# On the transitivity of the group of orbifold diffeomorphisms.

Federica Pasquotto  and  Thomas O. Rot

###### Abstract.

Consider a connected manifold of dimension at least two and the group of
compactly supported diffeomorphisms that are isotopic to the identity through
a compactly supported isotopy. This group acts $n$-transitively: any $n$-tuple
of points can be moved to any other $n$-tuple by an element of this group. The
group of diffeomorphisms of an orbifold is typically not $n$-transitive:
simple obstructions are given by isomorphism classes of isotropy groups of
points. In this paper we investigate the transitivity properties of the group
of compactly supported diffeomorphisms of orbifolds that are isotopic to the
identity through a compactly supported isotopy. We also study an example in
the category of area preserving mappings.

###### Key words and phrases:

Orbifolds, differential topology, diffeomorphisms, area preserving mappings

###### 2010 Mathematics Subject Classification:

57R18, 57R50, 58D05

##  1\. Introduction

Given a connected manifold $M$ of dimension at least two and a natural number
$n$, the group $\operatorname{Diff}_{\mathrm{c}}(M)$ of compactly supported
diffeomorphisms of $M$ that are isotopic to the identity through a compactly
supported isotopy acts $n$-transitively, meaning that given two $n$-tuples
$(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$ of distinct points in $M$
there exists a diffeomorphism $f\in\operatorname{Diff}_{\mathrm{c}}(M)$ such
that $f(x_{i})=y_{i}$ for all $i=1,\ldots,n$. This result is fundamental for
many applications in differential topology and holds also when the
diffeomorphisms are required to preserve certain structures, e.g. symplectic
forms and analytic structures. We refer to [14, 2, 1] for further discussion
on this topic.

The orbifold diffeomorphism group was first investigated in [3, 4], and it is
natural to wonder whether the orbifold diffeomorphism group is $n$-transitive.
It was observed in [4, Lemma 5.1] that an orbifold diffeomorphism necessarily
preserves (up to isomorphism) the isotropy groups of points, hence the group
of compactly supported orbifold diffeomorphisms does not act transitively, let
alone $n$-transitively. Nevertheless we show that
$\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$, the group of compactly
supported orbifold diffeomorphisms that are isotopic to the identity through a
compactly supported isotopy, does act transitively on the connected components
of each singular stratum.

All the notions which appear in the theorem below will be properly introduced
in the next section.

######  Theorem 1.1 ($n$-Transitivity).

Let $\mathcal{O}$ be an orbifold. Let $(x_{1},\ldots,x_{n})$ and
$(y_{1},\ldots,y_{n})$ be two $n$-tuples of pairwise distinct points. Assume
that for every $i$ the points $x_{i}$ and $y_{i}$ lie in the same connected
component of the singular stratification. Assume furthermore that each
connected component of $\Sigma_{1}$ contains at most one of the $x_{i}$’s.
Then there exists an orbifold diffeomorphism
$f:\mathcal{O}\rightarrow\mathcal{O}$, isotopic to the identity through a
compactly supported isotopy, such that $f(x_{i})=y_{i}$ for all $1\leq i\leq
n$.

This has also ramifications in other categories of automorphisms: in Section 4
we consider an example in the category of orbifolds equipped with an area
form. We also discuss a setting where $n$-transitivity might give rise to new
invariants of orbifold mappings.

### Acknowledgements

T.O. Rot is supported by NWO-NWA startimpuls - 400.17.608. We thank the
referees for carefully reading our manuscript and for their helpful comments.

##  2\. Orbifolds and orbifold maps

###  2.1. Orbifold charts

Let $O$ be a topological space. A (smooth) _orbifold chart_ is a triple
$(\tilde{U},\Gamma_{U},\phi_{U})$ which consists of

  * •

an open, connected subset $\tilde{U}\subseteq\mathbb{R}^{n}$;

  * •

a finite group $\Gamma_{U}$ acting smoothly and effectively on $\tilde{U}$;

  * •

a continuous map $\phi_{U}$ from $\tilde{U}$ onto an open subset $U\subseteq
O$, such that $\phi_{U}\circ\gamma=\phi_{U}$ for all $\gamma\in\Gamma_{U}$,
and $\phi_{U}$ induces a homeomorphism between $\tilde{U}/\Gamma$ and $U$.

An _embedding_ between two orbifold charts $(\tilde{U},\Gamma_{U},\phi_{U})$
and $(\tilde{V},\Gamma_{V},\phi_{V})$ is a smooth embedding
$\lambda:\tilde{U}\rightarrow\tilde{V}$ such that
$\phi_{V}\circ\lambda=\phi_{U}$. In particular, this means
$U=\phi_{U}(\tilde{U})\subseteq V$.

Two orbifold charts $(\tilde{U},\Gamma_{U},\phi_{U})$ and
$(\tilde{V},\Gamma_{V},\phi_{V})$ such that $U\cap V\neq\emptyset$ are called
_locally compatible_ if for every $p\in U\cap V$ there exists an orbifold
chart $(\tilde{W},\Gamma_{W},\phi_{W})$ such that $p\in W\subseteq U\cap V$
and $(\tilde{W},\Gamma_{W},\phi_{W})$ embeds into both
$(\tilde{U},\Gamma_{U},\phi_{U})$ and $(\tilde{V},\Gamma_{V},\phi_{V})$.

An _orbifold atlas_ on $O$ is a collection $\mathcal{U}$ of orbifold charts
such that $\mathcal{U}$ covers $O$, and the elements of $\mathcal{U}$ are
locally compatible (in the sense of the above definition). An orbifold atlas
$\mathcal{U}$ _refines_ and orbifold atlas $\mathcal{V}$ if every chart of
$\mathcal{U}$ admits an embedding into some chart of $\mathcal{V}$. Two
orbifold atlases are _equivalent_ if they admit a common refinement. An
effective _orbifold_ is a paracompact and Hausdorff space $O$ equipped with an
orbifold atlas $\mathcal{U}$. Just as in the manifold case, every orbifold
atlas lies in a unique maximal atlas and we will always assume our orbifolds
to be equipped with a maximal atlas.

We denote the pair $(O,\mathcal{U})$ by $\mathcal{O}$, to distinguish between
the orbifold and its underlying topological space $O$.

The Bochner-Cartan linearization theorem shows that we may always choose
charts in which the groups $\Gamma_{U}$ act linearly on
$\tilde{U}=\mathbb{R}^{n}$, i.e. the chart is a representation of
$\Gamma_{U}$. We call such a chart a linear orbifold chart.

###  2.2. Singular set and singular dimension

Let $\mathcal{O}$ be an orbifold and let $x$ be a point in the underlying
topological space $O$. If $(\tilde{U},\Gamma_{U},\phi_{U})$ is any orbifold
chart such that $x=\phi_{U}(\tilde{x})\in U$, we define the isotropy group at
$x$ to be

$\Gamma_{x}=\\{\gamma\in\Gamma_{U}\,:\,\gamma\cdot\tilde{x}=\tilde{x}\\}.$

Up to group isomorphism, this definition is independent of the choice of chart
and lift. The _singular set_ $\Sigma$ of $\mathcal{O}$ consists of the points
with non-trivial isotropy group:

$\Sigma=\\{x\in O\,:\,\Gamma_{x}\neq 1\\}.$

The singular set admits a stratification by singular dimension, which we
explain now. Each point $x\in O$ is contained in an open neighborhood $U_{x}$,
for which there exists an orbifold chart where the group acting is isomorphic
to $\Gamma_{x}$: we call this a _chart centered at $x$_ and write
$(\tilde{U}_{x},\Gamma_{x},\phi_{x})$ for such a chart. If
$(\tilde{U}_{x},\Gamma_{x},\phi_{x})$ is centered at $x$, then there exists a
unique $\tilde{x}\in\tilde{U}_{x}$ such that $\phi_{x}(\tilde{x})=x$. The
action of $\Gamma_{x}$ fixes $\tilde{x}$, and the differential induces an
action on $T_{\tilde{x}}\tilde{U}_{x}$. This action fixes a subspace
$T_{\tilde{x}}\tilde{U}_{x}^{\Gamma_{x}}$, cf. [10]. The _singular dimension_
of $x$ is defined to be $\operatorname{sdim}(x)=\dim
T_{\tilde{x}}\tilde{U}_{x}^{\Gamma_{x}}$ and does not depend on the choice of
orbifold chart. The singular set can be written as the union
$\Sigma=\bigcup_{k=0}^{n-1}\Sigma_{k}$ of _singular strata_ , where
$\Sigma_{k}=\\{x\in\Sigma\,|\,\operatorname{sdim}(x)=k\\}$.

The following proposition, which is for example proven in [9, Proposition
3.4], uses the fact that that the fixed point set of a smooth finite group
action on a manifold is a manifold.

######  Proposition 2.1.

Let $\mathcal{O}$ be an orbifold. Then each singular stratum $\Sigma_{k}$ is
naturally a manifold of dimension $k$, whose tangent space at $x\in\Sigma_{k}$
is modeled on $T_{\tilde{x}}\tilde{U}_{x}^{\Gamma_{x}}$, where $\tilde{U}_{x}$
is a chart centered at $x$ and $\tilde{x}$ is the lift of $x$ to this chart.

We denote by $\Sigma(x)$ the connected component of
$\Sigma_{\operatorname{sdim}(x)}$ containing $x$. The isomorphism class of the
isotropy group $\Gamma_{y}$ is constant for $y\in\Sigma(x)$, and if
$\mathcal{O}$ is compact there are finitely many connected components in each
singular stratum $\Sigma_{k}$.

Sometimes we need to consider multiple orbifolds at the same time. Whenever
confusion might arise, we decorate the symbols for the singular set etc. with
a sub/superscript specifying the orbifold in question.

###  2.3. Orbifold maps

Given the two orbifolds $\mathcal{O}$ and $\mathcal{P}$, a _smooth orbifold
map_ between $\mathcal{O}$ and $\mathcal{P}$ is defined by the following
data/conditions:

  * •

a continuous map $f:O\rightarrow P$ between the underlying topological spaces;

  * •

for every $x\in O$, there exist charts $(\tilde{U},\Gamma_{U},\phi_{U})$ and
$(\tilde{V},\Gamma_{V},\phi_{V})$ containing $x$ and $f(x)$, respectively,
with the property that $f$ maps $U$ into $V$ and can be lifted to a smooth map
$\tilde{f}:\tilde{U}\rightarrow\tilde{V}$ such that
$\phi_{V}\circ\tilde{f}=f\circ\phi_{U}$.

######  Remark 2.2.

There are several, non-equivalent definitions of (smooth) orbifold maps in the
literature, such as, for instance, good maps [8], complete orbifold maps [6],
and strong maps [16]. Conveniently for us, these differences tend to disappear
for orbifold diffeomorphisms. For instance, every orbifold diffeomorphism as
defined below is a regular map, in the sense of [8], so in particular it is a
good orbifold map, with a unique isomorphism class of compatible systems.

In this work we are dealing with a special type of orbifold maps, namely
orbifold diffeomorphisms, which behave in a particularly nice way: for
instance, the lifts to local charts are uniquely determined up to
multiplication by the local group. We now recall the definition and some basic
properties of orbifold diffeomorphisms.

An _orbifold diffeomorphism_ is a smooth orbifold map
$f:\mathcal{O}\rightarrow\mathcal{P}$ such that there exists another smooth
orbifold map $g:\mathcal{P}\rightarrow\mathcal{O}$ with $g\circ f=1_{O}$ and
$f\circ g=1_{P}$. An orbifold diffeomorphism is a homeomorphism between the
underlying topological spaces and the local lifts of an orbifold
diffeomorphism are local diffeomorphisms in the usual sense, as we prove
below.

######  Lemma 2.3.

If $f:\mathcal{O}\rightarrow\mathcal{P}$ is an orbifold diffeomorphism and
$\tilde{f}$ is a local smooth lifting of $f$, then $\tilde{f}$ is a local
diffeomorphism.

###### Proof.

Let $(\tilde{U},\Gamma_{U},\phi_{U})$ be an orbifold chart. Let
$\tilde{\operatorname{id}}:\tilde{U}\rightarrow\tilde{U}$ be a smooth lift of
the identity $\operatorname{id}:U\rightarrow U$. We claim that
$\tilde{\operatorname{id}}$ is a local diffeomorphism: We can assume that the
group $\Gamma_{U}$ acts linearly on $\tilde{U}=\mathbb{R}^{n}$. Let $x\in
U\setminus\Sigma$ be a smooth point and let $\tilde{x}$ be a lift to
$\tilde{U}$ of $x$. Then there exists a unique $\gamma\in\Gamma_{U}$ such that
$\tilde{\operatorname{id}}(\tilde{x})=\gamma\tilde{x}$. By continuity there
exists an open neighborhood $\tilde{V}$ of $\tilde{x}$ such that
$\tilde{\operatorname{id}}(\tilde{y})=\gamma\tilde{y}$ for all
$\tilde{y}\in\tilde{V}$. Thus $T_{\tilde{y}}\tilde{\operatorname{id}}=\gamma$
for $\tilde{y}\in\tilde{V}$ and
$\det(T_{\tilde{y}}\tilde{\operatorname{id}})=\det\gamma$ is constant and non-
vanishing on $\tilde{V}$. The non-smooth points are open and dense, so it
follows from the previous argument that
$\det(T_{\tilde{y}}\tilde{\operatorname{id}})$ is constant and non-vanishing
on each connected component of the regular set. By continuity,
$\det(T_{\tilde{y}}\tilde{\operatorname{id}})$ is constant and non-vanishing
on the whole of $\tilde{U}$. Thus $\tilde{\operatorname{id}}$ has an
invertible differential and is a local diffeomorphism.

Now suppose $f$ is an orbifold diffeomorphism and let $g$ be the inverse of
$f$. Let $\tilde{f}$ be a local lift of $f$ around $x\in U$ to orbifold charts
$(\tilde{U},\Gamma_{U},\phi_{U})$ and $(\tilde{V},\Gamma_{V},\phi_{V})$. Let
$\tilde{g}$ be a local lift of $g$ around $y=f(x)$ to orbifold charts
$(\tilde{V}^{\prime},\Gamma_{V^{\prime}},\phi_{V^{\prime}})$ and
$(\tilde{U^{\prime}},\Gamma_{U^{\prime}},\phi_{U^{\prime}})$. Note that we
may, by shrinking the charts if necessary, assume that there exist chart
embeddings $\lambda:\tilde{V}\rightarrow\tilde{V}^{\prime}$ and
$\mu:\tilde{U}^{\prime}\rightarrow\tilde{U}$. Then the composition
$\mu\circ\tilde{g}\circ\lambda\circ\tilde{f}:\tilde{U}\rightarrow\tilde{U}$ is
a lift of the identity, hence by the first part of the proof it is a local
diffeomorphism. The underlying spaces $O$ and $P$ are homeomorphic, thus
$\dim(\mathcal{O})=\dim(\mathcal{P})$. If the composition of smooth maps
between manifolds of the same dimension are local diffeomorphisms, then each
individual map is a local diffeomorphism. Therefore $\tilde{g}$ and
$\tilde{f}$ are local diffeomorphisms. In fact, by [9, Proposition 1.11] it
then follows that $\tilde{\operatorname{id}}(x)=\gamma x$ everywhere. ∎

The previous lemma implies in particular that by a suitable choice of orbifold
charts we may always assume the diffeomorphism $f$ to lift to a diffeomorphism
between the charts. That will be our standing assumption in what follows.

######  Lemma 2.4.

If $f$ is an orbifold diffeomorphism, and $\tilde{f}$ is a smooth lifting of
$f$ between the orbifold charts $(\tilde{U},\Gamma_{U},\phi_{U})$ and
$(\tilde{V},\Gamma_{V},\phi_{V})$, then there is an induced group isomorphism
$\Theta:\Gamma_{U}\rightarrow\Gamma_{V}$ such that
$\tilde{f}(\gamma\cdot\tilde{x})=\Theta(\gamma)\cdot\tilde{f}(\tilde{x})$ for
all $\tilde{x}\in\tilde{U}$ (in other words, $\tilde{f}$ is
$\Theta$-equivariant). If $\tilde{f}^{\prime}$ is another lift of $f$ between
the same orbifold charts, then $\tilde{f}=\mu\circ\tilde{f}^{\prime}$ for a
unique $\mu\in\Gamma_{V}$.

###### Proof.

We can construct a map between the groups associated to the orbifold charts in
the following way: given $\gamma\in\Gamma_{U}$, the composition
$\tilde{f}\circ\gamma\circ\tilde{f}^{-1}$ is a diffeomorphism of $\tilde{V}$.
Moreover, we have that

$\phi_{V}\circ(\tilde{f}\circ\gamma\circ\tilde{f}^{-1})=f\circ\phi_{U}\circ\gamma\circ\tilde{f}^{-1}=f\circ\phi_{U}\circ\tilde{f}^{-1}=\phi_{V}\circ\tilde{f}\circ\tilde{f}^{-1}=\phi_{V},$

so by [9, Proposition 1.11] it follows that there exists a unique element
$\mu\in\Gamma_{V}$ such that $\tilde{f}\circ\gamma\circ\tilde{f}^{-1}=\mu$. We
set $\Theta(\gamma)=\mu$, and this defines a map
$\Theta:\Gamma_{U}\rightarrow\Gamma_{V}$, which by construction is a group
isomorphism.

We note that $\tilde{f}\circ(\tilde{f}^{\prime})^{-1}$ is a lift of the
identity on $V$, and again by [9, Proposition 1.11], we have that there exists
a unique $\mu\in\Gamma_{V}$ such that
$\tilde{f}\circ(\tilde{f}^{\prime})^{-1}=\mu$, i.e.
$\tilde{f}=\mu\circ\tilde{f}^{\prime}$. ∎

The _support_ of a smooth orbifold diffeomorphism
$f:\mathcal{O}\rightarrow\mathcal{O}$ is the set

$\mathrm{supp}(f)=\overline{\\{x\in O\,|\,f(x)\neq x\\}}.$

In order to define homotopies of orbifold maps, we need to consider the
product orbifold structure on $\mathcal{O}\times[0,1]$ ([5]): its singular set
consists of points of the form $(x,t)$, with $x\in\Sigma$ and $t\in[0,1]$, and
for all such points the isotropy group $\Gamma_{(x,t)}$ is isomorphic to
$\Gamma_{x}$. The time $t$ inclusion
$\mathcal{i}_{t}:\mathcal{O}\rightarrow\mathcal{O}\times[0,1]$ is a smooth
orbifold map.

Two orbifold maps $f,g:\mathcal{O}\rightarrow\mathcal{P}$ are called _smoothly
homotopic_ if there exists a smooth orbifold map
$F:\,\mathcal{O}\times[0,1]\rightarrow\mathcal{P}$ such that

$F\mathcal{i}_{0}=f\quad\textrm{and}\quad F\mathcal{i}_{1}=g.$

Two orbifold diffeomorphisms are _isotopic_ if they are homotopic through
orbifold diffeomorphisms. We denote by
$\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$ the group of compactly
supported diffeomorphisms of the orbifold $\mathcal{O}$ that are isotopic to
the identity through a compactly supported isotopy.

##  3\. Transitivity

In this section we will prove our main result. We start with the observation
that an orbifold diffeomorphism preserves the singular dimension.

######  Lemma 3.1.

Suppose $f:\mathcal{O}\rightarrow\mathcal{P}$ is an orbifold diffeomorphism:
then for every $x\in O$ we have that
$\operatorname{sdim}(x)=\operatorname{sdim}(f(x))$.

###### Proof.

The isotropy group $\Gamma_{x}$ acts on $T_{\tilde{x}}\tilde{U}_{x}$ via the
differential. We denote this action by $\gamma\cdot\tilde{X}$, where
$\gamma\in\Gamma_{x}$ and $\tilde{X}\in T_{\tilde{x}}\tilde{U}_{x}$. If
$\tilde{X}\in T_{\tilde{x}}\tilde{U}_{x}^{\Gamma_{x}}$, then
$\gamma\cdot\tilde{X}=\tilde{X}$ and by the equivariance proved in Lemma 2.4
we find that $\Theta_{x}(\gamma)\cdot
T_{\tilde{x}}\tilde{f}_{x}(\tilde{X})=T_{\tilde{x}}\tilde{f}_{x}(\gamma\cdot\tilde{X})=T_{\tilde{x}}\tilde{f}_{x}(x)\tilde{X}$.
Since the map $\Theta_{x}:\Gamma_{x}\rightarrow\Gamma_{f(x)}$ is a group
isomorphism, every $\mu\in\Gamma_{f(x)}$ is of the form $\Theta_{x}(\gamma)$
for some $\gamma\in\Gamma_{x}$. Thus $\mu\cdot
T_{\tilde{x}}\tilde{f}_{x}\tilde{X}=T_{\tilde{x}}\tilde{f}_{x}(\tilde{X})$ for
all $\mu\in\Gamma_{f(x)}$. We conclude that
$T_{\tilde{x}}\tilde{f}_{x}(T_{\tilde{x}}\tilde{U}_{x}^{\Gamma_{x}})\subset
T_{\tilde{f}_{x}(\tilde{x})}\tilde{U}_{f(x)}^{\Gamma_{f(x)}}$, which implies
that $\operatorname{sdim}(x)\leq\operatorname{sdim}(f(x))$. The same argument
applied to the inverse map $f^{-1}$ then shows that
$\operatorname{sdim}(f(x))\leq\operatorname{sdim}(x)$, hence
$\operatorname{sdim}(x)=\operatorname{sdim}(f(x))$. ∎

The next proposition is the crucial step in the proof of our result: it shows
that locally we can carry a given point $x$ to any sufficiently close point
lying on the same connected component by a diffeomorphism which is isotopic to
the identity via an isotopy having compact support in a prescribed open
neighborhood of $x$.

######  Proposition 3.2.

Let $\mathcal{O}$ be an orbifold, and let $U$ be an open neighborhood of a
point $x\in O$. Then there exists an open subset $V$ of $\Sigma(x)$, such that
$x\in V$, $V\subseteq U$, and for each $y\in V$ there exists
$f\in\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$ such that $f(x)=y$ and $f$
is isotopic to the identity via an isotopy with compact support contained in
$U$.

###### Proof.

Choose an orbifold chart $(\tilde{U}_{x}=\mathbb{R}^{n},\Gamma_{x},\phi_{x})$
centered around $x$, and such that $U_{x}\subseteq U$. Let
$\tilde{x}\in\tilde{U}_{x}$ be the unique point such that
$\phi_{x}(\tilde{x})=x$, and let $\tilde{V}\subseteq\tilde{U}_{x}$ be the
connected component of the fixed set of $\Gamma_{x}$ containing $\tilde{x}$.
Define $V=\phi_{x}(\tilde{V})$: the restriction of the map $\phi_{x}$ to
$\tilde{V}$ is injective, and $\phi_{x}(\tilde{V})\subseteq\Sigma(x)\cap
U_{x}$. Let $\tilde{y}\in\tilde{V}$. We want to show that that exists a
diffeomorphism $\tilde{f}$ of $\tilde{U}_{x}$ which satisfies the following
three conditions:

  * (i)

$\tilde{f}$ is equivariant with respect to the action of $\Gamma_{x}$;

  * (ii)

$\tilde{f}$ is isotopic to the identity, via equivariant diffeomorphisms
compactly supported in $U$;

  * (iii)

$\tilde{f}(\tilde{x})=\tilde{y}$.

Since $\tilde{V}$ is a connected manifold, there exists a vector field
$\tilde{Y}$ on $\tilde{V}$, which is compactly supported111A vector field is
compactly supported if it vanishes outside a compact set and whose time-one
flow maps $\tilde{x}$ to $\tilde{y}$. We can extend $\tilde{Y}$ to a compactly
supported vector field defined everywhere on $\tilde{U}_{x}$, and we will
denote this vector field by the same symbol. Recall that the action of
$\Gamma_{x}$ on $\tilde{U}_{x}$ induces a linearized action on
$T\tilde{U}_{x}$, which we denote by $\gamma\cdot\tilde{X}$, with
$\gamma\in\Gamma_{x}$ and $\tilde{X}\in T\tilde{U}_{x}$. Note that
$\gamma\cdot:T_{\tilde{z}}\tilde{U}_{x}\rightarrow
T_{\gamma\tilde{z}}\tilde{U}_{x}$ and, as $\tilde{Y}$ is tangent to
$\tilde{V}$ and the action fixes $\tilde{V}$, that
$\gamma\cdot\tilde{Y}(\tilde{z})=\tilde{Y}(\tilde{z})$ for all
$\tilde{z}\in\tilde{V}$. We need to average over the action of the group
$\Gamma_{x}$ in order to make our vector field equivariant, that is, for all
$\tilde{z}\in\tilde{U}_{x}$ we define $\tilde{X}(\tilde{z})$ via the formula:

$\tilde{X}(\tilde{z})=\frac{1}{|\Gamma_{x}|}\sum_{\gamma\in\Gamma_{x}}\gamma\cdot\tilde{Y}(\gamma^{-1}\tilde{z}).$

Since $\tilde{Y}$ is compactly supported, and $\Gamma_{x}$ is finite, the
vector field $\tilde{X}$ is compactly supported. Then we check that, for
$\mu\in\Gamma_{x}$,

$\displaystyle\mu\cdot\tilde{X}(\tilde{z})$
$\displaystyle=\frac{1}{|\Gamma_{x}|}\sum_{\gamma\in\Gamma_{x}}\mu\cdot(\gamma\cdot\tilde{Y}(\gamma^{-1}\tilde{z}))$
$\displaystyle=\frac{1}{|\Gamma_{x}|}\sum_{\gamma\in\Gamma_{x}}(\mu\gamma)\cdot\tilde{Y}((\mu\gamma)^{-1}\mu\tilde{z})=\frac{1}{|\Gamma_{x}|}\sum_{\gamma^{\prime}\in\Gamma_{x}}\gamma^{\prime}\cdot\tilde{Y}(\gamma^{\prime-1}\mu\tilde{z})=\tilde{X}(\mu\tilde{z}),$

i.e. $\tilde{X}$ is indeed equivariant with respect to the action of
$\Gamma_{x}$. Note that for $\tilde{z}\in\tilde{V}$ we have that $\gamma\cdot
X(\tilde{z})=\tilde{Y}(\tilde{z})$, hence the flow of $\tilde{X}$ agrees with
that of $\tilde{Y}$ when restricted to $\tilde{V}$. In particular, the time-
one flow $\tilde{f}$ of $\tilde{X}$ maps $\tilde{x}$ to $\tilde{y}$. Clearly
the projection of $\tilde{f}$ to $U$ can be extended (by declaring it to be
the identity outside of $U$) to an orbifold diffeomorphism $f$, which is
isotopic to the identity through an isotopy with support in $U$. By
construction, $f$ maps $x$ to $y=\phi_{x}(\tilde{y})$. ∎

We can now move on to the proof of our main result.

######  Theorem 3.3 (Transitivity).

Let $\mathcal{O}$ be a connected orbifold. Let $(x_{1},\ldots,x_{n})$ and
$(y_{1},\ldots,y_{n})$ be two $n$-tuples of pairwise distinct points. Assume
that for every $i$ the points $x_{i}$ and $y_{i}$ lie in the same connected
component of the singular stratification. Assume furthermore that each
connected component of $\Sigma_{1}$ contains at most one of the $x_{i}$’s.
Then there exists an orbifold diffeomorphism
$f:\mathcal{O}\rightarrow\mathcal{O}$, isotopic to the identity through a
compactly supported isotopy, such that $f(x_{i})=y_{i}$ for all $1\leq i\leq
n$.

###### Proof.

We will start by reordering the $x_{i}$’s and $y_{i}$’s according to which
stratum they belong to. Up to a permutation, we may assume that there exist
numbers $1=l_{1}<l_{2}<\ldots<l_{k}\leq n$ such that for all $i\leq k-1$ one
has $x_{j}\in\Sigma^{\mathcal{O}}(x_{l_{i}})=:Y_{i}$ if and only if $l_{i}\leq
j\leq l_{i+1}-1$, and $x_{j}\in\Sigma^{\mathcal{O}}(x_{l_{k}})=:Y_{k}$ if and
only if $j\geq l_{k}$ (and analogously for the $y_{i}$’s). Consider the
$n$-fold cartesian product
$\mathcal{O}^{n}=\mathcal{O}\times\ldots\times\mathcal{O}$ and the (open)
suborbifold $\mathcal{O}^{(n)}\subseteq\mathcal{O}^{n}$, whose underlying
space consists of $n$-tuples of points which are pairwise distinct, i.e.

$O^{(n)}=\\{(x_{1},\ldots,x_{n})\in
O^{n}\,|\,x_{i}\not=x_{j}\quad\text{if}\quad i\not=j\\}.$

Let $n_{i}=l_{i+1}-l_{i}$ for all $1\leq i\leq k-1$ and $n_{k}=n-l_{k}+1$.
Then both $(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$ lie in the
singular stratum
$\Sigma^{\mathcal{O}^{n}}(x_{1},\ldots,x_{n})=Y_{1}^{n_{1}}\times\ldots\times
Y_{k}^{n_{k}}$. We claim that the following identity holds:

$\Sigma^{\mathcal{O}^{(n)}}(x_{1},\ldots,x_{n})=\Sigma^{\mathcal{O}^{n}}(x_{1},\ldots,x_{n})\cap\mathcal{O}^{(n)}.$
(1)

The inclusion of the left hand side in the right hand side is immediate, and
for the inclusion of the right hand side in the left hand side we should check
that $\Sigma^{\mathcal{O}^{n}}(x_{1},\ldots,x_{n})\cap\mathcal{O}^{(n)}$ is
connected. The fat diagonal in $Y_{i}^{n_{i}}$ is defined by

$\Delta_{i}=\\{(z_{1},\ldots,z_{n_{i}})\in Y_{i}^{n_{i}}\,|z_{j}=z_{k}\text{
for some }j\not=k\\}.$

By our assumption on $\Sigma_{1}$, the fat diagonal $\Delta_{i}$ is either
empty or a union of submanifolds of $Y_{i}^{n_{i}}$ of codimension greater
than or equal to two in $Y_{i}^{n_{i}}$. Thus
$Y_{i}^{n_{i}}\setminus\Delta_{i}$ is connected as $Y_{i}^{n_{i}}$ is
connected. By a similar argument

$\Sigma^{\mathcal{O}^{n}}(x_{1},\ldots,x_{n})\cap\mathcal{O}^{(n)}=Y_{1}^{n_{1}}\times\ldots\times
Y_{k}^{n_{k}}\setminus\bigcup_{i=1}^{k}\left(Y_{1}^{n_{1}}\times\ldots\times\Delta_{i}\times\ldots
Y_{k}^{n_{k}}\right)$

is connected and (1) is true.

The group $\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$ acts on
$\mathcal{O}^{(n)}$ diagonally and this action preserves each singular stratum
$\Sigma^{\mathcal{O}^{(n)}}(x_{1},\ldots,x_{n})$. We show next that each
$\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$-orbit is open in
$\Sigma^{\mathcal{O}^{(n)}}(x_{1},\ldots,x_{n})$. Let
$(z_{1},\ldots,z_{n})\in\Sigma^{\mathcal{O}^{(n)}}$ and choose for each
$z_{i}$ an open subset $U_{i}\ni z_{i}$ such that $U_{i}\cap U_{j}=\emptyset$
if $i\not=j$. Then $U_{1}\times\ldots\times U_{n}\subseteq O^{(n)}$. Let
$V_{i}$ be the open subset of $\Sigma^{\mathcal{O}}(z_{i})$ given by
Proposition 3.2, where we choose $x=z_{i}$ and $U=U_{i}$. Then for each
$w_{i}\in V_{i}$ there exists
$f_{i}\in\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$ such that
$f_{i}(z_{i})=w_{i}$. Let $g=f_{1}\circ\ldots\circ f_{n}$. Since all $f_{i}$’s
have distinct support, we have that $g(z_{i})=w_{i}$. It follows that the
$\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$-orbit through each point
$(z_{1},\ldots,z_{n})$ is open in
$\Sigma^{\mathcal{O}^{(n)}}(x_{1},\ldots,x_{n})$. Since
$\Sigma^{\mathcal{O}^{(n)}}(x_{1},\ldots,x_{n})$ is connected, there can be
only one orbit and hence there exists an element
$f\in\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})$ such that
$f(x_{1},\ldots,x_{n})=(y_{1},\ldots,y_{n})$. ∎

######  Remark 3.4.

The statement can be improved to include multiple points in each connected
component of $\Sigma_{1}$, but the result is more technical to state. The
reason is that even in the manifold case the diffeomorphism group does not act
$k$-transitively on points if the manifold is one dimensional and $k>1$.
Consider for example the one-dimensional manifold $\mathbb{R}$. Then two
tuples $(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$ are said to be
_ordered_ if $x_{1}<\ldots<x_{n}$ and $y_{1}<\ldots<y_{n}$. If the $n$-tuples
$(x_{1},\ldots,x_{n})$ and $(y_{1},\ldots,y_{n})$ are both ordered, then there
exists a diffeomorphism $f\in\operatorname{Diff}_{\mathrm{c}}(\mathbb{R})$
with $f(x_{i})=y_{i}$, whereas if only one of the $n$-tuples is ordered and
the other is not, such a diffeomorphism does not exist. If one of the tuples
has the reverse order, e.g. $y_{1}>y_{2}>\ldots>y_{n}$, a diffeomorphism still
exists, but it is not isotopic to the identity, as it reverses the orientation
of $\mathbb{R}$.

The case for the circle $S^{1}$ is similar: one can lift a diffeomorphism of
$S^{1}$ to a diffeomorphism of $\mathbb{R}$ by choosing a base-point in
$S^{1}$ and an initial lift to $\mathbb{R}$. The points $x_{1},\ldots,x_{n}$
and $y_{1},\ldots,y_{n}$ are then lifted to points
$\tilde{x}_{1},\ldots,\tilde{x}_{n}$ and $\tilde{y}_{1},\ldots,\tilde{y}_{n}$.
There exists a diffeomorphism that maps $x_{i}$ to $y_{i}$ if and only if
there exists cyclic permutations $\sigma$ and $\mu$ such that

$\tilde{x}_{\sigma(1)}<\ldots<\tilde{x}_{\sigma(n)}\quad\text{and}\quad\tilde{y}_{\mu(1)}<\ldots<\tilde{y}_{\mu(n)}.$

In the orbifold case each connected component of $\Sigma_{1}$ is a one
dimensional manifold, and we can still make sense of the orderability
condition above by orienting these manifolds. We do not think it worthwhile to
make a full statement.

######  Corollary 3.5.

Let $\mathcal{O}$ be an orbifold with $\dim\mathcal{O}\geq 2$ and let
$x\in\mathcal{O}$. Then the orbit
$\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})x$ coincides with $\Sigma(x)$.
Hence the orbit $\operatorname{Diff}_{\mathrm{c}}(\mathcal{O})x$ is dense in a
connected orbifold $\mathcal{O}$ if and only if
$\operatorname{sdim}(x)=\dim\mathcal{O}$, that is, $x$ is a non-singular
point.

######  Remark 3.6.

The referee posed the following interesting question: are non-effective
orbifolds also transitive, in the sense discussed in this paper? We do not
know the answer at the time of writing.

##  4\. Applications

###  4.1. Displacing curves on a two sphere, and its orbifold analogue.

The non-transitivity of the orbifold diffeomorphism group has consequences
outside of the smooth category, even if we are not concerned with the
displacement of points: for instance, in the setting of area preserving maps.
We give an example here, and we are confident that the reader can think of
more applications.

Let us recall the following property of area preserving maps of the two-
sphere. Equip the sphere $S^{2}$ with the standard area form $\omega$.
Consider an embedded simple closed curve $i:S^{1}\rightarrow S^{2}$, and let
$u_{1}:D^{2}\rightarrow S^{2}$ and $u_{2}:D^{2}\rightarrow S^{2}$ be the two
discs bounded by this curve222It is clear that the curve divides $S^{2}$ into
two compact orientable surfaces with boundary. A simple argument involving the
Euler characteristic and the classification of orientable surfaces then shows
that these must be discs. . If we denote by $D_{1}$ and $D_{2}$ their image on
the sphere, that is, $D_{i}=u_{i}(D^{2})$, $i=1,2$, then their area is given
by $A_{1}=\int_{D^{2}}u_{1}^{*}\omega$ and
$A_{2}=\int_{D^{2}}u^{*}_{2}\omega$, respectively. We are going to prove that
any such simple closed curve corresponds, under an area preserving
diffeomorphism of the sphere, to a circle of constant height. Consider then
such a circle of latitude $L$, chosen so that it divides the sphere into two
spherical caps $C_{1}$ and $C_{2}$, of area $A_{1}$ and $A_{2}$, respectively.
Identifying $S^{1}$ with $L$, and using Alexander’s trick, we find a
diffeomorphism $f:S^{2}\rightarrow S^{2}$ which carries the circle of latitude
$L$ to the image of $i:S^{1}\rightarrow S^{2}$, our original simple closed
curve. See Figure 1. The area form $\omega^{*}=f^{*}\omega$ is not standard,
but the areas of $C_{1}$ and $C_{2}$, with respect to $\omega$ and
$\omega^{*}$, are the same. Moreover, in view of the Lagrangian tubular
neighborhood theorem, we may assume $\omega$ and $\omega^{*}$ to already
coincide in an open neighborhood of $L$, that is, we may assume
$\omega-\omega^{*}$ to have support in the interior of $C_{1}$ and $C_{2}$.
Now an application of Moser’s stability argument for volume forms [17] shows
that there exists a diffeomorphism of the sphere, fixing the circle $L$, which
identifies $\omega^{*}$ and $\omega$.

If $A_{1}$ and $A_{2}$ are not equal, the original circle is mapped to a
circle latitude strictly above or below the equator. Via a rotation, which is
an area preserving diffeomorphism, the curve can be displaced from itself.
Note that the disc with smaller area is mapped into the disc with larger area.

If $A_{1}=A_{2}$, the embedded curve is called _monotone_ (or _balanced_), and
it cannot be displaced by an area preserving map. In fact, by the argument
above, we may always assume the image of the curve to be the equator. If the
equator could be displaced by an area preserving map, then one hemisphere
would have to be mapped onto a proper subset of its interior or of the
interior of the opposite hemisphere, contradicting in both cases the
assumption that the map is area preserving. 333A nice example of an area
preserving map is the antipodal map $-\operatorname{id}$. Thus any simple
closed curve dividing $S^{2}$ into two regions of the same area, must have
antipodal points. A nice visual explanation is given in the numberphile video
[12].

$g$$f$ Figure 1. On the right a simple closed curve on $S^{2}$ is drawn. Via a
diffeomorphism $f$ this is pulled back to a parallel curve. Pulling the
standard area form on the right back to the middle picture creates a non-
standard area form. Via an additional diffeomorphism the area form can be
brought back into standard form. The original curve is pulled back to a circle
of constant latitude. The height is completely determined by the ratio of
areas $A_{1}/A_{2}$.

This discussion can be summarized in the statement below:

######  Proposition 4.1.

A simple closed curve on the two sphere can be displaced from itself by an
area preserving diffeomorphism if and only if the curve is not monotone.

There are various ways to generalize this example. For instance, this result
fits into the framework of non-displaceable (monotone) Lagrangian
submanifolds, which is due to Floer [11] (in the exact case) and Oh [18] in
the monotone case. In this generality the proofs are much more difficult.

Let us now move on to the orbifold world. Let $p$ and $q$ be two coprime
natural numbers. The $(p,q)$-spindle is the orbifold obtained as
$\mathbb{CP}^{1}(p,q)=S^{3}//S^{1}$, where $S^{1}\subseteq\mathbb{C}$ acts on
$S^{3}\subseteq\mathbb{C}^{2}$ via
$t\cdot(z_{0},z_{1})=(t^{p}z_{0},t^{q}z_{1})$. As this action is isometric
with respect to the standard Riemannian metric, the $(p,q)$-spindle is
naturally a Riemannian orbifold, and therefore also carries an area form. The
argument that follows does not depend on the specific area form. We denote the
points on $\mathbb{CP}^{1}(p,q)$ by the equivalence class $[z_{0},z_{1}]$. The
spindle has two isolated singular points at $N=[0,1]$ and $S=[1,0]$, with
isotropy $\mathbb{Z}_{p}$ and $\mathbb{Z}_{q}$, respectively. Now let $\gamma$
be any simple closed curve that does not pass through the singular points $N$
and $S$. The curve divides the spindle into two parts $D_{1}$ and $D_{2}$.
There are a few cases to consider. Suppose $N\in D_{1}$ and $S\in D_{2}$. Then
there is no way to displace the curve from itself by an area preserving map:
if $f$ is a diffeomorphism, it must fix $N$ and $S$ by Lemma 3.1. The curve
must then be mapped to the interior of either $D_{1}$ or $D_{2}$. Without loss
of generality we can assume that it is mapped to the interior of $D_{1}$.
Since the singular point $N$ is fixed, $D_{1}$ should be mapped to its own
interior. But this is clearly not possible in an area preserving manner.

######  Proposition 4.2.

Suppose $p$ and $q$ are coprime. If a simple closed curve on the spindle
$\mathbb{CP}^{1}(p,q)$ divides it into two components, each containing one of
the singular points, then it cannot be displaced from itself by an area
preserving orbifold diffeomorphism.

The interesting thing to note is that we are trying to displace a curve that
lies entirely in the manifold part of the orbifold, but it still “feels” the
singularities far away.

If both singular points lie in one region, say $N,S\in D_{1}$, then we can
displace the curve by an area preserving map, provided the area of $D_{2}$ is
smaller than that of $D_{1}$.

###  4.2. Homotopy classes of maps

Distinguishing homotopy classes of maps between connected closed manifolds $M$
and $N$ is an interesting and challenging problem. One of the easiest
invariants which can be defined to approach this problem is the (embedded)
cobordism class of a regular value. A regular value $y$ of a map
$f:M\rightarrow N$ defines a submanifold $f^{-1}(y)$ of $M$. If
$F:M\times[0,1]\rightarrow N$ is a homotopy between $f$ and $g$, and $y$ is a
regular value $y$ of $F$, which is also a regular value of $f$ and $g$, then
$F$ defines an embedded cobordism $F^{-1}(y)$ in $M\times[0,1]$ between
$f^{-1}(y)$ and $g^{-1}(y)$.

It follows from the fact that regular values are open and dense, and that the
diffeomorphism group acts transitively on $N$, that the cobordism class of a
regular value of a map $f:M\rightarrow N$ is independent of the regular value
and the homotopy class of $f$.

But this invariant is not strong enough to distinguish some important maps.
Let us consider for instance the Hopf fibration, which is homotopically non-
trivial. If we view $S^{3}$ as the unit sphere in $\mathbb{C}^{2}$, and
$S^{2}$ as the unit sphere in $\mathbb{C}\times\mathbb{R}$, then the Hopf
fibration is the map $f:S^{3}\rightarrow S^{2}$ defined by
$f(z_{0},z_{1})=(2z_{0}\overline{z}_{1},|z_{0}|^{2}-|z_{1}|^{2})$. This map is
in fact a generator of $\pi_{3}(S^{2})\cong\mathbb{Z}$. Every value is regular
and the preimage of each regular value is an unknotted, embedded circle in
$S^{3}$. The unknot is the boundary of an embedded disc in $S^{3}\times[0,1]$,
hence the cobordism class of the preimage of a regular value does not
distinguish the homotopy class of the Hopf fibration from the homotopy class
of the constant map.

Still we can try to obtain further information by considering a pair of
distinct points and their preimages. In this setting, a cobordism will consist
of a pair of disjoint, embedded surfaces, whose boundary consist of the
preimages of different pairs of regular points. Given that the diffeomorphism
group acts $2$-transitively on $S^{2}$, the embedded cobordism class of two
regular values is independent of the chosen regular values, and it is
independent of the homotopy class. The preimages of two points $p,q\in S^{2}$
under the Hopf map are two embedded unknots in $S^{3}$, but they are _linked_.
In particular it can be shown that that we cannot find a pair of disjoint,
embedded surfaces in $S^{3}\times[0,1]$ whose boundary consists of these two
circles alone. It follows that the Hopf fibration is homotopically non-
trivial. In this case this invariant captures the same information as the
_framed cobordism_ sets of $S^{3}$, cf. [15], but in general these invariants
are different.

Let us formalize the argument above directly in the language of orbifolds.
Given a proper orbifold map $f:\mathcal{O}\rightarrow\mathcal{P}$, and a
choice of connected components of some singular strata
$\Sigma(x_{1}),\ldots\Sigma(x_{n})$, we would like to associate a proper
homotopy invariant to $f$. To make sure that regular values are well-defined,
and the preimages of regular values are well-defined orbifolds, we need to
impose additional structure on the orbifold mappings, and consider for
instance complete orbifold mappings [6] or good maps [8]. Moreover, we need to
define the notion of full suborbifolds. We will not discuss these notions
here, but we refer the reader to [20, 13, 7] for more details. A
$k$-dimensional _$n$ -colored suborbifold_, is a collection of $n$ disjoint
compact full suborbifolds (without boundary)
$(\mathcal{N}_{1},\ldots,\mathcal{N}_{n})$ of $\mathcal{O}$. A cobordism
between the $k$-dimensional $n$-colored suborbifolds
$(\mathcal{N}_{1},\ldots,\mathcal{N}_{n})$ and
$(\mathcal{M}_{1},\ldots,\mathcal{M}_{n})$ is a tuple
$(\mathcal{W}_{1},\ldots,\mathcal{W}_{n})$ of $(k+1)$-dimensional compact full
suborbifolds of $\mathcal{O}\times[0,1]$ with boundary
$(\mathcal{N}_{1}\times\\{0\\},\ldots,\mathcal{N}_{n}\times\\{0\\})\cup(\mathcal{M}_{1}\times\\{1\\},\ldots,\mathcal{M}_{n}\times\\{1\\})$.
We denote the set of $k$-dimensional $n$-colored suborbifolds modulo cobordism
by $\mathrm{CEmb}^{n}_{k}(\mathcal{O}).$ Note that the suborbifolds
$\mathcal{N}_{i}$, $\mathcal{M}_{i}$ and $\mathcal{W}_{i}$ are not required to
be connected. Now perturb $f$ to $g$ such that there exist regular values
$y_{i}\in\Sigma(x_{i})$, and define the $n$-colored suborbifold
$(\mathcal{N}_{1}=g^{-1}(y_{1}),\ldots,\mathcal{N}_{n}=g^{-1}(y_{n}))$. The
invariance, up to $n$-colored cobordism, under different choices of regular
values now depends on the $n$-transitivity of the compactly supported orbifold
diffeomorphism group. But in this orbifold setting, we have to deal with the
problem of equivariant transversality: even though the set of regular values
of an orbifold mapping is still dense [6], if a point $y$ lies in the singular
stratum of $P$ it is not clear that the map
$f:\mathcal{O}\rightarrow\mathcal{P}$ can be made transverse to $y$ by a
suitable perturbation. We managed to avoid these transversality issues when
$\dim(\mathcal{O})=\dim(\mathcal{P})$ and developed a degree theory for
orbifolds [19]. The general case remains open.

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