x}\cong(G{\rm N}_{x})/G$ denotes
the natural projection.

The frame bundle construction allows us to consider any reduced smooth
orbifold as a quotient space of an action by a compact Lie group. Namely, by
Theorem 1.23 in [1], there is the following result: Let $X$ be a reduced
smooth orbifold of dimension $n$. Then the frame bundle ${\rm Fr}(X)$ is a
smooth manifold with a smooth, effective and almost free action by the
orthogonal group ${\rm O}(n)$. The orbifold $X$ is naturally isomorphic to the
resulting quotient orbifold ${\rm Fr}(X)/{\rm O}(n)$.

######  Definition 3.3.

Let $X$ and $Y$ be smooth orbifolds. A map $f\colon X\to Y$ is called a smooth
orbifold map, if for every $x\in X$, there are charts $(\tilde{U},G,\varphi)$
around $x$ and $(\tilde{V},H,\psi)$ around $f(x)$, such that $f$ maps
$U=\varphi(\tilde{U})$ into $V=\psi(\tilde{V})$ and the restriction $f|U$ can
be lifted to a smooth equivariant map $\tilde{f}\colon\tilde{U}\to\tilde{V}$.
A smooth orbifold map $f\colon X\to Y$ is called an orbifold diffeomorphism,
if there is a smooth orbifold map $g\colon Y\to X$ such that $g\circ f={\rm
id}_{X}$ and $f\circ g={\rm id}_{Y}$.

The support ${\rm supp}(f)$ of an orbifold diffeomorphism $f\colon X\to X$ is
the closure of the set

$\\{x\in X\mid f(x)\not=x\\}.$

We give an orbifold structure to $X\times I$ as follows: If
$(\tilde{U}_{i},G_{i},\varphi_{i})$ is an orbifold chart of $X$, then
$(\tilde{U}_{i}\times I,G_{i},\varphi_{i}\times{\rm id}_{I})$ is an orbifold
chart of $X\times I$, where $G_{i}$ acts trivially on $I$ and diagonally on
$\tilde{U}_{i}\times I$. This kind of charts define an orbifold atlas for
$X\times I$.

Assume then $X$ and $Y$ are smooth orbifolds and assume $f_{0},f_{1}\colon
X\to Y$ are two smooth orbifold maps. If there is a smooth orbifold map

$F\colon X\times I\to Y,$

with $F_{0}=f_{0}$ and $F_{1}=f_{1}$, we say that $f_{0}$ and $f_{1}$ are
smoothly homotopic. Here, for every $t\in I$, $F_{t}\colon X\to Y$, $x\mapsto
F(x,t)$. If, in addition, $F_{t}$ is a smooth orbifold diffeomorphism for
every $t\in I$, we say that $f_{0}$ and $f_{1}$ are smoothly isotopic.

##  4\. Orbifold stratifications

Orbifolds admit stratifications by singular dimension. This stratification is
explained in detail in [3] and used in [10]: Let $X$ be a reduced smooth
$n$-dimensional orbifold, and let $x\in X$. Let
$(\tilde{V}_{x},G_{x},\varphi_{x})$ be an orbifold chart centered at $x$, and
let $V=\varphi_{x}(\tilde{V}_{x})$. There is a unique point
$\tilde{x}\in\tilde{V}_{x}$ with $\varphi_{x}(\tilde{x})=x$. If
$(\tilde{V}^{\prime}_{x},G^{\prime}_{x},\varphi^{\prime}_{x})$ is another
orbifold chart centered at $x$, then the groups $G_{x}$ and $G^{\prime}_{x}$
are isomorphic. The action of $G_{x}$ on $\tilde{V}_{x}$ fixes $\tilde{x}$ and
the differential induces a linear action on the tangent space ${\rm
T}_{\tilde{x}}\tilde{V}_{x}$. Denote by ${\rm
T}_{\tilde{x}}\tilde{V}_{x}^{G_{x}}$ the subspace of ${\rm
T}_{\tilde{x}}\tilde{V}_{x}$ consisting of the fixed points of the
$G_{x}$-action. The singular dimension of $x$ is then defined to be ${\rm
sdim}(x)={\rm dim}({\rm T}_{\tilde{x}}\tilde{V}_{x}^{G_{x}})$, and it does not
depend on the choice of the orbifold chart. The singular set

$\Sigma=\\{x\in X\mid G_{x}\not=1\\}$

of $X$ can be written as the union of the singular strata $\Sigma_{k}$, $0\leq
k\leq n-1$, where

$\Sigma_{k}=\\{x\in\Sigma\mid{\rm sdim}(x)=k\\}.$

Similarly, for a point $x\in X$ with trivial $G_{x}$, the singular dimension
${\rm sdim}(x)=n$, and $\Sigma_{n}$ is the union of such points. Then
$X=\bigcup_{i=0}^{n}\Sigma_{i}$. The connected component of $\Sigma_{{\rm
sdim}(x)}$ containing $x$ is a smooth manifold denoted by $\Sigma(x)$. The
isomorphism class of the isotropy group $G_{y}$ is constant for
$y\in\Sigma(x)$.

For a finite group $H$, let

$X_{H}=\\{x\in X\mid G_{x}\cong H\\}.$

For $x\in X$, denote by $\Sigma^{\prime}(x)$ the connected component of the
subset $X_{G_{x}}$ of $X$ containing $x$. Then
$\Sigma(x)\subset\Sigma^{\prime}(x)$. Cover $\Sigma^{\prime}(x)$ by open sets
$V_{i}$, where $(\tilde{V}_{i},G_{i},\varphi_{i})$ is a chart for $V_{i}$
centered at some point in $\Sigma^{\prime}(x)$, $\tilde{V}_{i}$ is
homeomorphic to ${\mathbb{R}}^{n}$ and $n$ is the dimension of $X$. The
following lemma then implies that also $\Sigma^{\prime}(x)\subset\Sigma(x)$,
and hence that $\Sigma(x)=\Sigma^{\prime}(x)$.

######  Lemma 4.1.

Let $H$ be a finite group acting smoothly on a euclidean space
${\mathbb{R}}^{n}$. Let $F$ be a connected component of the fixed point set of
the action. Assume $x,y\in F$. The group $H$ acts on the tangent spaces ${\rm
T}_{x}{\mathbb{R}}^{n}$ and ${\rm T}_{y}{\mathbb{R}}^{n}$ via the
differentials. Then $\dim({\rm T}_{x}{\mathbb{R}}^{n})^{H}=\dim({\rm
T}_{y}{\mathbb{R}}^{n})^{H}$.

###### Proof.

Notice that the fixed point set ${\mathbb{R}}^{n}_{H}$ is a $\Sigma$-manifold.
Thus the connected components of ${\mathbb{R}}^{n}_{H}$ equal the path
components. Denote the set of group homomorphisms from $H$ to the general
linear group ${\rm GL}_{n}({\mathbb{R}})$, equipped with the compact-open
topology, by

${\rm hom}\bigl{(}H,{\rm GL}_{n}({\mathbb{R}})\bigr{)}.$

Let $f\colon I\to F$ be a smooth path from $x$ to $y$. The map

$G\colon H\times I\to{\rm GL}_{n}({\mathbb{R}}),\,\,\,(h,t)\mapsto dh_{f(t)},$

is continuous. Since $H$ is compact, $G$ induces a continuous map

$G^{\\#}\colon I\to{\rm hom}\bigl{(}H,{\rm
GL}_{n}({\mathbb{R}})\bigr{)},\,\,\,t\mapsto G_{t},$

where $G_{t}(h)=G(h,t)$. By Lemma VIII 38.1 in [2], any group homomorphism
$H\to{\rm GL}_{n}({\mathbb{R}})$ that is sufficiently close to $G_{t}$ must be
conjugate to $G_{t}$. Since $I$ is compact and $G^{\\#}$ is continuous, the
homomorphisms $G_{0}$ and $G_{1}$ are conjugate. The claim follows, since
conjugate actions have isomorphic fixed point spaces. ∎

By using the frame bundle construction, we know that a reduced orbifold $X$
may be considered as a quotient space $M/G$, where $G$ is a compact Lie group
acting smoothly, effectively and almost freely on a smooth manifold $M$. Then
$M$ has a natural stratification by orbit types $M_{(H)}$, where $H$ is a
finite subgroup of $G$ (see [11], Theorem 4.3.7). If $M$ is compact, then the
action has only finitely many orbit types. We denote by $\tilde{\Sigma}(x)$
the connected component of $M_{(G_{x})}$ containing $x$. This stratification
of $M$ induces a stratification on the orbit space $M/G$: If $\pi\colon M\to
M/G$ is the natural projection, then the strata on $M/G$ are the images
$\pi(M_{(H)})$ of the strata of $M$ ([11], 4.3.9). Denote by
$\hat{\Sigma}\bigl{(}\pi(x)\bigr{)}$ the connected component of the stratum
containing the point $\pi(x)\in M/G$, where $x\in M$. Let ${\rm N}$ be a
linear slice at $z$, where $\pi(z)\in\Sigma^{\prime}\bigl{(}\pi(x)\bigr{)}$.
Then $G{\rm N}$ is open in $M$ and for any $y\in{\rm N}$, $G_{y}\subset
G_{z}$. Thus $\\{y\in M\mid G_{y}\cong G_{z}\cong G_{x}\\}\cap G{\rm N}\subset
M_{(G_{x})}$. Now $\pi(G{\rm N})$ is open in $M/G$, and
$\Sigma^{\prime}\bigl{(}\pi(x)\bigr{)}\cap\pi(G{\rm
N})=\hat{\Sigma}\bigl{(}\pi(x)\bigr{)}\cap\pi(G{\rm N})$. By covering $M/G$ by
sets of the form $\pi(G{\rm N})$, we see that in fact
$\Sigma^{\prime}\bigl{(}\pi(x)\bigr{)}=\hat{\Sigma}\bigl{(}\pi(x)\bigr{)}$.
Thus, although the concept of conjugacy of the local groups of an orbifold
does not make sense, it follows that when an orbifold is considered as an
orbit space, the local groups of points on a connected component of a stratum
$\Sigma^{\prime}$ are not just isomorphic but actually conjugate.

##  5\. Smooth maps between orbit spaces

In [12], G. Schwarz considered smooth isotopies between orbit spaces. In this
section we look at how his concept of smoothness is related to smoothness in
the orbifold sense. Thus, let $G$ be a compact Lie group and let $M$ be a
smooth $G$-manifold. Let $\pi\colon M\to M/G$ denote the natural projection.
We give the orbit space $M/G$ the quotient space topology and differentiable
structure as in [12]: If $U$ is an open subset of $M/G$, then ${\rm
C}^{\infty}(U)$ is the set of functions $f\colon U\to{\mathbb{R}}$ for which
the composition $f\circ\pi|\colon\pi^{-1}(U)\to{\mathbb{R}}$ is smooth. Assume
$N$ is another smooth $G$-manifold. Then a map $\psi\colon M/G\to N/G$ is
defined to be smooth, if $f\circ\psi\colon M/G\to{\mathbb{R}}$ is smooth for
every smooth $f\colon N/G\to{\mathbb{R}}$. In particular, a map $M/G\to N/G$
is smooth, if it is induced by a smooth $G$-equivariant map $M\to N$. The
notions of diffeomorphism and isotopy have their usual categorical meaning. In
[12], Schwarz proves the following isotopy lifting conjecture ([12], p. 38):

######  Theorem 5.1.

Let $G$ be a compact Lie group, and let $M$ be a smooth $G$-manifold. Let
$\bar{F}\colon M/G\times I\to M/G$ be a smooth isotopy starting at the
identity. Then there is a smooth $G$-equivariant isotopy $F\colon M\times I\to
M$ starting at the identity and inducing $\bar{F}$.

Let $G$ be a compact Lie group and let $M$ and $N$ be smooth $G$-manifolds.
Assume the actions are effective and almost free so that the quotient spaces
$M/G$ and $N/G$ are smooth orbifolds. Let $\pi\colon M\to M/G$ and
$\pi^{\prime}\colon N\to N/G$ denote the natural projections. Assume
$\psi\colon M/G\to N/G$ is a smooth orbifold map. Let $x\in M/G$. Then there
are charts $({\rm N}_{x},G_{x},\pi_{x})$ around $x$ and $({\rm
N}_{y},G_{y},\pi^{\prime}_{y})$ around $y=\psi(x)$, such that $\psi$ maps
$U={\rm N}_{x}/G_{x}$ into $V={\rm N}_{y}/G_{y}$ and the restriction $\psi|U$
can be lifted to a smooth equivariant map $\tilde{\psi}\colon{\rm
N}_{x}\to{\rm N}_{y}$. The sets $\pi^{-1}(U)=G\times_{G_{x}}{\rm N}_{x}$ and
$(\pi^{\prime})^{-1}(V)=G\times_{G_{y}}{\rm N}_{y}$ are open in $M$ and $N$,
respectively. Let $f\colon V\to{\mathbb{R}}$ and assume $f\in{\rm
C}^{\infty}(V)$. Then, by definition, the composed map
$f\circ\pi^{\prime}|\colon G\times_{G_{x}}{\rm N}_{y}\to{\mathbb{R}}$ is
smooth. But then

$f\circ\psi|_{U}\circ\pi_{x}=f\circ\pi^{\prime}_{y}\circ\tilde{\psi}\colon{\rm
N}_{x}\to{\mathbb{R}}$

is smooth. Consequently, $f\circ\psi|_{U}\circ\pi|\colon G\times_{G_{x}}{\rm
N}_{x}\to{\mathbb{R}}$ is smooth. It follows that $\psi|_{U}$ is smooth in the
sense on [12]. Since $x$ was chosen arbitrarily, it follows that also $\psi$
is smooth in the sense of [12].

We have proved the following:

######  Lemma 5.2.

Let $G$ be a compact Lie group and let $M$ and $N$ be smooth $G$-manifolds.
Assume the actions of $G$ on $M$ and $N$ are effective and almost free. Let
$f\colon M/G\to N/G$ be a smooth orbifold map. Then $f$ is smooth in the sense
of [12].

Theorem 5.1 and Lemma 5.2 imply the following:

######  Proposition 5.3.

Let $X$ be a reduced smooth orbifold of dimension $n$, and let ${\rm Fr}(X)$
be the frame bundle of $X$. Let $\bar{F}\colon X\times I\to X$ be a smooth
orbifold isotopy starting at the identity. Then $\bar{F}$ is induced by a
smooth ${\rm O}(n)$-equivariant isotopy $F\colon{\rm Fr}(X)\times I\to{\rm
Fr}(X)$ starting at the identity.

∎

##  6\. Results

Let $G$ be a Lie group and let $H$ be a compact subgroup of $G$. Let $M$ be a
proper smooth $G$-manifold. Assume $M_{H}\not=\emptyset$. Since $G$ acts
properly on $M$, it follows that the normalizer $N(H)$ of $H$ in $G$ acts
properly on $M_{H}$. Moreover, $N(H)_{x}=H$, for every $x\in M_{H}$. Let

$\phi\colon N(H)\times M_{H}\to M_{H}\times M_{H},\,\,\,(k,x)\mapsto(kx,x).$

By Lemma 1.2, $N(H)/H\times M_{H}$ is diffeomorphic to

$\phi\bigl{(}N(H)\times M_{H}\bigr{)}=\\{(kx,x)\mid k\in N(H),x\in M_{H}\\}.$

Notice that

$\\{(x,y)\in M_{H}\times M_{H}\mid Gx\not=Gy\\}=(M_{H}\times
M_{H})\setminus\phi\bigl{(}N(H)\times M_{H}\bigr{)}.$

Let $Y$ be a connected component of $M_{H}$. Then $Y$ is a closed submanifold
of $M_{H}$ and the diffeomorphism $f$ of Lemma 1.2 takes $N(H)/H\times Y$ onto
the submanifold

$\\{(ky,y)\mid k\in N(H),y\in Y\\}=\phi\bigl{(}N(H)\times Y\bigr{)}$

of $M_{H}\times M_{H}$. Since $Y\times Y$ is a connected component of
$M_{H}\times M_{H}$, it follows that $\phi\bigl{(}N(H)\times
Y\bigr{)}\cap(Y\times Y)$ is a submanifold of $Y\times Y$.

Let $N(H)_{0}$ denote the connected component of $N(H)$ containing the
identity element. Then $N(H)_{0}Y$ is connected, which implies that
$N(H)_{0}Y=Y$. Thus $\phi\bigl{(}N(H)_{0}\times Y\bigr{)}\subset Y\times Y$.
Then

$\displaystyle\dim\Bigl{(}\phi\bigl{(}N(H)\times Y\bigr{)}\cap(Y\times
Y)\Bigr{)}$ $\displaystyle=\dim\Bigl{(}\phi\bigl{(}N(H)_{0}\times
Y\bigr{)}\cap(Y\times Y)\Bigr{)}$
$\displaystyle=\dim\Bigl{(}\phi\bigl{(}N(H)_{0}\times Y\bigr{)}\Bigr{)}$
$\displaystyle=\dim\Bigl{(}\bigl{(}N(H)_{0}H\bigr{)}/H\times Y\Bigr{)}$
$\displaystyle=\dim\bigl{(}N(H)/H\times Y\bigr{)}$
$\displaystyle=\dim\bigl{(}N(H)/H\bigr{)}+\dim(Y).$

Notice that

$\\{(x,y)\in Y\times Y\mid Gx\not=Gy\\}=(Y\times
Y)\setminus\Bigl{(}\phi\bigl{(}N(H)\times Y\bigr{)}\cap(Y\times Y)\Bigr{)}.$

Thus the space $\\{(x,y)\in Y\times Y\mid Gx\not=Gy\\}$ is connected if the
codimension of $\phi\bigl{(}N(H)\times Y\bigr{)}\cap(Y\times Y)$ in $Y\times
Y$ is at least $2$. This happens, if and only if,

$\dim\bigl{(}N(H)/H\bigr{)}\leq\dim(Y)-2.$

Proof of Theorem 0.1. We consider $n$-tuples $(x_{1},\ldots,x_{n})$ of points
in $M$. There are finitely many compact subgroups $H$ of $G$ and finitely many
connected components $Y_{i}$ of the sets $M_{H}$, $1\leq i\leq k$, for some
$k\leq n$, such that each of the points $x_{1},\ldots,x_{n}$ lies in some
connected component $Y_{i}$, for some $i\in\\{1,\ldots,k\\}$. We may assume
that for all $i\not=j$ and for all $g\in G\setminus N(H)_{0}$,
$Y_{i}\not=gY_{j}$. This is because for any $G$-equivariant map $f\colon M\to
M$, the condition $f(y_{i})=z_{i}$ equals the condition $f(gy_{i})=gz_{i}$.
Thus the condition about points in the connected components $gY_{i}$ can all
be expressed by considering only points in $Y_{i}$. Just like in the proof of
the corresponding result for orbifolds ([10], Theorem 6) we begin by
reordering the points $x_{j}$ and $y_{j}$ according to which set $Y_{i}$ they
lie in. Thus, up to a permutation, we assume there exist numbers
$1=l_{1}<l_{2}\cdots<l_{k}\leq n$ such that for all $i\leq k-1$, $x_{j}\in
Y_{i}$, if and only if $l_{i}\leq j\leq l_{i+1}-1$, and $x_{j}\in Y_{k}$, if
and only if $l_{k}\leq j\leq n$ (and analogously for the points $y_{j}$).

Let $M^{n}$ denote the $n$-fold cartesian product $M\times\cdots\times M$, on
which $G$ acts diagonally. Let $M^{(n)}$ denote the subspace of $M^{n}$
consisting of $n$-tuples of points whose orbits are pairwise distinct. Thus

$M^{(n)}=\\{(z_{1},\ldots,z_{n})\in M^{n}\mid Gz_{i}\not=Gz_{j},\,\,{\rm
if}\,\,i\not=j\\}.$

Let $n_{i}=l_{i+1}-l_{i}$ for all $1\leq i\leq k-1$, and $n_{k}=n-l_{k}+1$.
Then

$(x_{1},\ldots,x_{n}),(y_{1},\ldots,y_{n})\in Y_{1}^{n_{1}}\times\cdots\times
Y_{k}^{n_{k}}.$

We show that $M^{(n)}\cap(Y_{1}^{n_{1}}\times\cdots\times Y_{k}^{n_{k}})$ is
connected. The fat $G$-diagonal in $Y_{i}^{n_{i}}$ is

$\Delta^{G}_{i}=\\{(z_{1},\ldots,z_{n_{i}})\in Y_{i}^{n_{i}}\mid
Gz_{p}=Gz_{q}\,\,{\rm for\,\,some}\,\,p\not=q\\}.$

By assumption, if $Y_{i}$ is a connected component of $M_{H}$ and if
$\dim\bigl{(}N(H)/H\bigr{)}=\dim(Y_{i})-1$, then $Y_{i}$ contains at most one
of the points $x_{1},\ldots,x_{n}$. Thus, in this case, $n_{i}=1$ and
$\Delta^{G}_{i}=\emptyset$. (Notice that if $\dim(Y_{i})=1$, then
$\dim\bigl{(}N(H)/H\bigr{)}\in\\{0,1\\}$. If $\dim(N(H)/H)=0$, then the
equation $\dim\bigl{(}N(H)/H\bigr{)}=\dim(Y_{i})-1$ is satisfied. If
$\dim(N(H)/H)=1$, then $Y_{i}$ consists of just one orbit, and again, by
assumption, $Y_{i}$ can contain at most one of the points
$x_{1},\ldots,x_{n}$.) If $\dim\bigl{(}N(H)/H\bigr{)}<\dim(Y_{i})-1$, then
$\Delta^{G}_{i}$ is a union of submanifolds of $Y_{i}^{n_{i}}$ of codimension
at least two. Since $Y_{i}^{n_{i}}$ is connected, this implies that the
complement $Y_{i}^{n_{i}}\setminus\Delta^{G}_{i}$ is connected. Similarly,

$M^{(n)}\cap(Y_{1}^{n_{1}}\times\cdots\times
Y_{k}^{n_{k}})=Y_{1}^{n_{1}}\times\cdots\times
Y_{k}^{n_{k}}\setminus\bigcup_{i=1}^{k}(Y_{1}^{n_{1}}\times\cdots\times\Delta^{G}_{i}\times\cdots\times
Y_{k}^{n_{k}})$

is connected. The above equality holds since we assumed $Y_{i}\not=gY_{j}$,
for all $i\not=j$ and for all $g\in G\setminus N(H)_{0}$.

The group ${\rm Diff}^{G}_{c}(M)$ acts diagonally on $M^{(n)}$ and this action
preserves each stratum $M^{(n)}\cap(Y_{1}^{n_{1}}\times\cdots\times
Y_{k}^{n_{k}})$. We will show that each ${\rm Diff}^{G}_{c}(M)$-orbit is open
in $M^{(n)}\cap(Y_{1}^{n_{1}}\times\cdots\times Y_{k}^{n_{k}})$. Therefore,
let $(z_{1},\ldots,z_{n})\in M^{(n)}\cap(Y_{1}^{n_{1}}\times\cdots\times
Y_{k}^{n_{k}})$. For each $z_{j}$, choose an open $G$-invariant neighborhood
$U_{j}$ such that $U_{i}\cap U_{j}=\emptyset$, if $i\not=j$. Then
$U_{1}\times\cdots\times U_{n}\subset M^{(n)}$. By Proposition 2.3, each
$z_{i}$ has a neighborhood $V_{i}$ in $Y_{i}$ with the property that
$GV_{i}\subset U_{i}$, and for each $w_{i}\in V_{i}$ there exists
$f_{i}\in{\rm Diff}^{G}_{c}(M)$ such that the $G$-compact support of the
isotopy from the identity to $f_{i}$ is contained in $U_{i}$ and
$f_{i}(z_{i})=w_{i}$. Let $g=f_{1}\circ\cdots\circ f_{n}$. Since ${\rm
supp}(f_{i})\cap{\rm supp}(f_{j})=\emptyset$, for all $i\not=j$, it follows
that $g(z_{i})=w_{i}$, for all $1\leq i\leq n$. This means that the ${\rm
Diff}^{G}_{c}(M)$-orbit of $(z_{1},\ldots,z_{n})$ is open in
$M^{(n)}\cap(Y_{1}^{n_{1}}\times\cdots\times Y_{k}^{n_{k}})$. Since
$M^{(n)}\cap(Y_{1}^{n_{1}}\times\cdots\times Y_{k}^{n_{k}})$ is connected,
there is exactly one orbit. Thus there exists $f\in{\rm Diff}^{G}_{c}(M)$ such
that $f(x_{1},\ldots,x_{n})=(y_{1},\ldots,y_{n})$. ∎

######  Corollary 6.1.

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. Let
$n\in{\mathbb{N}}$, and let $(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$
be two $n$-tuples of points in $M$. Assume the Conditions $(1)-(3)$ in Theorem
0.1 are satisfied. Then there exists a diffeomorphism $f\colon M/G\to M/G$
isotopic to the identity through a compactly supported isotopy, such that
$f(Gx_{i})=Gy_{i}$ for all $1\leq i\leq n$.

###### Proof.

Since the $n$-tuples $(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$ satisfy
the conditions of Theorem 0.1, there is an equivariant $G$-compactly supported
isotopy $F\colon M\times I\to M$ with $F_{0}={\rm id}\colon M\to M$ and
$F_{1}(x_{i})=y_{i}$, for all $i\in\\{1,\ldots,n\\}$. Then $F$ induces a
compactly supported isotopy $\tilde{F}\colon(M/G)\times I\to M/G$, with
$\tilde{F}_{0}={\rm id}\colon M/G\to M/G$ and $\tilde{F}_{1}(Gx_{i})=Gy_{i}$
for all $i\in\\{1,\ldots,n\\}$. The isotopy $\tilde{F}$ is smooth in the sense
of [12], since it is induced by the equivariant isotopy $F$. ∎

Proof of Theorem 0.2. Let $X$ be a reduced smooth orbifold. Using the frame
bundle construction, we identify $X$ with an orbit space $M/G$, where $G$ is a
compact Lie group acting smoothly on $M$ by an effective, almost free action.
Let $\pi\colon M\to M/G$ denote the natural projection. For every
$i\in\\{1,\ldots,n\\}$ there exists $\tilde{x}_{i}\in M$ with
$\pi(\tilde{x}_{i})=x_{i}$. Denote the isotropy subgroup of $\tilde{x}_{i}$ by
$H_{i}$, for every $i$. Let $Y_{i}$ be the connected component of $M_{H_{i}}$
containing $\tilde{x}_{i}$. Since $\Sigma(x_{i})=\hat{\Sigma}(x_{i})$ (see
Section 4), and by Lemma 1.4, we may choose points $\tilde{y}_{i}\in M$ with
$\pi(\tilde{y}_{i})=y_{i}$ in such a way that $\tilde{y}_{i}\in Y_{i}$, for
each $i$.

We now have two $n$-tuples $(\tilde{x}_{1},\ldots,\tilde{x}_{n})$ and
$(\tilde{y}_{1},\ldots,\tilde{y}_{n})$ in $M$. We will show that these
$n$-tuples satisfy the Conditions $(1)-(3)$ in Theorem 0.1. Condition $(1)$
holds by assumption, and Condition $(2)$ holds by the way the points
$\tilde{y}_{i}$ were chosen.

It remains to verify that Condition $(3)$ holds. Let $x\in M_{H}$, and let $Y$
be the connected component of $M_{H}$ containing $x$. Then the normalizer
$N(H)$ acts properly on $N(H)Y$. Let ${\rm N}$ be a linear slice at $x$ in
$N(H)Y$. Then

${\rm T}_{x}\bigl{(}N(H)Y\bigr{)}\cong{\rm
T}_{eH}\bigl{(}N(H)/H\bigr{)}\oplus{\rm T}_{x}({\rm N}).$

Since $N(H)_{0}Y=Y$, and since $N(H)Y$ is a union of some connected components
of $M_{H}$, it follows that

$\displaystyle\dim(Y)$ $\displaystyle=\dim\bigl{(}N(H)_{0}Y\bigr{)}$
$\displaystyle=\dim\bigl{(}N(H)/H\bigr{)}+\dim({\rm N})$
$\displaystyle>\dim\bigl{(}N(H)/H\bigr{)}+1,$

when $\dim({\rm N})>1$. Assume $\dim({\rm N})=1$. Then
$\dim\bigl{(}\pi(Y)\bigr{)}=1$ and, by assumption, $\pi(Y)$ can contain at
most one of the points $x_{i}$. Consequently, $Y$ can contain at most one of
the points $\tilde{x}_{i}$. Thus Condition $(3)$ is satisfied. The claim of
the theorem can now be proved the same way Corollary 6.1 was proved. ∎

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