# Existence and uniqueness for triangulations of volume forms

Mélanie Bertelson & Julie Distexhe

###### Contents

  1. Introduction
  2. Acknowledgments
  3. 1 Preliminaries
  4. 2 Triangulations of volume forms
    1. 2.1 Agreement near the skeleton
      1. 2.1.1 First step
      2. 2.1.2 Induction step
    2. 2.2 Interpolation
    3. 2.3 Several Consequences
  5. 3 Uniqueness for triangulations of volume forms

## Introduction

This paper stems from Eliashberg’s idea to develop a theory of piecewise
linear symplectic topology [Eli]. Central objects of this theory are piecewise
linear manifolds endowed with piecewise constant symplectic structures (or
equivalently piecewise constant non-degenerate Whitney forms), called
hereafter _piecewise linear (or PL) symplectic manifolds_. One of the very
first questions is how these compare to smooth symplectic manifolds. For
instance, does any smooth symplectic manifold $(M,\omega)$ admit a smooth
triangulation, that is, does there exists a Whitehead triangulation $h:|K|\to
M$ of $M$ for which $h^{*}\omega$ is piecewise constant ? It is surprisingly
difficult to find such triangulations, except in dimension two, where it
follows from the results presented here, or in very special cases (cf. [D19]
and [BD]).

This question is closely related to another “very first” question which is to
decide whether a PL symplectic manifold $(W,\omega)$ satisfies Darboux, in the
sense that for any point $w$ of $W$, there exists a PL homeomorphism $h$
between some open neighborhood of $w$ in $W$ and an open subset of
$\mathbb{R}^{2n}$, through which $\omega$ corresponds to the standard
symplectic form $\omega_{o}=\sum_{i=1}^{n}dx^{i}\wedge dy^{i}$ on
$\mathbb{R}^{2n}$. Indeed, a PL symplectic manifold which triangulates a
smooth one necessarily satisfies Darboux.

On the other hand, as proven in [D19] and [BD], it is possible to construct,
from a smooth symplectic manifold $(M,\omega)$ endowed with a smooth
triangulation _in general position with respect to the symplectic structure_
(and those exist), piecewise linear symplectic manifolds that
$C^{0}$-approximate $(M,\omega)$ arbitrarily closely. But it is unclear
whether any of those triangulates $\omega$ or even satisfies Darboux. It might
be necessary to invent another procedure to produce PL symplectic structures
that would enforce Darboux from the start.

At some point during our wandering, it became unclear whether even volume
forms could be triangulated. If not, that would bury our hopes for symplectic
forms. We prove in this paper the following theorem.

######  Theorem 0.1.

For a smooth volume form $\Omega$ on a manifold $M$ and a Whitehead
triangulation $h:|K|\to M$, there exists a triangulation
$\overline{h}:|\overline{K}|\to M$, defined on a refinement $\overline{K}$ of
$K$, such that $\overline{h}^{*}\Omega$ is constant on each simplex of
$\overline{K}$, yielding a _triangulation of $(M,\Omega)$_. Moreover, two
triangulations of the same volume form are PL equivalent.

Observe, first of all, that, given a smooth triangulation $h:K\to M$ of $M$,
there is a natural candidate for the role of the triangulating volume form,
namely, the unique piecewise constant $(\dim M)$-form on $K$ attributing to
each simplex of $K$ the same orientation and volume as $h^{*}\Omega$. Denote
it by $\Omega_{1}$ and denote the form $h^{*}\Omega$ by $\Omega_{0}$. If there
exists a piecewise smooth homeomorphism $\varphi:\overline{K}\to\overline{K}$,
defined on a refinement $\overline{K}$ of $K$, that pulls back $\Omega_{0}$ to
$\Omega_{1}$, then $\overline{h}=h\circ\varphi:\overline{K}\to M$ triangulates
$\Omega$.

In the smooth category, the standard argument to prove equivalence of two
cohomologous volume forms on a closed manifold is Moser’s. When trying to
adapt it to the above-described context, one runs into severe difficulties for
the following reason. The equivalence is the time-on map of the vector field
defined by the equation $\iota(X_{t})\Omega_{t}=\alpha$, where
$\Omega_{t}=\Omega_{0}+t(\Omega_{1}-\Omega_{0})$ and $\alpha$ is some
primitive of $\Omega_{0}-\Omega_{1}$. Since the forms $\Omega_{t}$ and
$\alpha$ are Whitney forms, the vector field $X_{t}$ is multi-valued along the
skeleton of $K$ and its flow is therefore not obviously well-defined.
Moreover, some phenomena might occur that render any attempt to define it
doomed.

Another avenue should therefore be explored. The one followed here is to
arrange for the form $\alpha$ to vanish along the skeleton of some subdivision
$\overline{K}$ of $K$, so that the flow of $X_{t}$ is stationary there. It
preserves therefore any simplex and is thus well-defined on $\overline{K}$.
Observe that a piecewise smooth homeomorphism $\varphi:L\to L$ of a simplicial
complex $L$ that restricts to the identity along the skeleton of $L$ satisfies
$\varphi_{*_{x}}=\rm Id$ at all points $x$ in the codimension $2$ skeleton of
$L$. So that in order to be able to find such a primitive $\alpha$, it is
necessary to modify the initial triangulation in order to achieve that
$\Omega_{0}$ and $\Omega_{1}$ coincide along the codimension $2$ skeleton, and
this is the delicate point of the proof. We prove in fact more : for a
suitable choice of $h$, the forms $\Omega_{0}$ and $\Omega_{1}$ agree on a
neighborhood of the entire skeleton of some subdivision $\overline{K}$ of $K$.

The next question is that of uniqueness : given a closed manifold $M$ endowed
with a volume form $\Omega$ and two triangulations $g:|K|\to M$ and $h:|L|\to
M$, does there necessarily exist a PL equivalence $\psi:K\to L$ such that
$\psi^{*}(h^{*}\Omega)=g^{*}\Omega$ ? The answer is yes. In fact any two
piecewise constant volume forms on a same PL manifold that have the same total
volume are PL equivalent.

Let us mention that the main result of the unpublished paper [HP04] implies
uniqueness in a certain particular case. More precisely, the authors show that
two equivalent PL manifolds in $\mathbb{R}^{n}$ obtained by gluing affine
$n$-simplices along their boundaries and equipped with the restriction of
$\mathbb{R}^{n}$’s standard volume form are PL homeomorphic provided they have
the same total volume.

The literature on PL symplectic topology is quite scare. We would like to
mention Bernhard Gratza ’s unpublished thesis [Gra98], which shows that for
any compactly supported Hamiltonian diffeomorphism $\varphi$ of
$(\mathbb{R}^{2n},\omega_{o})$, there exists two triangulations $K_{1}$ and
$K_{2}$ of $\mathbb{R}^{2n}$ and a piecewise linear symplectic diffeomorphism
$\psi:K_{1}\to K_{2}$ that $C^{0}$-approximates $\varphi$ (of course, this
result does not permit to replace a smooth atlas by a PL one). Panov in
[Pan09] studied polyhedral Kähler manifold but from a different viewpoint.
More recently, Jouberteau, Rollin and Tapie have proven in [JRT18] that an
isotropic immersed torus $\mathbb{T}^{k}$ in $(\mathbb{R}^{2n},\omega_{o})$
can be approximated by isotropic piecewise linear maps
$\mathbb{T}^{k}\to\mathbb{R}^{2n}$.

The paper is subdivided into three sections. The first one presents the
definitions and notations adopted in the text. The second one contains the
proof of Theorem 0.1, as well as some of its consequences. It is itself
divided into a first subsection that explains the induction procedure to
construct a triangulation making $\Omega_{0}$ agrees with $\Omega_{1}$ near
the skeleton. A technical lemma in relegated in a second subsection and
corollaries appear in the third subsection. The last section is devoted to
proving uniqueness.

## Acknowledgments

We are particularly grateful to Yakov Eliashberg, who initiated the project to
investigate PL/PS symplectic and contact topology and with whom we have had
numerous very stimulating and interesting conversations. We also thank
François Laudenbach, whom we have consulted on several occasions.

##  1 Preliminaries

The conventions, notations and definitions adopted in the text are described
hereafter.

By a $n$-simplex $\sigma$, we mean a closed affine simplex of dimension $n$ in
some Euclidean space $\mathbb{R}^{N}$, or, equivalently, the convex hull of a
collection of $n+1$ independent points $v_{0},\ldots,v_{n}\in\mathbb{R}^{N}$
(_independent_ means that none is contained in the affine plane generated by
the other ones). We write $\sigma=\langle v_{0},\ldots,v_{n}\rangle$. In the
text, a simplex $\sigma$ is sometimes also thought of as the simplicial
complex that consists $\sigma$ plus all its faces.

We think of a simplicial complex $K$ as being a collection of simplices in
some Euclidean space $\mathbb{R}^{N}$ that satisfies the usual properties : a
face of a simplex in $K$ belongs to $K$ and the intersection of two simplices
$\sigma_{1}$ and $\sigma_{2}$ in $K$ is either empty or consists of a common
face of $\sigma_{1}$ and $\sigma_{2}$. If $K$ is not a finite collection then
we require, in addition, that $K$ be locally finite. For $n\in{\mathbb{N}}{}$,
the collection of all closed $n$-simplices of $K$ is denoted by $K^{(n)}$.

The _star_ of a simplex $\sigma\in K$, denoted by $\rm st(\sigma)$ or $\rm
st_{K}(\sigma)$, is the subcomplex of $K$ consisting of all simplices that
contain $\sigma$, together with their faces. The _link_ of $\sigma$, denoted
by $\rm lk(\sigma)$ or $\rm lk_{K}(\sigma)$, consists in the simplices of $\rm
st(\sigma)$ that do not contain $\sigma$. The _body_ of a simplicial complexe
$L$, denoted by $|L|$, is the union $|L|=\cup_{\sigma\in K}\sigma$. A
_polyhedron_ is defined to be the body of some simplicial complexe. In turn a
simplicial complexe $K$ is a _triangulation_ of the polyhedron $|K|$.

A _subdivision_ or _refinement_ of a simplicial complex $K$ is a simplicial
complex $L$ such that each simplex of $L$ is contained in a simplex of $K$ and
each simplex of $K$ is a union of simplices of $L$. In particular their
polyhedra $|K|$ and $|L|$ are identical. The simplicial complexes $K$ and $L$
are called triangulations of this polyhedron.

A _piecewise linear (PL) embedding_ of a simplicial complex $K$ into some
Euclidean space $\mathbb{R}^{M}$ is a continuous embedding of its body into
$\mathbb{R}^{M}$ whose restriction to each simplex $\sigma$ is an affine map.
Given two simplicial complexes $K$ and $L$, a _PL homeomorphism_ from $K$ to
$L$, denoted $h:K\to L$, is a homeomorphism $h:|K|\to|L|$ whose restriction to
each simplex $\sigma\in K$ is an affine map onto a simplex $h(\sigma)\in L$.
If $K$ and $L$ admit subdivisions $K^{\prime}$ and $L^{\prime}$ and a PL
homeomorphism $h:K^{\prime}\to L^{\prime}$, then $K$ and $L$ are said to be
_PL equivalent_. Similarly a homeomorphism $h:|K|\to|L|$ is said to be
_piecewise smooth (PS)_ if each simplex of $K$ is smoothly embedded onto a
simplex of $L$. A _piecewise smooth equivalence_ is defined accordingly.

A PL manifold of dimension $n$ is a polyhedron $P$ which is locally flat, in
the sense that for some (in fact, any) triangulation $K$ of $P$, the star of
any simplex $\sigma\in K$ admits a PL embedding into $\mathbb{R}^{n}$. Two PL
manifolds $P$ and $P^{\prime}$ are said to be PL (respectively PS) equivalent
if they admit PL (respectively PS) equivalent triangulations. The collection
of PL manifolds with PL equivalences form a category (see [L]).

Given a simplex $\sigma$, its barycenter is denoted by $b$ or $b_{\sigma}$.
The _stellar subdivision_ of a $n$-simplex $\sigma$, denoted by
$\mathcal{S}\sigma$, is the simplicial complex consisting of all proper faces
of $\sigma$, together with the vertex $b_{\sigma}$ and the simplices
$<\tau,b_{\sigma}>$, where $\tau$ runs through the collection of proper faces
of $\sigma$. The stellar subdivision of a PL manifold $K$ of dimension $n$,
denoted by ${\mathcal{S}}{}K$, is the union over all $\sigma\in K^{(n)}$ of
the $\mathcal{S}\sigma$’s. Thus only the top-dimensional faces are subdivided.

A Whitney form of degree $k$ or a PS $k$-form on a simplicial complex $K$ is a
collection ${\overline{\alpha}}=(\alpha_{\sigma})_{\sigma\in K^{(n)}}$, where
$\alpha_{\sigma}$ is a smooth $k$-form on $\sigma$, such that

${\alpha_{\sigma_{1}}}\Bigl{|}_{\sigma_{1}\cap\sigma_{2}}={\alpha_{\sigma_{2}}}\Bigl{|}_{\sigma_{1}\cap\sigma_{2}},$

for all $\sigma_{1},\sigma_{2}\in K^{(n)}$ (see [Sul77] Section 7). Observe
that the defining condition above is empty in case $k=n$. A Whitney $n$-form
is thus simply a collection of smooth $n$-forms on the simplices of dimension
$n$. A Whitney form of degree $k$ on a PL manifold $P$ of dimension $n$ is a
Whitney $k$-form on a simplicial complex $K$ that triangulates $P$. Of course,
if $K^{\prime}$ is a refinement of $K$, the form $\alpha$ induces a Whitney
$k$-form, also denoted by $\alpha$ on $K^{\prime}$.

If $\alpha$ is a smooth $k$-form on a smooth manifold $M$ of dimension $n$
endowed with a Whitehead triangulation $h:|K|\to M$, then the pullback of
$\alpha$ through $h$ is defined to be the Whitney $k$-form
$h^{*}\alpha=(h|_{\sigma}^{*}\alpha)_{\sigma\in K^{(n)}}$.

A _piecewise constant (PC) $k$-form_ on a PL manifold $P$ of dimension $n$ is
a Whitney $k$-form $\alpha=(\alpha_{\sigma})$ on some triangulation $K$ of $P$
such that each $\alpha_{\sigma}$ is a constant form. Notice that the condition
to be constant for a $k$-form defined on a simplex $\sigma=\langle
v_{0},\ldots,v_{n}\rangle$ is of course well-defined.

######  Definition 1.1.

A _piecewise constant_ or _piecewise smooth volume form_ on a PL manifold $P$
of dimension $n$ is a PC or PS $n$-form on $P$ which does not vanish and
induces a well-defined orientation of $P$. Given a smooth volume form $\Omega$
on a smooth manifold $M$, a _triangulation of $\Omega$_ is a Whitehead
triangulation $h:|K|\to M$ ($h|_{\sigma}$ is a smooth embedding for all
$\sigma\in K$) such that $h^{*}\Omega$ is piecewise constant.

Two Whitney forms $\alpha_{1}$ and $\alpha_{2}$ on a simplicial complex $K$
are said to be PS (respectively PL) equivalent when there exists two
subdivisions $K^{\prime}$ and $K^{\prime\prime}$ of $K$ and a PS (respectively
PL) homeomorphism $h:K^{\prime}\to K^{\prime\prime}$ such that
$h^{*}\alpha_{1}=\alpha_{2}$. The notion of equivalence between $k$-forms on a
PL manifold is defined accordingly.

Given an oriented PL manifold $P=|K|$ of dimension $n$ endowed with a PS
$n$-form $\Omega=(\Omega_{\sigma})_{\sigma\in K^{(n)}}$, the integral of
$\Omega$ along $P$ is denoted by $\int_{P}\Omega$ and defined to be the sum

$\sum_{\sigma\in K^{(n)}}\int_{\sigma}\Omega|_{\sigma}.$

The _volume cocycle of $\Omega$_ is the map $V_{\Omega}:\sigma\in
K^{(n)}\to\int_{\sigma}\Omega|_{\sigma}$.

Finally we adopt Misha Gromov’s notation $\mathcal{O}p(A)$ or
$\mathcal{O}p_{X}(A)$ for an unspecified neighborhood of a subset $A$ in a
topological space $X$.

##  2 Triangulations of volume forms

Consider an oriented smooth manifold $M$ of dimension $n\geq 2$ equipped with
a positive volume form $\Omega$. If $h:|K|\to M$ is a smooth triangulation of
$M$, consider the Whitney $n$-form $\Omega_{0}=h^{*}\Omega$ and some piecewise
constant $n$ form $\Omega_{1}$ on $K$. As explained in the introduction, we
would like to select $\Omega_{1}$ and $h$ so that a primitive of
$\Omega_{0}-\Omega_{1}$ may be chosen that vanish along the skeleton. This
ensures that the flow of the time-dependent vector field $X_{t}$ defined by
$i(X_{t})\Omega_{t}=\alpha$, for
$\Omega_{t}=\Omega_{0}+t(\Omega_{1}-\Omega_{1})$, is stationary along the
skeleton and induces therefore a well-defined PS isotopy $(\varphi_{t})$ of
$M$ which will achieve $\varphi_{1}^{*}\Omega_{0}=\Omega_{1}$. (Observe that
it is not sufficient that $\alpha|_{\partial\sigma}=0$ as this latter
condition only implies that $X_{t}$ is tangent to $\partial\sigma$.)

Let us investigate now the precise conditions to be satisfied by $\Omega_{1}$
and $h$.

  1. 1.

In order to be able to apply Moser’s argument to volume forms $\Omega_{0}$ and
$\Omega_{1}$ on an oriented $n$-simplex $\sigma$, it is necessary for them to
be cohomologous, that is, to have the same total volume :
$\int_{\sigma}\Omega_{1}=\int_{\sigma}\Omega_{0}.$ In other words,
$\Omega_{0}$ and $\Omega_{1}$ must have same volume cocycle.

  2. 2.

Suppose that there exists a diffeomorphism $\varphi$ of $\sigma$ which
coincides with the identity on $\partial\sigma$ and such that
$\varphi^{*}({\Omega_{1}}|_{\sigma})={\Omega_{0}}|_{\sigma}$. Then
$\varphi_{*_{x}}={\rm Id}$ at any point $x$ of the codimension-two skeleton
since, at such a point, the tangent space to $\sigma$ is generated by the
tangent spaces to the faces passing through $x$. Therefore, a second necessary
condition is that the forms $\Omega_{0}$ and $\Omega_{1}$ coincide at points
of $\sigma^{(n-2)}$.

Given some PS volume form $\Omega_{0}=h^{*}\Omega$, a piecewise constant form
$\Omega_{1}$ that satisfies the first condition is easily constructed : for
each $n$-simplex $\sigma$ endowed with the orientation induced by
$\Omega_{0}$, let $\Omega_{1}^{\sigma}$ denote the constant $n$-form on
$\sigma$ whose total volume is $\int_{\sigma}\Omega_{0}$. These forms gather
into a piecewise constant volume form
$\Omega_{1}={\mathcal{C}}_{K}(\Omega_{0})=(\Omega_{1}^{\sigma})_{\sigma\in
K^{(n)}}$ on $K$ that has the same volume cocycle as $\Omega_{0}$.

The rest of this chapter consists in constructing a triangulation
$h^{\prime}:|K^{\prime}|\to M$, defined on some refinement $K^{\prime}$ of $K$
for which $h^{\prime*}\Omega$ and $\Omega_{1}$ agree near the skeleton of
$K^{\prime}$. This is done by induction on the dimension of the simplices and
allows us to construct a primitive of $\Omega_{0}-\Omega_{1}$ that vanishes in
a neighborhood of the skeleton. We may then use Moser’s argument, which we
briefly recall now, inside each $n$-simplex .

######  Proposition 2.1.

[Relative Moser] Let $\Omega_{0}$ and $\Omega_{1}$ be smooth volume forms on
an oriented $n$-simplex $\sigma$ that agree near its boundary and have the
same volume. Then there exists a diffeomorphism $\varphi$ of $\sigma$ such
that $\varphi^{*}\Omega_{1}=\Omega_{0}$ and
$\varphi|_{\mathcal{O}p(\partial\sigma)}=\mathrm{Id}$.

###### Proof.

Since the $n$-form $\Omega_{0}-\Omega_{1}$ is compactly supported in $\rm
Int(\sigma$ and has volume zero, it represents the trivial cohomology class in
$H^{n}_{c}({\rm Int}(\sigma);\mathbb{R})$. It admits therefore a compactly
supported primitive $\alpha$. The time-dependent vector field $X_{t}$ defined
by

$\iota(X_{t})(\Omega_{0}+td\alpha)=\alpha$

has compact support. Its flow provides us with an isotopy $\varphi_{t}$ that
is stationary near $\partial\sigma$ and satisfies
$\varphi_{t}^{*}\Omega_{t}=\Omega_{0}$ for each $t\in[0,1]$. ∎

######  Remark 2.2.

If we only assume that $\Omega_{1}$ and $\Omega_{0}$ coincide at points of the
codimension two skeleton, it is also possible to construct a primitive
$\alpha$ of $\Omega_{1}-\Omega_{0}$ which vanishes on $\partial\sigma$. This
result is shown in [BMPR18]. More precisely, the authors prove that if
$\Omega_{0}$ and $\Omega_{1}$ have same total volume on a $n$-simplex $\sigma$
and agree on $\sigma^{(n-2)}$ then a primitive that vanishes along
$\partial\sigma$ can be constructed. Of course this hypothesis has no reason
to be satisfied here and the main body of our argument consists in showing
that $\Omega_{0}$ is PS equivalent to a volume form $\Omega^{\prime}_{0}$ that
coincides with $\Omega_{1}$ along the codimension-two skeleton. We prove more
: these forms can be made to agree on a neighborhood of the skeleton. Their
difference admits therefore a primitive vanishing there (for trivial reasons).

###  2.1 Agreement near the skeleton

We are given a smooth volume form $\Omega$ on a manifold $M$ endowed with a
triangulation $h:|K|\to M$. We consider the PS form $\Omega_{0}=h^{*}\Omega$
and the PC form $\Omega_{1}={\mathcal{C}}_{K}(\Omega_{0})$ on $K$. The purpose
of this section is to replace the triangulation $h:|K|\to M$ by
$\widetilde{h}$ of the type $\psi\circ h\circ\varphi$, where $\varphi$ is a PS
homeomorphism of some refinement ${K^{\prime}}$ of $K$ and $\psi$ is a
diffeomorphism of $M$, such that $\widetilde{h}^{*}\Omega$ coincides with
$\Omega_{1}$ near the skeleton of the new simplicial complex ${K^{\prime}}$.
We first handle the $0$-skeleton $K^{(0)}$ and then treat successively the
higher-dimensional skeleta by means of parametric versions of the $0$-skeleta
case. Obtaining such a parametric versions is the subtle part of the argument.

####  2.1.1 First step

Fix a $n$-simplex $\sigma$. The first step is to deform $\Omega_{0}$ via a
piecewise smooth homeomorphism of $\sigma$ to make it equal to $\Omega_{1}$
near $|\sigma^{(0)}|$. The idea is to consider the stellar subdivision
$\mathcal{S}\sigma$ of $\sigma$ and to construct a piecewise smooth
homeomorphism of $\mathcal{S}\sigma$ which fixes $\partial\sigma$ and moves
the points in a neighborhood of $v\in\sigma^{(0)}$ in the directions of the
edge $<v,b>$. As the property that the volumes of $\Omega_{0}$ and
$\Omega_{1}$ agree on the $n$-simplices has no reason to be true on
$\mathcal{S}\sigma$, we first attend to that and, at the same time, take care
of the new vertex $b_{\sigma}$.

######  Lemma 2.3.

Let $\Omega_{0}$ and $\Omega_{1}$ be smooth volume forms on a linear simplex
$\sigma$ with identical total volume and let $\mathcal{S}\sigma$ be the
stellar subdivision of $\sigma$. There exists a smooth volume form
$\overline{\Omega}$ on $\sigma$ and a diffeomorphism $\varphi$ of $\sigma$
such that

  * •

$\varphi^{*}\Omega_{0}=\overline{\Omega}$ and $\varphi$ fixes
$\mathcal{O}p(\partial\sigma)$,

  * •

$\int_{\tau}\overline{\Omega}=\int_{\tau}\Omega_{1}$ for all
$\tau\in\mathcal{S}\sigma^{(n)}$,

  * •

$\overline{\Omega}=\Omega_{1}$ near $b_{\sigma}$.

###### Proof.

Define $\overline{\Omega}=f\Omega_{0}$, where $f$ is a smooth positive
function on $\sigma$ which is identically equal to one near $\partial\sigma$,
and such that $f\Omega_{0}$ and $\Omega_{1}$ have the same volume cocycle on
$\mathcal{S}\sigma$ and agree near the new vertex $b_{\sigma}$. Now observe
that $\overline{\Omega}$ and $\Omega_{0}$ satisfy the assumptions of
Proposition 2.1, so that there exists a diffeomorphism $\varphi$ of $\sigma$
which coincides with the identity near $\partial\sigma$ and for which
$\varphi^{*}\Omega_{0}=\overline{\Omega}$. ∎

The next step is a deformation of $\sigma$ which makes $\overline{\Omega}$
equal to $\Omega_{1}$ near the vertices of $\sigma$. Notice that the proof of
the following lemma uses a technical result which is shown later, in Section
2.2.

######  Lemma 2.4.

Consider smooth volume forms $\overline{\Omega}$ and $\Omega_{1}$ on a linear
simplex $\sigma$ inducing the same orientation. Then there exists a PS
homeomorphism $\rho$ of $\mathcal{S}{\sigma}$ such that

  * •

$\rho$ is the identity on $\sigma\backslash\mathcal{O}p(|\sigma^{(0)}|)$,

  * •

$\rho$ preserves $|\mathcal{S}\sigma^{(n-1)}|$ and fixes $\partial\sigma$,

  * •

$\rho^{*}\overline{\Omega}=\Omega_{1}$ on $\mathcal{O}p(|\sigma^{(0)}|)$,

  * •

If $\overline{\Omega}$ and $\Omega_{1}$ coincide near a vertex $v$, then
$\rho$ is the identity near $v$.

  * •

The forms $\overline{\Omega}$ and $\rho^{*}\overline{\Omega}$ have identical
volume cocycles on $\mathcal{S}{\sigma}$.

If, in addition, $\overline{\Omega}$ and $\Omega_{1}$ depend smoothly on a
parameter $\lambda$ running in some smooth manifold $\Lambda$, then so does
$\rho$.

###### Proof.

Fix a vertex $v\in\sigma^{(0)}$ and denote by $b$ the barycenter of $\sigma$.
Let $\partial_{v}\sigma$ denote the boundary of $\sigma$ minus the interior of
the $(n-1)$-face opposite to $v$, and let $\pi$ be the projection of $\sigma$
onto $\partial_{v}\sigma$, parallel to the edge $<b,v>$. Observe that any
point $x$ in $\sigma$ can be written as

$x=\pi(x)+t(x)(b-v)$

for a certain $t(x)\in\mathbb{R}^{+}$. Now let $F$ be the smooth positive
function defined by the relation $\overline{\Omega}=F\Omega_{1}$ and define a
piecewise smooth map $\varphi_{F}$ on $\sigma$ (smooth on each
$\tau\in{\mathcal{S}}{}\sigma$), with values in the affine plane spanned by
$\sigma$, by

$\varphi_{F}(x)=\pi(x)+\left(\int_{0}^{t(x)}F(\pi(x)+s(b-v))ds\right)(b-v).$

Clearly, $\varphi_{F}$ coincides with the identity on $\partial_{v}\sigma$. If
$\tau$ is a $(n-1)$-face of $\sigma$ which contains $v$, we can choose linear
coordinates $x_{1},\ldots,x_{n-1}$ in $\tau$ and complete them with $t$ to
obtain coordinates on $\sigma\cap\pi^{-1}(\tau)$. Then

$\varphi_{F}(x_{1},\ldots,x_{n-1},t)=\left(x_{1},\ldots,x_{n-1},\int_{0}^{t}F(x_{1},\ldots,x_{n-1},s)ds\right)$

on $\sigma\cap\pi^{-1}(\tau)$. Whence $|\mathrm{Jac}(\varphi_{F})|=F$ and
$\varphi_{F}^{*}\Omega_{1}=\overline{\Omega}$, on all of $\sigma$.

The idea is now to modify $\varphi_{F}$ away from $v$ in such a way that it
becomes the identity out of an arbitrary small neighborhood of $v$. In the
affine plane spanned by $\sigma$, consider a ball $B$ centered at $v$ that
does not contain $b$. Choose also a non-negative function $\chi$ on $\sigma$
that vanishes near $v$, equals $1$ outside $\frac{1}{2}B$ (the ball centered
at $v$ of half the radius) and that is non-decreasing along the radial
directions. Define $\overline{F}=(1-\chi)F+\chi$ and consider the map
$\varphi_{\overline{F}}$. Since $\overline{F}$ equals $1$ out of
$\frac{1}{2}B$, any point $x$ such that $\pi(x)\notin\frac{1}{2}B$ satisfies
$\varphi_{\overline{F}}(x)=x$. Altogether $\varphi_{\overline{F}}$ satisfies
the assumptions of Lemma 2.10 below, and there exists thus a piecewise smooth
homeomorphism $\overline{\varphi}_{\overline{F}}$ of ${\mathcal{S}}{}\sigma$
which coincides with $\varphi_{\overline{F}}$ near $v$ and with the identity
near $\partial B$. The same procedure can be repeated near each vertex and
this completes the proof.

Moreover, the procedure just described, being explicite and relying on Lemma
2.10, provides us with a smooth dependence of $\rho$ on a parameter $\lambda$
whenever the initial forms $\overline{\Omega}$ and $\Omega_{1}$ depend
smoothly on $\lambda$. ∎

Thus, given a smooth volume form $\Omega$ on a manifold $M$ endowed with a
triangulation $h:|K|\to M$ and the associated forms $\Omega_{0}=h^{*}\Omega$
and $\Omega_{1}={\mathcal{C}}_{K}(\Omega_{0})$, Lemma 2.3 and Lemma 2.4
provide, for each $n$-simplex $\sigma$, piecewise smooth homeomorphisms
$\varphi_{\sigma}$ and $\rho_{\sigma}$ of ${\mathcal{S}}{}\sigma$ which, among
may other properties, are the identity on $\partial\sigma$. When $\sigma$ runs
through $K^{(n)}$, the maps $\varphi_{\sigma}$ and $\rho_{\sigma}$ form thus
global piecewise smooth homeomorphisms of ${\mathcal{S}}{}K$, denoted
respectively by $\varphi$ and $\rho$. The associated triangulation
$\tilde{h}=h\circ\varphi\circ\rho:|\mathcal{S}K|\to M$ satisfies the relation

$\tilde{h}^{*}\Omega=\rho^{*}(\varphi^{*}(h^{*}\Omega))=\rho^{*}(\varphi^{*}\Omega_{0})=\rho^{*}\overline{\Omega}.$

The latter form coincides with the piecewise constant form $\Omega_{1}$ near
the vertices of $\mathcal{S}K$ and admits the same volumes cocycle as
$\Omega_{1}$ on $\mathcal{S}K$. We have thus proven the following result.

######  Proposition 2.5.

Given a smooth volume form $\Omega$ on a manifold $M$, there exists a
triangulation $h:|K|\to M$ such that $h^{*}\Omega$ and
$\Omega_{1}={\mathcal{C}}_{K}(\Omega_{0})$ have the same volume cocycle on $K$
and coincide near the vertices of $K$.

####  2.1.2 Induction step

We will henceforth assume that $h:|K|\to M$ and $\Omega_{1}$ satisfy the
conclusion of Proposition 2.5. We fix again a $n$-simplex $\sigma$. The goal
now is to extend the area where $\Omega_{1}$ and $h^{*}\Omega$ agree from
neighborhood of the vertices to neighborhoods of $|\sigma^{(1)}|$,
$|\sigma^{(2)}|$, $\ldots$, $|\sigma^{(n)}|$ successively. The procedure is
the same as for the $0$-skeleton : assuming that $\Omega_{0}=\Omega_{1}$ on
$\mathcal{O}p(|\sigma^{(k-1)}|)$, we first show that $\Omega_{0}$ can be
replaced by a form which admits the same volume cocycle as $\Omega_{1}$ on
$\mathcal{S}\sigma$, and which coincides with $\Omega_{1}$ on a neighborhood
of the new $k$-simplices (those in
$\mathcal{S}\sigma^{(k)}\setminus\sigma^{(k)}$). We then handle the
$k$-simplices of $\sigma$.

######  Lemma 2.6.

Let $\Omega_{0}$ and $\Omega_{1}$ be smooth volume forms of equal total volume
on a linear simplex $\sigma$. Assume moreover that $\Omega_{1}=\Omega_{0}$ on
a neighborhood of $|\sigma^{(k-1)}|$ for a certain $k\in\\{1,\ldots,n\\}$.
Then there exists a smooth volume form $\overline{\Omega}$ and a
diffeomorphism $\varphi$ of $\sigma$ such that

  * •

$\varphi^{*}\Omega_{0}=\overline{\Omega}$ and $\varphi$ fixes
$\mathcal{O}p(\partial\sigma)$,

  * •

$\int_{\tau}\overline{\Omega}=\int_{\tau}\Omega_{1}$ for any
$\tau\in\mathcal{S}\sigma^{(n)}$,

  * •

$\overline{\Omega}=\Omega_{1}$ on a neighborhood of
$|\mathcal{S}\sigma^{(k)}\backslash\sigma^{(k)}|$.

###### Proof.

The proof is exactly the same as that of Lemma 2.3. The additional assumption
that $\Omega_{1}=\Omega_{0}$ near $|\sigma^{(k-1)}|$ allows us to choose a
function $f$ that achieves the value one near $\partial\sigma$ and satisfies
$f\Omega_{0}=\Omega_{1}$ near
$|\mathcal{S}\sigma^{(k)}\backslash\sigma^{(k)}|$, in addition to the
condition that $\int_{\tau}f\Omega_{0}=\int_{\tau}\Omega_{1}$ for each
$\tau\in\mathcal{S}\sigma^{(n)}$. ∎

Now fix a $k$-simplex $\tau$ of $\sigma$ and assume that
$\overline{\Omega}=\Omega_{1}$ near $\partial\tau$. The idea, to extend the
equality to a neighborhood of $\tau$, is to consider a collection of parallel
$(n-k)$-planes along $\tau$, transverse to $\tau$, and to apply Lemma 2.4 in
each of them.

######  Lemma 2.7.

Consider smooth volume forms $\overline{\Omega}$ and $\Omega_{1}$ on a linear
simplex $\sigma$ such that $\overline{\Omega}=\Omega_{1}$ on
$\mathcal{O}p(|\sigma^{(k-1)}|)$ for a certain $k\in\\{1,\ldots,n\\}$. Then
there exists a piecewise smooth homeomorphism $\rho$ of $\mathcal{S}{\sigma}$
such that

  * •

$\rho$ is the identity on $\sigma\backslash\mathcal{O}p(|\sigma^{(k)}|)$ and
fixes $\partial\sigma$,

  * •

$\rho^{*}\overline{\Omega}=\Omega_{1}$ on $\mathcal{O}p(|\sigma^{(k)}|)$,

  * •

$\rho^{*}\overline{\Omega}$ and $\Omega_{1}$ have the same volume cocycle on
$\mathcal{S}{\sigma}$.

###### Proof.

Let $v_{0},\ldots,v_{n}$ denote the vertices of $\sigma$. Consider the
$k$-face $\tau$ of $\sigma$ generated by the vertices
$v_{i_{0}},\ldots,v_{i_{k}}$, where $0\leq i_{0}<\ldots<i_{k}\leq n$. On a
neighborhood $U$ of $\tau$ in $\sigma$, define the constant vector fields

$X_{l}=v_{l}-v_{i_{0}}$

for any $l\neq i_{0}$. Observe that $X_{l}$ is tangent to $\tau$ if and only
if $v_{l}$ is a vertex of $\tau$. For any $x\in\mathrm{int}(\tau)$, consider
the $(n-k)$-dimensional affine plane $\mu_{x}$ passing through $x$ and
generated by the $X_{l}$’s with $l\notin\\{i_{1},\ldots,i_{k}\\}$. If $b$
denotes the barycenter of $\sigma$, let $\nu$ be the $(k+1)$-simplex $\nu$ of
$\mathcal{S}\sigma$ generated by $\tau$ and $b$, and notice that the
intersection $\nu\cap\mu_{x}$ is a segment for any $x\in\mathrm{int}(\tau)$.
Now consider the $(n-k)$-forms

$\displaystyle\overline{\omega}$
$\displaystyle=\iota(X_{i_{1}},\ldots,X_{i_{k}})\;\overline{\Omega},$
$\displaystyle\omega_{1}$
$\displaystyle=\iota(X_{i_{1}},\ldots,X_{i_{k}})\;\Omega_{1}.$

Notice that their restrictions to the simplices $\sigma\cap\mu_{x}$,
$x\in\mathrm{int}(\tau)$, are volume forms. Now, for each
$x\in\mathrm{int}(\tau)$, we apply Lemma 2.4 to obtain a PS homeomorphism
$\rho_{x}$ of $\sigma\cap\mu_{x}$ which is the identity outside of $U$, moves
points in the direction of the segment $\nu\cap\mu_{x}$, fixes
$\partial\sigma\cap\mu_{x}$ and satisfies
$(\rho_{x})^{*}\overline{\omega}=\omega_{1}$ in
$\mathcal{O}p(\tau)\cap\mu_{x}$. We may also assume that the family of maps
$\rho_{x}$ depends smoothly on $x\in{\rm Int}(\tau)$ and coincides with the
identity map for $x$ near $\partial\tau$.

Notice that although we may not strictly speaking directly apply Lemma 2.4
because the simplices $\sigma\cap\mu_{x}$ vary, we could reparameterize them
affinely by a standard simplex $\Delta_{0}$, work there and push-forward the
resulting maps to $\sigma\cap\mu_{x}$.

This procedure provides us with a smooth map $\rho_{\tau}:\sigma\to\sigma$
which fixes $\tau$ and $\partial\sigma$, preserves each simplex in
$\mathcal{S}\sigma$, is the identity outside $U$ and satisfies
$\rho_{\tau}^{*}\overline{\Omega}=\Omega_{1}$ near $\tau$. Finally, the
various $\rho_{\tau}$ for $\tau\in\sigma^{(k)}$ may be glued into a PS
homeomorphism $\rho$ of $\mathcal{S}\sigma$ that satisfies the desired
properties.

$v_{3}$

$v_{2}$

$v_{1}$

$v_{0}$

$b$

Figure 1: Stellar subdivision of a 3-simplex and the simplex $\nu$.

∎

Notice that, for $k=n-1$, it is not necessary to subdivide $\sigma$. Once
again, the piecewise smooth homeomorphisms resulting from Lemmas 2.6 and 2.7
fix the boundaries of the $n$-simplices and thus glue into global piecewise
smooth homeomorphisms of $K$. The following proposition summarizes what has
been done so far.

######  Proposition 2.8.

let $h:|K|\to M$ be a smooth triangulation. Consider a smooth volume form
$\Omega$ on $M$, and a piecewise smooth volume form $\Omega_{1}$ on $K$ such
that $h^{*}\Omega$ and $\Omega_{1}$ have the same volume cocycle. Assume that
$h^{*}\Omega$ coincides with $\Omega_{1}$ on a neighborhood of $|K^{(k-1)}|$
for a certain $1\leq k\leq n$. Then there exists a smooth triangulation
$\overline{h}:|\mathcal{S}K|\to M$ for which $\overline{h}^{*}\Omega$ and
$\Omega_{1}$ have the same volume cocycle on $\mathcal{S}K$ and coincide on a
neighborhood of $|\mathcal{S}K^{(k)}|$.

###  2.2 Interpolation

In this section we prove Lemma 2.10 needed in the proof of Lemma 2.4. The
fundamental ingredient is an explicit procedure to build from a smooth
increasing function $\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ with
$\varphi(0)=0$, a smooth increasing interpolation $\overline{\varphi}$ between
$\varphi$ near the origin and the identity near $R$. This result is of course
far from surprising but the point is the explicit character of the
construction which yields a parametric version of the interpolation process.

To build such an interpolation, we use the convolution product with
regularizing functions. Consider the smooth even function

$\rho_{0}:\mathbb{R}\to\mathbb{R}^{+}:x\mapsto\rho_{0}(x)=\left\\{\begin{array}[]{ll}e^{\left(-\frac{1}{1-x^{2}}\right)}&\mathrm{if}~{}~{}|x|<1\\\
0&\mathrm{if}~{}~{}|x|\geq 1.\end{array}\right.$

Let us denote its integral by $A$ and define the functions $\rho$ and
$\rho_{\delta}$, for $\delta>0$, by $\rho=\frac{1}{A}\rho_{0}$ and
$\rho_{\delta}(x)=\frac{1}{\delta}\rho(\frac{x}{\delta})$ respectively. Then
$\int\rho_{\delta}=1$ for all $\delta$.

As is well known, the convolution product $f*\rho_{\delta}$ of a continuous
function $f:\mathbb{R}\to\mathbb{R}$ and $\rho_{\delta}$ is a smooth function
that $C^{0}$ converges to $f$ when $\delta$ tends to $0$. The following basic
properties will be useful later on :

  * •

If $f$ is an increasing function, then $f*\rho_{\delta}$ is also increasing.
Indeed, observe that

$\displaystyle(f*\rho_{\delta})^{\prime}(x)$
$\displaystyle=\int_{-\delta}^{\delta}\rho^{\prime}_{\delta}(y)f(x-y)dy$
$\displaystyle=\int_{0}^{\delta}\rho^{\prime}_{\delta}(y)\left(f(x-y)-f(x+y)\right)dy$

which is non-negative since both $\rho^{\prime}_{\delta}(y)$ and
$f(x-y)-f(x+y)$ are non-positive for any $y\geq 0$.

  * •

If $f$ is a continuous piecewise linear function of the following simple type
:

$f(x)=\left\\{\begin{array}[]{ll}ax+b&\mathrm{if}~{}~{}x<\epsilon\\\
cx+d&\mathrm{if}~{}~{}x\geq\epsilon\end{array}\right.$

for some $a,b,c,d,\epsilon\in\mathbb{R}$ with $a\epsilon+b=c\epsilon+d$, then
$(f*\rho_{\delta})(x)=f(x)$ if $|x-\epsilon|\geq\delta$. Indeed, if
$x\geq\epsilon+\delta$, then $x-y\geq\epsilon$ for any $y\in[-\delta,\delta]$,
so that

$\displaystyle(f*\rho_{\delta})(x)$
$\displaystyle=\int_{-\delta}^{\delta}f(x-y)\rho_{\delta}(y)dy$
$\displaystyle=\int_{-\delta}^{\delta}\bigl{(}c(x-y)+d\bigr{)}\rho_{\delta}(y)dy$
$\displaystyle=(cx+d)\int_{-\delta}^{\delta}\rho_{\delta}(y)dy-c\int_{-\delta}^{\delta}y\rho_{\delta}(y)dy$
$\displaystyle=cx+d,$

since $y\rho_{\delta}(y)$ is an odd function. The case $x\leq\epsilon-\delta$
is close to identical.

######  Lemma 2.9.

Consider a smooth non-decreasing map $\varphi:[0,R]\to\mathbb{R}^{+}$ such
that $\varphi(0)=0$. Then there exists $r>0$ and a smooth non-decreasing map
$\overline{\varphi}:[0,R]\to\mathbb{R}^{+}$ such that

  * •

$\overline{\varphi}$ coincides with $\varphi$ on $[0,r]$,

  * •

$\overline{\varphi}(x)=x$ for any $x\in[R-r,R]$,

Moreover, if $\varphi=\mathrm{Id}$, then $\overline{\varphi}=\mathrm{Id}$.
Furthermore, if $\lambda\to R_{\lambda}$ is a smooth map defined on some
manifold $\Lambda$ and if $\varphi_{\lambda}:[0,R_{\lambda}]\to\mathbb{R}^{+}$
is a smooth family of maps enjoying the same properties as $\varphi$, then
$r=r(\lambda)$ and $\overline{\varphi}_{\lambda}$ depend smoothly on $\lambda$
as well.

$\varepsilon$

$R$

$x$

$y$

$\mathrm{Id}$

$\varphi(x)$

###### Proof.

Consider the graph of $\varphi$ in $\mathbb{R}^{2}$ and fix $\epsilon>0$ such
that the point $(\epsilon,\varphi(\epsilon))$ is at distance $\frac{R}{4}$
from the origin. Of course, the number $\epsilon$ depends smoothly on $R$.
Define $g:[\epsilon,\frac{3R}{4}]\to\mathbb{R}$ to be the affine map whose
graph is the segment joining $(\epsilon,\varphi(\epsilon))$ to
$(\frac{3R}{4},\frac{3R}{4})$, and let $h$ be the piecewise smooth function
given by

$h(x)=\left\\{\begin{array}[]{ll}\varphi(x)&\mathrm{if}~{}~{}x\in[0,\epsilon],\\\
g(x)&\mathrm{if}~{}~{}x\in]\epsilon,\frac{3R}{4}[,\\\
x&\mathrm{if}~{}~{}x\in[\frac{3R}{4},R].\end{array}\right.$

$\varepsilon$

$R$

$x$

$h(x)$

$y$

Figure 2: The piecewise smooth map $h$.

This map provides a non-decreasing interpolation between $\varphi$ and the
identity, but it is smooth neither at $x=\epsilon$ nor at $x=\frac{3R}{4}$
(cf. Figure 2). Now fix $\delta=\frac{\epsilon}{2}$ and consider the
regularizing function $\rho_{\delta}$ defined previously. If
$x\in[\frac{R}{2},R]$, define $\overline{\varphi}(x)=(h*\rho_{\delta})(x)$.
Since $h$ is piecewise linear and non-decreasing on $[\frac{R}{2},R]$, the
function $\overline{\varphi}$ is a smooth monotone map which coincides with
$h$ outside the interval $(\frac{3R}{4}-\delta,\frac{3R}{4}+\delta)$. On the
interval $[0,\frac{R}{2})$, the idea is also to replace $h$ by the convolution
product $h*\rho_{\delta}$, but we have to modify the convolution slightly so
that $h$ and $h*\rho_{\delta}$ can be smoothly glued. Let $\chi$ be a smooth
non-decreasing function such that $\chi(x)=0$ on $[0,\frac{\epsilon}{3}]$,
$\chi(x)=1$ if $x\geq\frac{2\epsilon}{3}$ and which depends smoothly on
$\epsilon$. If $x\in[0,\frac{R}{2}]$, define

$\overline{\varphi}(x)=\int_{-\delta}^{\delta}h\bigl{(}x-\chi(x)y\bigr{)}\rho_{\delta}(y)dy.$

Then $\overline{\varphi}(x)$ is the convolution product of $h$ and
$\rho_{\delta}$ for $x\geq\frac{2\epsilon}{3}$. Moreover
$\overline{\varphi}(x)=h(x)$ when $x\in[0,\frac{\epsilon}{3}]$. Observe
finally that $\overline{\varphi}$ is increasing by a similar argument as for
the convolution product. Now if $\varphi=\mathrm{Id}$, then $h=\mathrm{Id}$ by
construction and, therefore $\overline{\varphi}(x)=\mathrm{Id}$. ∎

Now, let $\sigma$ be a linear $n$-simplex of $\mathbb{R}^{n}$ and $v$ a vertex
of $\sigma$. Consider the stellar subdivision $\mathcal{S}\sigma$ of $\sigma$
and take a closed ball $B$ in $\mathbb{R}^{n}$ centered at $v$ that does not
contain the barycenter $b$ of $\sigma$. Denote by
$\pi:B\cap\sigma\to\partial\sigma$ the projection onto $B\cap\partial\sigma$
parallel to $b-v$.

######  Lemma 2.10.

Assume that $\varphi:B\cap\sigma\to\varphi(B\cap\sigma)\subset\sigma$ is a
homeomorphism which is smooth on the traces in $B$ of the simplices of
$\mathcal{S}\sigma$ and enjoys the following properties :

  * •

$\varphi$ fixes the points of $B\cap\partial\sigma$,

  * •

$\varphi$ is parallel to the edge $b-v$, in the sense that
$\pi(\varphi(x))=\pi(x)$ for any $x\in B\cap\sigma$,

  * •

$\varphi$ is the identity near $\partial B\cap\partial\sigma$.

Then there exists a PS homeomorphism $\overline{\varphi}$ of
$\mathcal{S}\sigma$ which coincides with $\varphi$ near $v$ and with the
identity outside $B\cap\sigma$. If $\varphi$ depends smoothly on an additional
parameter $\lambda$ that varies in a smooth manifold $\Lambda$, then so does
$\overline{\varphi}$.

###### Proof.

Observe that $\varphi$ can be seen as a parametric family of maps

$\varphi_{\lambda}:[0,R_{\lambda}]\to\mathbb{R}^{+},$

with $\varphi_{\lambda}(0)=0$, where the parameter $\lambda$ varies in
$B\cap\partial\sigma$ and $R_{\lambda}$ denotes the length of the segment
$\pi^{-1}(\lambda)\cap B$. Note that

$B\cap\partial\sigma=\bigcup_{\tau\in\sigma^{(n-1)}}B\cap\tau,$

and $\varphi_{\lambda}$ depends smoothly on $\lambda\in B\cap\tau$, for any
$\tau\in\sigma^{(n-1)}$. It follows from Lemma 2.9 that there exists a family
of diffeomorphisms $\overline{\varphi}_{\lambda}$ which coincide with
$\varphi_{\lambda}$ near 0, with the identity near $R_{\lambda}$, and depends
smoothly on $\lambda$ in $B\cap\tau$ for any $\tau\in\sigma^{(n-1)}$. When
$\lambda$ approaches $\partial B\cap\partial\sigma$, the maps
$\varphi_{\lambda}$ and $\overline{\varphi}_{\lambda}$ are the identity.
Consequently, the piecewise smooth family
$\\{\overline{\varphi}_{\lambda}\\}_{\lambda\in B\cap\partial\sigma}$
corresponds to a homeomorphism $\overline{\varphi}$ of $B\cap\sigma$, smooth
on the traces of the simplices of $\mathcal{S}\sigma$ in $B$ and which
satisfies the other desired properties. ∎

###  2.3 Several Consequences

Let us now describe a few consequences of the previous results.

######  Proposition 2.11.

Consider a smooth manifold $M$ equipped with a smooth volume form $\Omega$.
Let $h:|K|\to M$ be a smooth triangulation and let
$\Omega_{1}={\mathcal{C}}_{K}(h^{*}\Omega)$. Then there exists a subdivision
$\overline{K}$ of $K$ and a triangulation $\overline{h}:|\overline{K}|\to M$
such that $\overline{h}^{*}\Omega=\Omega_{1}$.

###### Proof.

By Proposition 2.5 and Proposition 2.8 applied successively for $k=1,...,n-1$
, there exists a subdivision $\overline{K}$ of $K$ (the result of $n+1$
successive stellar subdivisions) and a smooth triangulation
$h_{o}:|\overline{K}|\to M$ for which $h_{o}^{*}\Omega$ and $\Omega_{1}$ have
the same volume cocycle on $\overline{K}$ and coincide on a neighborhood of
$|\overline{K}^{(n-1)}|$. Now Proposition 2.1 can be applied in each
$n$-simplex of $\overline{K}$ to provide us with a piecewise smooth
homeomorphism $\varphi$ of $\overline{K}$ supported away from the skeleton of
$\overline{K}$ and such that $\varphi^{*}(h_{o}^{*}\Omega)=\Omega_{1}$. The
triangulation $\overline{h}=h_{o}\circ\varphi:|\overline{K}|\to M$ satisfies
$\overline{h}^{*}\Omega=\Omega_{1}$. ∎

Consider two volume forms on a closed manifold $M$, one is smooth and the
other one is piecewise smooth with respect to some triangulation of $M$.
Assume only that they have the same total volume. Then it is possible to
construct an equivalence between them.

######  Corollary 2.12 (Moser PS – $C^{\infty}$).

Let $(M,\Omega)$ be a closed manifold equipped with a smooth volume form.
Consider a triangulation $h:|K|\to M$ and a PS volume form $\Omega_{1}$ on
$K$. Assume that

$\int_{M}\Omega=\int_{K}\Omega_{1}.$

Then there exists a subdivision $\overline{K}$ of $K$ and a triangulation
$\overline{h}:|\overline{K}|\to M$ such that
$\overline{h}^{*}\Omega=\Omega_{1}$.

###### Proof.

Observe first that we can choose a positive smooth function $f$ on $M$ such
that

$\int_{h(\sigma)}f\Omega=\int_{\sigma}\Omega_{1}$

for all $\sigma\in K^{(n)}$. The smooth volume forms $\Omega$ and $f\Omega$
lie in the same cohomology class since they have the same total volume. Thus
the standard Moser’s argument implies that there exists a diffeomorphism
$\rho$ of $M$ such that $\rho^{*}\Omega=f\Omega$. Now the volume cocycles of
$f\Omega$ and $\Omega_{1}$ coincide so that Proposition 2.11 yields a
subdivision $\overline{K}$ of $K$ and a triangulation $h_{o}:|\overline{K}|\to
M$ such that $h_{o}^{*}(f\Omega)=\Omega_{1}$. The composition
$\overline{h}=\rho\circ h_{o}$ is a triangulation of $M$ for which

$\overline{h}^{*}\Omega=h_{o}^{*}(\rho^{*}\Omega)=h_{o}^{*}(f\Omega)=\Omega_{1}.$

∎

If $(M,\Omega)$ admits another triangulation $g:|L|\to M$, with the property
that $L$ is equipped with a PS form $\Omega_{2}$ which has the same total
volume on $L$ as $g^{*}\Omega$, Corollary 2.12 says that there is a
subdivision $\overline{L}$ of $L$ and a triangulation
$\overline{g}:|\overline{L}|\to M$ such that
$\overline{g}^{*}\Omega=\Omega_{2}$. It is therefore tempting to conclude that
$(\overline{g})^{-1}\circ\overline{h}$ gives a PS equivalence between
$\Omega_{1}$ and $\Omega_{2}$, but this is not necessarily the case since the
composition of PS maps is not well defined as a piecewise smooth map between
$|\overline{K}|$ and $|\overline{L}|$. This problem is similar to the problem
of uniqueness of triangulations and can perhaps be solved with a similar
procedure.

######  Corollary 2.13 (Smoothing of a piecewise constant volume form).

Let $K$ be a closed PL manifold equipped with a piecewise constant volume form
$\Omega_{K}$. Assume that $K$ admits a smoothing, that is, there exists a
smooth manifold $M$ and a triangulation $h:|K|\to M$. Then there exists a
smooth volume form $\Omega$ on $M$, a subdivision $\overline{K}$ of $K$ and a
triangulation $\overline{h}:|\overline{K}|\to M$ such that
$\overline{h}^{*}\Omega=\Omega_{K}$.

###### Proof.

Consider a smooth volume form $\Omega$ on $M$ such that $(M,\Omega)$ and
$(K,\Omega_{1})$ have the same total volume (notice that $M$ is necessarily
orientable). We may even assume that $\Omega$ and $\Omega_{1}$ have the same
volume cocycle and therefore invoke Proposition 2.11 to conclude. ∎

##  3 Uniqueness for triangulations of volume forms

Let us assume that we have two triangulations $g:|K|\to M$ and $h:|L|\to M$ of
a smooth volume form $\Omega$ on a $n$-dimensional manifold $M$. We know by
Whitehead’s Theorem that there exists a PL equivalence
$\varphi:|{K}|\to|{L}|$. Of course $\varphi$ does not necessarily satisfy
$\varphi^{*}\Omega_{L}=\Omega_{K}$, for $\Omega_{K}=g^{*}\Omega$ and
$\Omega_{L}=h^{*}\Omega$. (The map $\varphi$ is, in fact, a composition
$(g^{\prime})^{-1}\circ h^{\prime}$ for maps $g^{\prime}$ and $h^{\prime}$
approximating $g$ and $h$ respectively). Nevertheless $\varphi^{*}\Omega_{L}$
is another piecewise constant volume form on $K$ and its total volume agrees
with that of $\Omega_{K}$. We explain below how to construct a PL
homeomorphism $\psi:K_{1}\to K_{2}$ between two subdivisions of $K$ for which
$\psi^{*}(\varphi^{*}\Omega_{L})=\Omega_{K}$. This can be considered as a PL
version of Moser’s argument for volume forms.

######  Theorem 3.1.

Any two PC volume forms $\Omega_{1}$ and $\Omega_{2}$ on a closed oriented PL
manifold $P$, with identical total volume, are PL equivalent.

Here is the idea of the proof. Let $n$ denote the dimension of $K$. Let us
also consider the difference of volume cocycles

${\mathcal{D}}{}_{\Omega_{1},\Omega_{2}}={\mathcal{D}}{}:K^{(n)}\to\mathbb{R}:\sigma\mapsto\int_{\sigma}(\Omega_{2}-\Omega_{1})=V_{\Omega_{2}}(\sigma)-V_{\Omega_{1}}(\sigma).$

The idea is to remove the "excess of volume" from simplices $\sigma$ for which
${\mathcal{D}}{}(\sigma)>0$ by transferring it into simplices with a "deficit
of volume" (a negative ${\mathcal{D}}{}$). Here _transferring_ really means
pulling-back $\Omega_{2}$ via a PL homeomorphism between two (perhaps
distinct) subdivisions of $K$. This is done by choosing a path of adjacent
simplices between a simplex $\sigma_{\rm max}$ with maximal ${\mathcal{D}}{}$
to another one, $\sigma_{\rm min}$, with minimal ${\mathcal{D}}{}$ and then
transferring the excess of volume from $\sigma_{\rm max}$ to the next simplex
and repeating until reaching $\sigma_{\rm min}$, without changing the volume
of intermediary simplices. In the end, the simplex $\sigma_{\rm max}$ has same
$\Omega_{1}$-volume and $\Omega_{2}$-volume and $\sigma_{\rm min}$ has reduced
its deficit, perhaps to the point of having an excess but the later is lower
than the original one of $\sigma_{\rm max}$. We may repeat this process until
having suppressed all excess in volume. This yields a sequence
$\psi_{1},...,\psi_{k}$ of PL self-equivalences of $K$ that we may compose to
obtain the desired $\psi$. (It is indeed a standard fact that the collection
of PL manifolds, together with PL maps form a category.)

We will use the following subdivision of a pair of adjacent simplices $\sigma$
and $\tau$. Add to the collection of vertices $\sigma^{(0)}\cup\tau^{(0)}$ a
vertex $v$ inside $\sigma$ and a vertex $u$ inside $\sigma\cap\tau$ (not
necessarily barycenters) and consider the stellar subdivision
${\mathcal{S}}{}(\sigma\cup\tau)$ centered at $u$. It induces a _conical_
subdivision of $\tau$ obtained by considering the simplices generated by all
possible collection of vertices in $\tau^{(0)}\cup\\{u\\}$, except
$\sigma\cap\tau$. For $\sigma$, we first consider its stellar subdivision
${\mathcal{S}}{}\sigma$ centered at $v$ and replace the $n$-simplex
$<v,\sigma\cap\tau>$ by the conical subdivision associated to
${\mathcal{S}}{}(\sigma\cap\tau)$. Denote that new simplicial complex by
$\mathcal{S}_{v,u}(\sigma,\tau)$ and the induced subdivision of $K$ by
$\mathcal{S}_{v,w}(K)$.

######  Lemma 3.2.

[Transfering volume between adjacent simplices.] Let $\sigma,\tau\in K^{(n)}$
be two adjacent $n$-simplices of $K$ and let $0<A<\int_{\tau}\Omega_{2}$. Then
for some choice of $v\in{\rm Int}\sigma$, $w\in{\rm Int}\tau$,
$u_{\sigma},u_{\tau}\in{\rm Int}(\sigma\cup\tau)$, the PL map

$\psi:{\mathcal{S}}_{v,u_{\tau}}(\sigma,\tau)\to{\mathcal{S}}_{w,u_{\sigma}}(\tau,\sigma)$

which maps $v$ to $u_{\sigma}$, $u_{\tau}$ to $w$ and fixes the other vertices
satisfies the following conditions :

  1. (i)

$\psi=\rm Id$ on $|\partial(\sigma\cup\tau)|$,

  2. (ii)

$(\psi^{*}\Omega_{2})|_{\tau}$ and $(\psi^{*}\Omega_{2})|_{\sigma}$ are
constant,

  3. (iii)

$\int_{\tau}\psi^{*}\Omega_{2}=A$ (and
$\int_{\sigma}\psi^{*}\Omega_{2}=V_{\Omega_{2}}(\sigma)+V_{\Omega_{2}}(\tau)-A$).

###### Proof.

Let us denote by $\theta$ the $(n-1)$-simplex $\sigma\cap\tau$. We introduce
the notation ${\mathcal{A}}{}_{\sigma}$ for the affine plane spanned by a
simplex $\sigma\subset\mathbb{R}^{N}$.

We first discuss how to obtain $A$ as total volume of
$(\psi^{*}\Omega_{2})|_{\tau}$. The PL map $\psi$ removes the
$\Omega_{2}$-volume carried by $<w,\theta>$ from $\sigma$ and transfer it to
$<v,\theta>$. Adjusting $w$ allows to choose the size of
$V_{\Omega_{2}}(<w,\theta>)$. The relation to be satisfied is

$V_{\Omega_{2}}(<w,\theta>)=V_{\Omega_{2}}(\sigma)-A.$

It determines a $(n-1)$-simplex $\theta_{A}^{\tau}$ in $\tau$, parallel to
$\theta$, along which to choose $w$.

Concerning the condition for $(\psi^{*}\Omega_{2})|_{\tau}$ to be constant,
observe that, with respect to the subdivision
${\mathcal{S}}_{v,u_{\tau}}(\sigma,\tau)$, the simplex $\tau$ is divided into
$n$ $n$-simplices : the $<u_{\tau},\nu_{i}>$’s, where $\nu_{1},...,\nu_{n}$
are the $(n-1)$-faces of $\tau$ distinct from $\theta$. Set
$\nu^{*}_{i}=<u_{\tau},\nu_{i}>$. Of course, $\psi(\nu_{i}^{*})=<w,\nu_{i}>$.
To have $(\psi^{*}\Omega_{2})|_{\tau}$ constant, the ratios
$V_{\Omega_{2}}(\psi(\nu_{i}^{*}))/V_{\Omega_{2}}(\nu_{i}^{*})$ must all be
equal (to $A/V_{\Omega_{2}}(\sigma)$).

To simplify the discussion, observe that we may even choose $u_{\tau}$ and $w$
so that all $V_{\Omega_{2}}(\nu_{i}^{*})$ are equal and idem for the
$V_{\Omega_{2}}(\psi(\nu_{i}^{*}))$’s. Indeed, for $u_{\tau}$, this condition
amounts to a system of $n-1$ independent equations
$V_{\Omega_{2}}(<u_{\tau},\nu_{i}>)=V_{\Omega_{2}}(\tau)/n$, $i=1,...,n-1$ on
$u_{\tau}$ (since the relation
$V_{\Omega_{2}}(<u_{\tau},\nu_{n}>)=V_{\Omega_{2}}(\tau)/n$ is redundant),
each of which determines a hyperplane in ${\mathcal{A}}{}_{\tau}$ parallel to
${\mathcal{A}}{}_{\nu_{i}}$ and situated between ${\mathcal{A}}{}_{\nu_{i}}$
and its parallel copy passing through the vertex opposite to $\nu_{i}$ in
$\tau$. They all intersect into a unique point in $\theta$. The discussion for
$w$ is almost identical. Indeed, the relations to be satisfied by $w$ are :
$V_{\Omega_{2}}(<w,\nu_{i}>)=A/n$ for $i=1,...,n-1$, which determine a unique
point in $\theta_{A}^{\tau}$.

The next task is to ensure that $\psi^{*}\Omega_{2}$ is constant on $\sigma$.
Here too, it is possible to choose $v$ and $u_{\sigma}$ so that the
$\psi^{*}(\Omega_{2})$-volume of $<v,\theta>$ agrees with that of the
simplices $<v,\mu_{j}>$ for $\mu_{1},...,\mu_{n}$, the collection of elements
in $\sigma^{(n-1)}\setminus\\{\theta\\}$. This is very similar to the previous
discussion and amounts to choosing $v$ in the intersection of the
$(n-1)$-simplex $\theta_{A}^{\sigma}\subset\sigma$ parallel to $\theta$
determined by the relation

$V_{\Omega_{2}}(<w,\theta>)=\frac{1}{n}V_{\Omega_{2}}(\sigma)$

with the $n-1$ independent affine hyperplanes

$V_{\Omega_{2}}(<u_{\sigma},\mu_{i}>)=V_{\Omega_{2}}(\tau)-A.$

There remain one thing to do : to ensure that $\psi^{*}\Omega_{2}$ is constant
on $<v,\theta>$. The latter simplex is cut into $n$ $n$-simplices : the
$<v,u_{\tau},\gamma_{k}>$, with
$\\{\gamma_{1},...,\gamma_{n}\\}=\theta^{(n-2)}$. Of course
$V_{\psi^{*}\Omega_{2}}(<v,u_{\tau},\gamma_{k}>)=V_{\Omega_{2}}(\psi(<v,u_{\tau},\gamma_{k}>))=V_{\Omega_{2}}(<u_{\sigma},w,\gamma_{k}>)$.
Observe that our choice for $u_{\sigma}$ (respectively $u_{\tau}$) guarantees
that the $\Omega_{2}$-volume of the various $<u_{\sigma},w,\gamma_{k}>$’s
(respectively $<v,u_{\tau},\gamma_{k}>$’s) are identical. ∎

Proof of Theorem 3.1 There are many ways to proceed. We describe one that
requires few words but a lot of steps. There are more “economical procedures".

Observe that because $K$ is finite, the difference cocycle ${\mathcal{D}}{}$
is necessarily bounded and let $\sigma_{\rm max}$ denote a $n$-simplex in $K$
for which ${\mathcal{D}}{}(\sigma)$ is maximal and another one, $\sigma_{\rm
min}$, for which ${\mathcal{D}}{}(\sigma_{\rm min})$ is minimal. Then find a
path of adjacent $n$-simplices starting at $\sigma_{\rm max}$ and ending at
$\sigma_{\rm min}$, say $\sigma_{0}=\sigma_{\rm
max},\sigma_{1},...,\sigma_{k}=\sigma_{\rm min}$ and use Proposition 3.2 to
find $v\in\sigma_{1}$, $w\in\sigma_{0}$,
$u_{\sigma_{0}},u_{\sigma_{1}}\in\sigma_{0}\cap\sigma_{1}$ such that
$\psi_{0}:{\mathcal{S}}_{v,u_{\sigma_{1}}}(K)\to{\mathcal{S}}_{w,u_{\sigma_{0}}}(K)$
achieves
$V_{{\psi_{0}}^{*}\Omega_{2}}(\sigma_{0})=V_{\Omega_{1}}(\sigma_{0})$.

We redo this transfer with $\sigma_{0}$ (respectively $\sigma_{1}$) replaced
by $\sigma_{1}$ (respectively $\sigma_{2}$), constructing $\psi_{1}$ such that
$V_{\psi_{1}^{*}(\psi_{0}^{*}\Omega_{2})}(\sigma_{1})=V_{\Omega_{2}}(\sigma_{1})$.
Now the excess of $\Omega_{2}$-volume of $\sigma_{0}$ has been transferred,
via $\psi_{0}\circ\psi_{1}$, to $\sigma_{2}$, without modifying the volume of
the intermediary simplex $\sigma_{1}$. We repeat this step until having
transferred ${\mathcal{D}}{}(\sigma_{2})$ of the initial volume of
$\sigma_{0}$ to $\sigma_{k}$ via a composition
$\Psi=\psi_{0}\circ...\circ\psi_{k}$ of PL self-equivalences of $K$.

Observe that $\Omega_{2}^{\prime}=\Psi^{*}\Omega_{2}$ is piecewise constant on
$K$, the $\Omega^{\prime}_{2}$-volume of $\sigma_{\rm max}$ agrees with its
$\Omega_{1}$-volume and the $\Omega^{\prime}_{2}$-volume of $\sigma_{\rm min}$
is $V_{\Omega_{2}}(\sigma_{\rm min})+{\mathcal{D}}{}(\sigma_{\rm max})$, which
means that ${\mathcal{D}}{}_{\Omega^{\prime}_{2},\Omega_{1}}(\sigma_{\rm
min})<{\mathcal{D}}{}(\sigma_{\rm max})$. We now replace $\Omega_{2}$ by
$\Omega^{\prime}_{2}$ and repeat from the beginning. This will not damage what
has been achieved so far, since even if our new path includes $\sigma_{\rm
max}$ as an intermediary simplex, its new volume will remain identical.

Since we do not increase the total number of simplices, after finitely many
steps (at most the cardinality of $K^{(n)}$), we have equally distributed the
volume of $\Omega_{2}$ with respect to $\Omega_{1}$. $\square$

## References

  * [BD] M. Bertelson and J. Distexhe.  Piecewise linear approximation of smooth symplectic structures.
  * [BMPR18] M. Bruveris, P.W. Michor, A. Parusiński, and A. Rainer.  Moser’s theorem on manifolds with corners.  Proc. Amer. Math. Soc., 146(11):4889–4897, 2018.
  * [D19] PhD thesis.
  * [Eli] Yakov Eliashberg.  Private communication.
  * [Gra98] B. Gratza.  Piecewise linear approximations in symplectic geometry.  1998\.  [PhD Thesis (ETH Zurich)].
  * [HP04] A. Henriques and I. Pak.  Volume preserving pl maps between polyhedra.  2004\.  [https://www.math.ucla.edu/ pak/papers/henri4.pdf].
  * [JRT18] F. Jauberteau, Y. Rollin, and S. Tapie.  Discrete geometry and isotropic surfaces.  2018\.  [arXiv:1802.08712v2].
  * [Pan09] D. Panov.  Polyhedral Kähler manifolds.  Geom. Topol., 13(4):2205–2252, 2009.
  * [Sul77] D. Sullivan.  Infinitesimal computations in topology.  Inst. Hautes Études Sci. Publ. Math., (47):269–331 (1978), 1977\.