1806038 [Mathematics]Matematica Scuola di dottorato Vito Volterra XXXIII March
2021 2021 Prof. Paolo Piazza<EMAIL_ADDRESS>26 July 2021 Prof.
Daniele Guido Prof. Éric Leichtnam Prof. Paolo Antonini.

# Lipschitz-homotopy invariants:
$L^{2}$-cohomology, Roe index and $\rho$-class

Stefano Spessato

## Introduction

Cohomology Theories are one of the most powerful tools in Geometry. In this
P.h.D. thesis we will concentrate in particular on one of them: the
$L^{2}$-cohomology.
The $L^{2}$-cohomology of a complete Riemannian manifold $(M,g)$ is defined as

$H^{l}_{2}(M,g):=\frac{ker(d_{l})}{im(d_{l-1})}$ (1)

where $d_{l}$ is the closure in $\mathcal{L}^{2}(M)$ of the exterior
derivative defined on compactly supported differential $l$-forms. Moreover one
can also define the reduced $L^{2}$-cohomology as

$\overline{H}^{l}_{2}(M,g):=\frac{ker(d_{l})}{\overline{im(d_{l-1})}}.$ (2)

It’s a well-known fact that if two metrics $g$ and $h$ on $M$ are quasi-
isometric, i.e. there are two constants $C_{1}$ and $C_{2}$ such that

$C_{1}h\leq g\leq C_{2}h,$ (3)

then the (un)-reduced $L^{2}$-cohomologies coincides. In particular if $M$ is
a compact manifold we have that both, reduced or not, $L^{2}$-cohomologies
coincide with the de Rham Cohomology and so they don’t depend on the choice of
the metric $g$ in $M$.
Differently from the case of de Rham cohomology, the pullback along a
differentiable map $f$ between Riemannian manifolds, in general, doesn’t
induce a morphism between the $L^{2}$-cohomology groups (see Subsection 1.3.4
in Chapter 1).
Given a uniformly proper, lipschitz map between manifolds of bounded geometry
we shall define an operator $T_{f}$ such that it induces a morphism in
(un)-reduced $L^{2}$-cohomology. In particular, we will show that the usual
functorial properties hold and that if $f^{*}$ does induces a map in
(un)-reduced $L^{2}$-cohomology, then

$f^{*}[\alpha]=[T_{f}\alpha]$ (4)

for each square-integrable differential form.
Let us suppose, moreover, that there is a group $\Gamma$ of isometries that
acts on the manifolds. Then, if $f$ is $\Gamma$-equivariant, also $T_{f}$ will
be $\Gamma$-equivariant and the same will hold for the morphism in
$L^{2}$-cohomology induced by $T_{f}$.
The idea of how to define $T_{f}$ comes from the operator $T_{f}$ defined by
Hilsum and Skandalis in [13]. The authors use it in order to build a
perturbation for the signature operator of the disjoint union of compact
manifold which makes it invertible. We can understand our $T_{f}$ as a bounded
geometry version of the operator of Hilsum and Skandalis.

As consequence of our analysis, it follows that (un-reduced
)$L^{2}$-cohomology is a lipschitz-homotopy invariant for manifolds of bounded
geometry, i.e. if a map $f$ is a homotopy equivalence such that

  * •

$f$ is lipschitz,

  * •

there is a homotopy inverse $g$ of $f$ which is lipschitz and

  * •

the homotopies between $g\circ f$ and $f\circ g$ with their respective
identities are lipschitz,

then $T_{f}$ induces an isomorphism between the (un)-reduced
$L^{2}$-cohomologies.

In Chapter 2 we see an application of the existence of $T_{f}$. Our setting is
the same as above. We have two Riemannian manifolds $(M,g)$ and $(N,h)$ of
bounded geometry, a lipschitz-homotopy equivalence $f$ which preserves the
orientation between them and a group $\Gamma$ acting by isometries so that
$\frac{M}{\Gamma}$ and $\frac{N}{\Gamma}$ are manifolds of bounded geometry.
Our goal, in this second Chapter, will be to prove that, given the Roe
algebras $C^{*}(M)^{\Gamma}$ and $C^{*}(N)^{\Gamma}$, then

$f_{\star}(Ind_{Roe}(D_{M}))=Ind_{Roe}(D_{N}).$ (5)

In words: the Roe Index of the signature operator is a lipschitz-homotopy
invariant for manifolds of bounded geometry.
In order to establish (5) we check the existence of the Hilsum-Skandalis
perturbation for the signature operator in our setting and we will adapt the
work of Piazza and Schick [17], the work of Wahl [23], of Fukumoto [5] and
Zenobi [24] to the non-compact case.
Finally we will define a class

$\rho(M,f)\in K_{n+1}(D^{*}(N)^{\Gamma})$ (6)

that only depends on $M$ and on the class of lipschitz-homotopy of $f$.
In what follows there is a more detailed description of the Sections of this
Thesis.

#### Maps between manifolds of bounded geometry

In the first Section of the first Chapter, we introduce the setting of this
Thesis.
The objects of our study are manifolds of bounded geometry, which are
manifolds with strictly positive injectivity radius and some uniform bounds on
the covariant derivatives of the curvature. Some examples of manifolds of
bounded geometry can be compact manifolds, coverings of manifolds of bounded
geometry (so, for example, the euclidean space and the hyperbolic space), and
products of manifolds of bounded geometry. Moreover Greene, in [8], proved
that every differential manifold admits a metric of bounded geometry. The main
results that we will use about manifolds of bounded geometry can be found in
[4] and in [20].
On the other hand, the maps that we consider are the uniformly proper
lipschitz maps between metric space. These are lipschitz maps such that the
diameter of the preimage of a subset $A$ is controlled by the diameter of $A$
itself. We show, in particular, that every lipschitz-homotopy equivalence is
an uniformly proper map.
We conclude the first section by introducing a particular family of actions of
a group $\Gamma$ on a metric space. We call these uniformly proper,
discontinuous and free actions (or FUPD actions). We will see that if the
metric space is a manifold of bounded geometry, then an action of a group
$\Gamma$ is FUPD if and only if the quotient is a manifold of bounded
geometry.

####  $C^{k}_{b}$-maps

In this second section of first Chapter we introduce the notion of
$C^{k}_{b}$-map between manifolds of bounded geometry. One can also find the
definition and the main properties of these maps in [4]. A $C^{k}_{b}$-map is
a map such that, in local normal coordinates, has uniformly bounded
derivatives of order $l\leq k$. In particular one can check that
$C^{1}_{b}$-maps are lipschitz maps. The most important property, for us,
about the $C^{k}_{b}$-map is proved in subsection 1.2.2: we use a result of
Eldering [4] to prove that every lipschitz map $f$ between manifolds of
bounded geometry can be approximated by a$C^{\infty}$-map $F$ which is a
$C^{k}_{b}$-map. Moreover if we have two FUPD actions of a group $\Gamma$ on
$M$ and $N$ and $f$ is $\Gamma$-equivariant, then also the approximation is
$\Gamma$-equivariant. In particular, we have that the morphisms induced
between the K-theory groups of the Roe algebra by $f$ and $F$ are the same.
This means that, in the next sections, we consider $f$ as a smooth
$C^{k}_{b}$-map for each $k\in\mathbb{N}$.

####  $L^{2}$-cohomology and pull-back

In this third section of the first Chapter we introduce the notions of
$L^{2}$-cohomology and reduced $L^{2}$-cohomology. In subsection 1.3.4 we show
that the pullback along a map $f$, in general, doesn’t induce a map in
(un)-reduced $L^{2}$-cohomology, even if it is a lipschitz-homotopy
equivalence. Then, given a map $f:(M,g)\longrightarrow(N,h)$, we define the
Fiber Volume of $f$ as the Radon-Nicodym derivative

$Vol_{f}:=\frac{\partial f_{\star}\mu_{M}}{\partial\mu_{N}},$ (7)

where $\mu_{M}$ and $\mu_{N}$ are the measures on $M$ and $N$ induced by their
metrics. We call R.-N.-lipschitz map, a lipschitz map $f$ which has bounded
Fiber Volume. Then we show that the pullback along a R.-N.-lipschitz map
induces an $\mathcal{L}^{2}$-bounded operator. In particular, we also show
that R.-N.-lipschitz maps are exactly the v.b.-maps defined by Thomas Schick
in his Ph.D. Thesis.
We conclude the third section of first Chapter by showing that the Fiber
Volume of a submersion can be expressed as the integration along the fibers of
some particular differential forms. In particular we choose the name Fiber
Volume because in the case of a totally geodesics Riemannian submersion we
have that $Vol_{f}$ in a point $q$ in $N$ is exactly the volume of its fiber
$F_{q}$ respect to the metric induced by $g$.

#### Submersion related to lipschitz maps

This is the fourth section of first Chapter. Consider a uniformly proper
lipschitz map $f:(M,g)\longrightarrow(N,h)$, where $(N,h)$ is an oriented
manifold of bounded geometry and $(M,g)$ is an oriented manifold with bounded
scalar curvature. Let $\Gamma$ be a group which acts by isometries on $M$ and
$N$ and suppose that $f$ is $\Gamma$-equivariant. Let us define $T^{\delta}N$
as the subset of $TN$ given by the vectors with norm less or equal to
$\delta:=inj_{N}$.
The main goal of this section is to define a $\Gamma$-equivariant lipschitz
submersion

$p_{f}:(f^{*}T^{\delta}N,g_{S})\longrightarrow(N,h)$ (8)

such that $p_{f}(x,0)=f(x)$. The metric $g_{S}$ on $f^{*}T^{\delta}N$ is
called Sasaki metric and it depends on the metrics $g$ and $h$. Moreover we
also have that if $f$, in local normal coordinates, has some uniform bounds on
the derivatives of order less or equal $j$, then also $p_{f}$ has some bounds
on the derivatives of the same order. In particular these bounds are important
in Chapter 2.

#### The pull-back functor

The last section of the first Chapter starts by proving that the submersion
$p_{f}$ has bounded Fiber Volume and so it is a R.-N.-lipschitz map. Then we
define the operator
$T_{f}:\mathcal{L}^{2}(N)\longrightarrow\mathcal{L}^{2}(M)$ as

$T_{f}(\alpha):=\int_{B^{\delta}}p_{f}^{*}\alpha\wedge\omega,$ (9)

where $\omega$ is a particular Thom form of $f^{*}(TN)$. It follows the proof
of the $\mathcal{L}^{2}$-boundedness of $T_{f}$: it will be easily proved
since $p_{f}$ is a R.-N.-lipschitz map.
In the subsection 1.5.5 we prove two important lemmas: the first one gives a
formula which relates $f$ and $p_{f}$ when $f$ is a R.-N.-lipschitz map. The
second one gives a formula which relates the submersion $p_{f\circ g}$ with
the submersions $p_{f}$ and $p_{g}$, where $f:(M,m)\longrightarrow(N,h)$ and
$g:(S,l)\longrightarrow(M,m)$ are uniformly proper lipschitz map between
manifolds of bounded geometry.
We conclude the section by showing that, for every $z$ in $\mathbb{N}$

$\begin{cases}\mathcal{F}_{z}(M,g)=H^{z}_{2}(M)\\\
\mathcal{F}_{z}((M,g)\xrightarrow{f}(N,h))=H^{z}_{2}(N)\xrightarrow{T_{f}}H^{z}_{2}(M)\end{cases}$
(10)

is a controvariant functor between the category $\mathcal{C}$ of the manifolds
of bounded geometry with uniformly proper lipschitz maps and the category Vec
of the vector spaces. In particular, we will show that if two maps $f_{1}$ and
$f_{2}$ are uniformly proper lipschitz maps such that they are homotopic with
a lipschitz homotopy $H$, then $T_{f_{1}}=T_{f_{2}}$ in $L^{2}$-cohomology.
Moreover if $f$ is a R.-N.-lipschitz map, then $f^{*}=T_{f}$ in
$L^{2}$-cohomology. Finally we show that the same holds considering the
reduced $L^{2}$-cohomology instead of the $L^{2}$-cohomology.

#### Coarse geometry

In the first section of the second Chapter, we introduce the notions of coarse
structure and coarse map. This definitions and the main properties come from
the book Analytic K-homology of Higson and Roe [10]. In particular we use two
coarse structures. The first one is a metric coarse structure, which is a
coarse structure which can be defined on a metric space. It easily follows
that uniformly proper lipschitz map are coarse maps respect to metric coarse
structures.
The second one is a coarse structure which can be defined on the disjoint
union of metric spaces $M$ and $N$ such that there is a map
$f:M\longrightarrow N$. Indeed, in the following sections, we will need a
coarse structure on $M\sqcup N$, which is not a metric space.
After that, given a coarse space $X$ and given an action of a group $\Gamma$
on $X$, we define the Roe (or coarse) algebras $C^{*}(X)^{\Gamma}$ and the
structure algebra $D^{*}(X)^{\Gamma}$. In the case $X=M\sqcup N$ we will
denote the algebras by $C^{*}_{f}(M\sqcup N)^{\Gamma}$ and by
$D^{*}_{f}(M\sqcup N)^{\Gamma}$ to recall the dependence on $f$. We have that
$C^{*}(X)^{\Gamma}$ is an ideal of $D^{*}(X)^{\Gamma}$ and so this means that
the sequence

$0\longrightarrow C^{*}(X)^{\Gamma}\longrightarrow
D^{*}(X)^{\Gamma}\longrightarrow\frac{D^{*}(X)^{\Gamma}}{C^{*}(X)^{\Gamma}}\longrightarrow
0$ (11)

is exact. Thus we obtain the long exact sequence in K-Theory

$...K_{n}(C^{*}(X)^{\Gamma})\longrightarrow
K_{n}(D^{*}(X)^{\Gamma})\longrightarrow
K_{n}(\frac{D^{*}(X)^{\Gamma}}{C^{*}(X)^{\Gamma}})\longrightarrow
K_{n-1}(C^{*}(X)^{\Gamma})\longrightarrow...$ (12)

Finally, if $dim(M)=n$, we define the fundamental class $[D_{M}]\in
K_{n+1}(\frac{D^{\star}(M)^{\Gamma}}{C^{\star}(M)^{\Gamma}})$ of the signature
operator $D_{M}$ of a Riemannian manifold and we conclude the section defining
the Roe Index of a connected Riemannian manifold as the class

$Ind_{Roe}(D_{M}):=\delta[D_{M}]$ (13)

in $K_{n}(C^{*}(M)^{\Gamma})$, where $\delta$ the connecting homomorphism in
K-Theory.

#### Smoothing operators between manifolds of bounded geometry

This is the second section of second Chapter. Consider
$f:(M,g)\longrightarrow(N,h)$ a lipschitz-homotopy equivalence with preserves
the orientations. Then, in this section we prove that there is an
$\mathcal{L}^{2}$-bounded, $\Gamma$-invariant, operator $y$ such that

$1-T_{f}^{\dagger}T_{f}=dy+yd$ (14)

on $dom(d)$ where $T_{f}^{\dagger}=\tau_{N}T_{f}^{*}\tau_{M}$ and $\tau_{X}$
is the so-called chiral operator of the Riemannian manifold $X$.
After that we prove that $T_{f}$ is a smoothing operator, i.e. an operator $A$
such that for each smooth $\alpha$

$A\alpha(p)=\int_{N}K(p,q)\alpha(q)d\mu_{N},$ (15)

where $K$ is a smooth section of the fiber bundle
$\Lambda^{*}(M)\boxtimes\Lambda(N)$. Then, using the results reported in
Appendix D, we prove that also $dT_{f}$ is a smoothing operator, and, in
particular, we can see that if $f$ is a $C^{k}_{b}$-map where $k\geq 2$, then
$dT_{f}$ is also $\mathcal{L}^{2}$-bounded.
We conclude the section proving that if $f$ is a $C^{k\geq 2}_{b}$-map, then
the operators $T_{f}y$, $yT^{\dagger}_{f}$, $dT_{f}y$ and all their adjoints
and their algebraic combinations are all operators in $C^{*}_{f}(M\sqcup
N)^{\Gamma}$.

#### Lipschitz-homotopy invariance of the Roe Index

In the third section of Chapter 2 we use $T_{f}$ and $y$ to define the Hilsum-
Skandalis perturbation $\mathcal{P}$ of [13] for the signature operator on the
manifold of bounded geometry $M\sqcup N$. It follows, since $T_{f}$ induces an
isomorphism in $L^{2}$-cohomology, that the perturbed signature operator
$D_{M\sqcup N}+\mathcal{P}$ is $L^{2}$-invertible. Moreover, since $T_{f}$,
$T_{f}y$, $dT_{f}y$ etc.., are all operators in $C^{*}_{f}(M\sqcup
N)^{\Gamma}$, we are able to prove that also $\mathcal{P}$ is in
$C^{*}_{f}(M\sqcup N)^{\Gamma}$.
Similarly to what we did for Riemannian connected manifolds, we define the
fundamental class $[D_{M\sqcup N}]$ and the Roe Index $Ind_{Roe}(D_{M\sqcup
N})$. The using that $\mathcal{P}$ is in $C^{*}_{f}(M\sqcup N)^{\Gamma}$ and a
result of [11], we can find a class $\rho_{f}$ in $K_{n+1}(D_{f}^{*}(M\sqcup
N)^{\Gamma})$ such that its image in the long exact sequence in K-Theory
coincides with $[D_{M\sqcup N}]$. This fact implies that

$Ind_{Roe}(D_{M\sqcup N})=0,$ (16)

and, using some lemmas proved in Appendix C, we conclude that

$f_{\star}(Ind_{Roe}(\mathcal{D}_{M}))=Ind_{Roe}(\mathcal{D}_{N}).$ (17)

#### Lipschitz-homotopy stability of $\rho_{f}$ and $\rho(f,M)$

In the last section of this Thesis we prove that the class $\rho_{f}$ just
depend on the class of lipschitz-homotopy of $f$, i.e. if $f$ and $f^{\prime}$
are two $C^{k}_{b}$ lipschitz-homotopy equivalences and there is a lipschitz
homotopy $H$ between them, then

$D^{*}_{f}(M\sqcup N)^{\Gamma}=D^{*}_{f^{\prime}}(M\sqcup N)^{\Gamma}\mbox{
and }C^{*}_{f}(M\sqcup N)^{\Gamma}=C^{*}_{f^{\prime}}(M\sqcup N)^{\Gamma}.$
(18)

Moreover we also have that

$\rho_{f}=\rho_{f^{\prime}}.$ (19)

In order to prove this fact we consider for each $s\in[0,1]$ the
$f_{s}:=H(\cdot,s)$ and we prove that $\mathcal{P}_{s}$, which is the Hilsum-
Skandalis perturbation related to $f_{s}$ defines a curve in
$C^{*}_{f}(M\sqcup N)^{\Gamma}$.
We conclude this Thesis defining the projected $\rho$-class in
$K_{n+1}(D^{*}(N)^{\Gamma})$

$\rho(f,M):=f_{\star}[\rho_{f}]$ (20)

and showing that it depends only on $M$ and on the lipschitz-homotopy class of
$f$.

### Acknowledgments

I am grateful to Paolo Piazza, my advisor, for the interesting reasearch
project he proposed to me and for his assistance at every stage of the
research project.
I wish to thank Vito Felice Zenobi for the several discussions that we had,
for his help and for its competence.
I also would like to thank Francesco Bei and Thomas Schick for their
insightful comments and suggestions.
Another thanks goes to the referees for patiently reported all glitches and
imperfections to me.
Finally I would like to thank my family and Laura for their constant support.

###### Contents

  1. Introduction
  2. 1 A pull-back functor for (un)-reduced $L^{2}$-cohomology
    1. 1.1 Maps between manifolds of bounded geometry
      1. 1.1.1 Lipschitz maps
      2. 1.1.2 $\Gamma$-Lipschitz-homotopy equivalence
      3. 1.1.3 Uniformly proper maps
      4. 1.1.4 Manifolds of bounded geometry
      5. 1.1.5 Uniformly proper and discontinuous actions
    2. 1.2 $C^{k}_{b}$-maps
      1. 1.2.1 Maps with bounded derivatives
      2. 1.2.2 $C^{k}_{b}$-approximation of a lipschitz map
    3. 1.3 $L^{2}$-cohomology and pull-back
      1. 1.3.1 $\mathcal{L}^{2}$-forms
      2. 1.3.2 Exterior derivative
      3. 1.3.3 $L^{2}$-cohomology and reduced $L^{2}$-cohomology
      4. 1.3.4 Problems with pullback
      5. 1.3.5 Radon-Nicodym derivative and Fiber Volume
      6. 1.3.6 Radon-Nicodym-Lipschitz maps
      7. 1.3.7 Quotients of differential forms
      8. 1.3.8 Fiber Volume of a submersion
    4. 1.4 Submersion related to lipschitz maps
      1. 1.4.1 The Sasaki metric
      2. 1.4.2 Pull-back bundle and pull-back connection
      3. 1.4.3 The map $p_{f}$
    5. 1.5 The pull-back functor
      1. 1.5.1 The Fiber Volume of $p_{f}$
      2. 1.5.2 A Thom form for $f^{*}(TN)$
      3. 1.5.3 The $T_{f}$ operator
      4. 1.5.4 R.-N.-lipschitz equivalences of vector bundles
      5. 1.5.5 Lemmas about homotopy
      6. 1.5.6 Lipschitz-homotopy invariance of (un)-reduced $L^{2}$-cohomology
  3. 2 Roe index and $\rho_{f}$-class
    1. 2.1 Coarse geometry
      1. 2.1.1 Coarse structures
      2. 2.1.2 Roe and Structure Algebras
      3. 2.1.3 Coarse maps
      4. 2.1.4 Roe Index of the signature operator
    2. 2.2 Smoothing operators between manifolds of bounded geometry
      1. 2.2.1 The operator $y$
      2. 2.2.2 $T_{f}$ as smoothing operator
      3. 2.2.3 Some operators in $C^{*}_{f}(M\sqcup N)^{\Gamma}$
    3. 2.3 Lipschitz-homotopy invariance of the Roe Index
      1. 2.3.1 The perturbed signature operator
      2. 2.3.2 $L^{2}$-invertibility of $D_{\alpha,1}$
      3. 2.3.3 The perturbation in $C^{*}_{f}(M\sqcup N)^{\Gamma}$
      4. 2.3.4 Involutions
      5. 2.3.5 Invariance of the Roe Index
    4. 2.4 Lipschitz-homotopy stability of $\rho_{f}$
      1. 2.4.1 Lipschitz-homotopy stability of $D^{*}_{f}(M\sqcup N)^{\Gamma}$ and $C^{*}_{f}(M\sqcup N)^{\Gamma}$
      2. 2.4.2 The curves $T_{s}$ and $dT_{s}$
      3. 2.4.3 The curves $T_{s}y_{s}$ and $dT_{s}y_{s}$
      4. 2.4.4 Lipschitz-homotopy stability of $\rho_{f}$
  4. A Inequalities between norms of differential forms
  5. B Orientation and integration along the fibers
  6. C K-Theory
  7. D Smoothing operators

##  Chapter 1 A pull-back functor for (un)-reduced $L^{2}$-cohomology

###  1.1 Maps between manifolds of bounded geometry

####  1.1.1 Lipschitz maps

In this section our goal is to introduce families of maps between metric
spaces. The first one is the one of lipschitz maps. Let $(X,d_{X})$ and
$(Y,d_{Y})$ be two metric spaces. Let us consider a map
$f:(X,d_{X})\longrightarrow(Y,d_{Y})$.

######  Definition 1.1.1.

We say that $f$ is lipschitz if for all $p$ and $q$ in $X$ there is a constant
$C_{f}$ such that

$d_{Y}(f(p),f(q))\leq C_{f}d_{X}(p,q).$ (1.1)

######  Remark 1.

If $f$ is a smooth map between Riemannian manifolds there are also two other
equivalent definitions: the first one is that $f$ is a lipschitz map if there
is a constant $C_{f}$ such that for all $p$ in $M$ an for all $v_{p}$ in
$T_{p}M$

$||f_{\star,p}(v_{p})||\leq C_{f}||v_{p}||.$ (1.2)

The second one is the following: $f$ is a lipschitz map if there is a constant
$C_{f}$ such that for every $p$ in $M$, fixed some normal coordinates
$\\{x^{i}\\}$ centered in $p$ and $\\{y^{j}\\}$ centered in $f(p)$, then

$|\frac{\partial y^{j}\circ f}{\partial x^{i}}(p)|\leq C_{f}.$ (1.3)

The above inequality doesn’t depend on the choice of normal coordinates,
indeed we have that

$|\frac{\partial y_{2}^{j}}{\partial y_{1}^{i}}(f(p))|=|\frac{\partial
x_{2}^{j}}{\partial x_{1}^{i}}(p)|=1.$ (1.4)

####  1.1.2 $\Gamma$-Lipschitz-homotopy equivalence

Let us consider now a group $\Gamma$ acting on $M$ and $N$. We will
concentrate on the maps which preserves the action of $\Gamma$.

######  Definition 1.1.2.

Two maps $f_{0}$ and $f_{1}:(X,d_{X})\longrightarrow(Y,d_{Y})$ are
$\Gamma$-lipschitz-homotopic, denoted by

$f_{0}\sim_{\Gamma}f_{1},$ (1.5)

if they are $\Gamma$-equivariant maps and there is a $\Gamma$-invariant
lipschitz homotopy111Here we are considering $d_{X}\times d_{[0,1]}$ as the
product distance between $d_{X}$ and the euclidean distance on $[0,1]$.
$H:(X\times[0,1],d_{X}\times d_{[0,1]})\longrightarrow(Y,d_{Y})$ such that

$H(x,0)=f_{0}(x)\mbox{ and }H(x,1)=f(x)$ (1.6)

######  Definition 1.1.3.

A map $f:(X,d_{X})\longrightarrow(Y,d_{Y})$ is a $\Gamma$-lipschitz-homotopy
equivalence if $f$ is $\Gamma$-equivariant, lipschitz and there is a
$\Gamma$-equivariant map $g$ such that

  * •

$g$ is a homotopy inverse of $f$,

  * •

$g$ is lipschitz,

  * •

$f\circ g\sim_{\Gamma}id_{Y}$ and $g\circ f\sim_{\Gamma}id_{X}$.

####  1.1.3 Uniformly proper maps

Let us introduce the notion of uniformly proper map.

######  Definition 1.1.4.

Consider two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$. Let
$f:(X,d_{X})\longrightarrow(Y,d_{Y})$ be a map. Then $f$ is uniformly
(metrically) proper, if there is a continuous function
$\alpha:[0,+\infty)\longrightarrow[0,+\infty])$ such that for every subset
$A\subseteq Y$ we have

$diam(f^{-1}(A))\leq\alpha(diam(A))$ (1.7)

######  Remark 2.

If $f$ is a uniformly proper lipschitz map, then $\alpha$ can be chosen such
that for all $s,t\geq 0$

$\alpha(t)<\alpha(t+s).$ (1.8)

Indeed if $A\subseteq B$ then $f^{-1}(A)\subseteq f^{-1}(B)$. This means that
the composition of two uniformly proper maps is still uniformly proper, indeed

$\begin{split}diam((g\circ f)^{-1}(A))&=diam(f^{-1}(g^{-1}(A)))\\\
&\leq\alpha(diam(g^{-1}(A)))\\\
&\leq\alpha(\beta(diam(A)))=(\alpha\circ\beta)(diam(A)).\end{split}$ (1.9)

######  Remark 3.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be two metric spaces and let $\Gamma$ be a
group acting on $M$ and on $N$. Consider two lipschitz maps
$F,f:(X,d_{X})\longrightarrow(Y,d_{Y})$ such that $f\sim_{\Gamma}F$ trought a
$\Gamma$-equivariant lipschitz homotopy $h$. If $f$ is a uniformly proper map,
then also the homotopy $h$ is uniformly proper. Then, in particular, also $F$
is uniformly proper.
In order prove this fact we have to observe that

$d(f(x),h(x,t))=d(h(x,0),h(x,t))\leq C_{h}\cdot t.$ (1.10)

Then, if $A$ is a subset of $Y$ and $h_{t}:X\longrightarrow Y$ is defined as
$h_{t}(p):=h(p,t)$, we have that

$\begin{split}h_{t}^{-1}(A)&=\\{p\in X|h(p,t)\in A\\}\\\ &\subseteq\\{p\in
X|f(p)\in B_{C_{h}\cdot t}(A)\\}\\\ &=f^{-1}(B_{C_{h}\cdot t}(A))\end{split}$
(1.11)

where $B_{C_{h}\cdot t}(A)$ are the points $y$ of $Y$ such that $d(y,A)\leq
C_{h}\cdot t$. Then we have that

$\begin{split}h^{-1}(A)&=\bigsqcup\limits_{t\in[0,1]}h_{t}^{-1}(A)\times\\{t\\}\\\
&\subseteq f^{-1}(B_{C_{h}}(A))\times[0,1]\end{split}$ (1.12)

and so

$diam(h^{-1}(A))\leq
diam(f^{-1}(B_{1}(A)))+diam([0,1])\leq\alpha(diam(A)+2{C_{h}})+1.$ (1.13)

######  Lemma 1.1.1.

Let $f:(X,d_{X})\longrightarrow(Y,d_{Y})$ be a $\Gamma$-lipschitz-homotopy
equivalence between metric spaces and let us denote by $g$ a lipschitz-
homotopy inverse of $f$. Moreover we denote by $H$ the lipschitz homotopy
between $g\circ f$ and $id_{X}$. Then $f$ is uniformly proper.

###### Proof.

Since $H$ is a lipschitz map we have that if $y$ is a point in $Y$, then

$\begin{split}d_{X}(x,g\circ f(x))&=d_{X}(H(x,0),H(x,1))\\\ &\leq
C_{H}d_{X\times[0,1]}((x,0),(x,1))=C_{H}.\end{split}$ (1.14)

Now, let $x_{1}$ and $x_{2}$ be two points in $f^{-1}(A)$, then

$\begin{split}d_{X}(x_{1},x_{2})&\leq d_{X}(x_{1},g\circ
f(x_{1}))+d_{X}(g\circ f(x_{1}),g\circ f(x_{2}))+d_{X}(x_{1},g\circ
f(x_{1}))\\\ &\leq 2C_{H}+C_{g}d_{Y}(f(x_{1}),f(x_{2}))\\\ &\leq
2C_{H}+C_{g}diam(A).\end{split}$ (1.15)

∎

####  1.1.4 Manifolds of bounded geometry

In this section we will introduce the notion of manifolds of bounded geometry.
All the definitions and propositions which follow can be found in [4].
Let $(M,g)$ be a Riemannian manifold.

######  Definition 1.1.5.

The Riemannian manifold $(M,g)$ has $k$-bounded geometry if:

  * •

the sectional curvature $K$ of $(M,g)$ and its first $k$-covariant derivatives
are bounded, i.e. for each $i=0,...,k$ there is a constant $V_{i}$ such that
$\forall x\in M$

$|\nabla^{i}K(x)|\leq V_{i}.$ (1.16)

  * •

there is a number $C>0$ such that for all $p$ in $M$ the injectivity radius
$i_{g}(p)$ satisfies

$i_{g}(p)\geq C.$ (1.17)

The maximal $C$ which satisfies this inequality will be denoted by
$r_{inj}(M)$.

When we talk about a manifold $M$ with bounded geometry, without specifying
the $k$, we mean that $M$ has $k$-bounded geometry for all $k$ in
$\mathbb{N}$.

######  Proposition 1.1.2.

Let $(M,g)$ be a Riemannian manifold of $k$-bounded geometry. Then there
exists a $\delta>0$ such that the metric up to its $k$-th order derivatives
and the Christoffel symbols up to its $(k-1)$-th order derivatives are bounded
in normal coordinates of radius $\delta$ around each $x\in M$, with bounds
that are uniform in $x$.

######  Proposition 1.1.3.

Let $(M,g)$ be a Riemannian manifold of $k\geq 1$-bounded geometry. For every
$C>1$ there exists a $\delta>0$ such that the normal coordinate charts
$\phi_{x}$ are defined on $B_{\delta}(x)$ for each $x\in M$ and the Euclidean
distance $d_{E}$ on the normal coordinates is uniformly $C$-equivalent to the
metric distance $d$ induced by $M$, that is, $\forall x_{1},x_{2}\in
B_{\delta}(x)$

$C^{-1}d(x_{1},x_{2})\leq d_{E}(\phi_{x}(x_{1}),\phi_{x}(x_{2}))\leq
Cd(x_{1},x_{2}).$ (1.18)

######  Proposition 1.1.4.

Let $(M,g)$ be a Riemannian manifold of $k$-bounded geometry with $k\geq 2$.
There exists a $\delta$ with $0<\delta<r_{inj}(M)$ and a constant $C>0$ such
that for all $x_{1},x_{2}\in M$ with $d(x_{1},x_{2})<\delta$ we have that the
coordinate transition map

$\phi_{2,1}=exp^{-1}_{x_{2}}\circ exp_{x_{1}}:U\longrightarrow
T_{x_{2}}M\text{ with }U=exp_{x_{1}}(B_{\delta}(x_{1})\cap
B_{\delta}(x_{2}))\subset T_{x_{1}}M.$ (1.19)

is $C^{k-1}$-bounded222it means that all the derivatives of degree less or
equal to $k-1$ are bounded. with $|\phi_{2,1}|_{k-1}\leq C$.

######  Remark 4.

Consider a Riemannian manifold $(M,g)$ such that

$Ric(M)\geq(n-1)K$ (1.20)

where $K$ is a constant. In particular, if $M$ has $k_{\geq 0}$-bounded
geometry we have that (1.20) is satisfied. Then, as consequence of Bishop-
Gromov inequality, if $(M,g)$ satisfies (1.20) for each $p\in M$ and for each
$r\geq 0$ we have that

$\mu_{M}(B_{r}(p))\leq C(r)$ (1.21)

for some function $C:\mathbb{R}_{\geq 0}\longrightarrow\mathbb{R}_{\geq 0}$.
This means that if $A\subset M$ has $diam(A)=r$, then, once we fix a $p$ in
$A$, we have that

$\mu_{M}(A)\leq\mu_{M}(B_{diam(A)}(p))\leq C(diam(A)).$ (1.22)

####  1.1.5 Uniformly proper and discontinuous actions

In this section we will introduce a particular family of actions of a group
$\Gamma$ over a metric space $(X,d_{X})$.

######  Definition 1.1.6.

Consider $\Gamma$ a group which acts by isometries on a metric space
$(X,d_{X})$. We say that the action of $\Gamma$ is uniformly properly
discontinuous and free (FUPD) if

  * •

the action of $\Gamma$ is free and properly discontinuous,

  * •

there exists a number $\delta>0$ such that

$d_{X}(p,\gamma p)\leq\delta\implies p=\gamma p.$ (1.23)

######  Proposition 1.1.5.

Let us suppose that $(M,g)$ is a manifold of bounded geometry. Then the
following are equivalent:

  1. 1.

$\Gamma$ induce a FUPD action,

  2. 2.

the quotient $\frac{M}{\Gamma}$ has bounded geometry.

###### Proof.

$1\implies 2$
If $\delta$ is the constant of the FUPD action of $\Gamma$, then for each $p$
in $M$ we have that $B_{\delta}(p)$ is a trivializing open of
$\frac{M}{\Gamma}$ and so $inj_{\frac{M}{\Gamma}}\geq\min\\{inj_{M},\delta\\}$
and the curvature of $\frac{M}{\Gamma}$ has the same bounds of the curvature
of $M$.
$2\implies 1$
Let us suppose that for each $\delta>0$ there is a point $p$ and a
$\gamma\in\Gamma$ such that $d_{M}(p,\gamma p)\leq\delta$. Consider
$\delta<\min\\{inj_{M},inj_{\frac{M}{\Gamma}}\\}$. Then there is a vector $v$
in $T_{p}M$ with norm less then $\delta$ such that $exp_{p}(v)=\gamma p$. So,
if we denote by $s$ the Riemannian covering
$s:M\longrightarrow\frac{M}{\Gamma}$, we can apply the formula

$s\circ exp_{p}=exp_{s(p)}\circ ds$ (1.24)

to $v$. We have that

$s(p)=s\circ exp_{p}(v)=exp_{s(p)}\circ ds(v).$ (1.25)

Then, since $ds$ is an isometry, we have that the norm of $ds(v)$ is less or
equal to the injectivity radius of $\frac{M}{\Gamma}$. This means that $ds(v)$
has to be null, then in particular $v$ is null and so

$\gamma p=exp_{p}(v)=exp_{p}(0)=p.$ (1.26)

∎

###  1.2 $C^{k}_{b}$-maps

####  1.2.1 Maps with bounded derivatives

Consider $(M,g)$ be a Riemannian manifold of bounded geometry.

######  Definition 1.2.1.

We define $\delta>0$ to be $M$-small if the Propositions 1.1.2, 1.1.3 and
1.1.4 hold on all normal coordinate charts of radius $\delta$.

######  Definition 1.2.2.

Let $X,Y$ be Riemannian manifolds of $k+1$-bounded geometry and $f\in
C^{k}(X,Y)$. We say that $f$ is of class $C^{k}_{b}$ when there exist
$X,Y$-small $\delta_{X}$ , $\delta_{Y}>0$ such that for each $x\in X$ we have
$f(B_{\delta_{X}}(x))\subset B_{\delta_{Y}}(f(x))$ and the composition

$F_{x}=exp^{-1}_{f(x)}\circ f\circ exp_{x}:B_{\delta_{X}}(0)\subset
T_{x}X\longrightarrow T_{y}Y$ (1.27)

in normal coordinates is of class $C^{k}_{b}$ and its $C^{l}$-norms
($l=0,...,k$) as function from $T_{x}X$ to $T_{y}Y$ are bounded uniformly in
$x\in X$.

######  Remark 5.

There are four remarks we wish to make:

  * •

The condition of bounded geometry is necessary to define $C^{k}_{b}$-maps. One
can see more detail in [4].

  * •

Composition of two $C^{k}_{b}$-maps is a $C^{k}_{b}$-map.

  * •

Lipschitz maps between manifolds of bounded geometry are $C^{0}_{b}$-maps. In
order to prove this, observe that, since $M$ and $N$ are manifolds of bounded
geometry, then the exponential maps and their inverses are lipschitz maps. In
particular their lipschitz constants are uniformly bounded (see Proposition
2.5 of [4]). Then, in local normal coordinates,

$|F_{x}(y)|=|F_{x}(y)-F_{x}(0)|\leq C|y|\leq C\cdot\delta_{X}$ (1.28)

where $C$ is a constant which uniformly bounds in $x$ the lipschitz constants
of $F_{x}:=exp^{-1}_{f(x)}\circ f\circ exp_{x}$. This means that the
$C^{0}_{b}$-norm of $F_{x}$ is uniformly bounded in $x$.

  * •

we have that if $f$ is a $C^{k}_{b}$-map, then, in particular, it is a
$C^{k-1}_{bu}$-map.

####  1.2.2 $C^{k}_{b}$-approximation of a lipschitz map

Consider a lipschitz-homotopy equivalence $f:(M,g)\longrightarrow(N,h)$
between manifolds of bounded geometry. Let us suppose, moreover, that there is
a group $\Gamma$ on $M$ and $N$ acting by isometries FUPD and assume that $f$
is $\Gamma$-equivariant. We want to show that for all $k$ there is a
lipschitz-homotopy equivalence $F:(M,g)\longrightarrow(N,h)$ which is
$C^{k}_{b}$ and that is $\Gamma$-lipschitz-homotopic to $f$.
In order to prove this fact we need the following two proposition of [4].

######  Proposition 1.2.1.

Let $(M,g)$ be a Riemannian manifold of $k$-bounded geometry for some $k$.
Then for $\delta_{2}>0$ small enough and any $0<\delta_{1}\leq\delta_{2}$, $M$
has a countable cover $\\{B_{\delta_{1}}(x_{i})\\}$ such that

  * •

$\forall i\neq j\implies d(x_{i},x_{j})\geq\delta_{1}$

  * •

There exists an explicit global bound $K$ such that for each $x\in M$ the ball
$B_{\delta_{2}}(x)$ intersects at most $K$ of the $B_{\delta_{2}}(x_{i})$.

######  Proposition 1.2.2.

Let $r,\delta r>0$, $g\in C^{k\geq 0}_{b}(B_{r+2\delta
r}(0)\subset\mathbb{R}^{m},\mathbb{R}^{n})$ such that $g$ is uniformly
continuous and fix $l>k$, and $\epsilon>0$. Consider now a mollifier 333A
mollifier is a compactly supported, smooth, positive function
$\phi:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ such that its integral over
$\mathbb{R}^{n}$ equals to 1. $\phi$ with support in the euclidean ball of
$\mathbb{R}^{m}$ and let us define
$\phi_{\nu}(x):=\nu^{-m}\phi(\frac{x}{\nu})$. Then $g$ can be approximated by
a function $G_{\nu}$ such that

  * •

$G_{\nu}=g$ outside $B_{r+\delta r}(0)$;

  * •

$G_{\nu}\in C^{l}_{b,u}\cap C^{\infty}$ on $B_{r}(0)$ and $G_{\nu}\in
C^{l}_{b,u}\cap C^{\infty}$ on an open $A$ if $g\in C^{l}_{b,u}\cap
C^{\infty}$ on $A$;

  * •

$|g-G_{\nu}|_{k}\leq\epsilon$;

  * •

$|G_{\nu}|_{l}\leq C(\nu,l)|g|_{0}$ on $B_{r}(0)$ for some $C_{\nu,l}>0$.

Note that $C(\nu,l)$ may grow unboundedly as $\nu\longrightarrow 0$ or
$l\longrightarrow+\infty$.

######  Remark 6.

The approximation above is defined in the following way. Let us consider a
smooth map $\chi:\mathbb{R}\longrightarrow[0,1]$ which is $1$ if $x\leq r$ and
$0$ if $x\geq r+\delta r$. Then we have that

$G_{\nu}(x):=(1-\chi(||x||))g(x)+\chi(||x||)\int_{\mathbb{R}^{m}}g(x-y)\phi_{\nu}(y)dy$
(1.29)

where $\nu<\frac{\delta r}{2}$.
moreover $G_{\nu}$ can be also written as

$\begin{split}G_{\nu}(x)&=(1-\chi(||x||))g_{i,\gamma}(x)+\chi(||x||)\int_{\mathbb{R}^{m}}g_{i,\gamma}(x-y)\phi_{\nu}(y)dy\\\
&=(1-\chi(||x||))g_{i,\gamma}(x)+\chi(||x||)\int_{B^{m}}g_{i,\gamma}(x-\nu
y)\phi(y)dy.\end{split}$ (1.30)

Then, using this expression of $G_{\nu}$, it is also possible to consider
$\nu=0$.

######  Remark 7.

In particular following the proof of Lemma 2.34 of[4],we obtain that if $g$ is
a lipschitz function of constant $C_{g}$ then

$d(g(p),G_{\nu}(p))\leq C_{g}\nu.$ (1.31)

We are ready to prove the existence of a $C^{k}_{b}$-approximation of a
lipschitz map.

######  Proposition 1.2.3.

Consider two Riemannian manifolds $(M,g)$ and $(N,h)$ of bounded geometry and
let $f:(M,g)\longrightarrow(N,h)$ be a lipschitz map. Fix $\nu>0$. Then there
is map $F:(M,g)\longrightarrow(N,h)$ such that

  * •

$d(F(p),f(p))\leq K\nu$ where $K$ is a constant which depends on the lipschitz
constant $C_{f}$ and by the uniform covering we choose to define $F$,

  * •

for all $l\geq 1$ the approximation $F$ is a $C^{l}_{b,u}\cap C^{\infty}$-map,

  * •

Consider $\Gamma$ a group that acts FUPD by isometries on $M$ and $N$. Assume
that $f$ is $\Gamma$-equivariant. Then also $F$ is $\Gamma$-equivariant.

  * •

$f\sim_{\Gamma}F$.

###### Proof.

Observe that since $f$ is lipschitz, then in particular is uniformly
continuous. Then there are two positive numbers $\sigma_{1}<\sigma_{2}$ both
$N$-small and there is a $M$-small number $\delta_{f}$ such that
$f(B_{\delta_{f}}(q)\subseteq B_{\sigma_{1}}(f(q))$.
Let us consider the Riemannian manifold $X:=\frac{M}{\Gamma}$. Let us define

$R_{0}:=\frac{1}{4}min\\{\delta_{f},r_{inj}(X)\\}.$ (1.32)

Following the Remark 1.1.5, we can see that $R_{0}\leq inj_{M}$ and
$R_{0}\leq\delta_{0}$ where $\delta_{0}$ is the constant given by the FUPD
action of $\Gamma$. Fix, then, $\delta_{1}<\delta_{2}<R_{0}$ where
$\delta_{2}$ is $X$-small.
So, applying Proposition 1.2.1, we know that there is a number $K$ and a
countable cover of $X$ given by $\\{B_{\delta_{1}}(x_{i})\\}$ such that for
all $x$ in $X$ we have that $B_{\delta_{2}}(x)$ intersect at most $K$ of
$\\{B_{\delta_{2}}(x_{i})\\}_{i}$.
Consider the preimage of this cover on $M$. Since $\delta_{2}<\delta_{0}$, it
has the form $\\{\bigsqcup_{\gamma\in\Gamma}B_{\delta_{2}}(\gamma
x_{i})\\}_{i}$, where $\gamma x_{i}$ is an element of the fiber of $x_{i}$.
Moreover we can observe that $f(B_{\delta_{2}}(q))\subseteq
B_{\sigma_{1}}(f(q))$ for all $q$ in $M$.
Now, let us define $F_{0}:=f$ and for all $i$ in $\mathbb{N}_{>0}$ we have

$F_{i+1}(p):=\begin{cases}\exp_{f(\gamma x_{i})}\circ G_{i,\gamma,\nu}\circ
exp_{\gamma x_{i}}^{-1}\mbox{ if }p\in B_{R_{0}}(\bigcup_{\gamma}\\{\gamma
x_{i}\\})\\\ F_{i}(p)\mbox{ otherwise}\end{cases}$ (1.33)

where $G_{i,\gamma,\nu}$ is the $C^{k}_{b}$-approximation of
$g_{i,\gamma}=\exp_{f(\gamma x_{i})}^{-1}\circ F_{i}\circ exp_{\gamma x_{i}}$
using $\nu\leq(\frac{\sigma_{2}-\sigma_{1}}{4K})$. Observe that $F_{i}$ is
well-defined for all $i$ using the gluing lemma.
Finally we can define

$F(p):=\lim_{i\rightarrow+\infty}F_{i}(p).$ (1.34)

Let us check all the properties of $F$:

  * •

$F$ is well-defined: consider $p$ a point in $M$. Let $s$ be the covering
$s:M\longrightarrow X$, then $s(B_{\delta_{2}}(p))$ intersect at most $K$ ball
$B_{\delta_{2}}(x_{i})$. Moreover we have that, since
$\delta_{2}<\frac{1}{4}\delta$, then for all of these $x_{i}$ there exists
only a $\gamma$ in $\Gamma$ such that $B_{\delta_{2}}(p)$ intersects
$B_{\delta_{2}}(\gamma x_{i})$.
Then for all $p$ there are at most $K$ indexes $i_{j}$ such that
$F_{i_{j}}(p)\neq F_{i_{j}+1}(p)$ and so the limit exists since the sequence
$\\{F_{i}(p)\\}_{i\geq i_{K}+1}$ is constant.

  * •

We have that

$d(F(p),f(p))\leq C_{f}K\nu.$ (1.35)

Indeed, applying the Proposition 1.2.2 and Remark 7, we have that

$d(f(p),F(p))\leq
d(f(p),F_{i_{1}+1}(p))+...+d(F_{i_{K}}(p),F_{i_{K}+1}(p))\leq K\cdot C_{f}\nu$
(1.36)

  * •

$F$ is a $C^{k}_{bu}$-map. Consider a point $p$. As consequence of Proposition
1.2.1, there is a neighborhood $U$ of $p$ and there is a finite sequence
$x_{i_{1}},...,x_{i_{k}}$ of the $x_{i}$s such that $F_{i+1}\neq F_{i}$ on
$U$. So, we have to study the boundedness of the derivatives of

$\begin{split}F_{p}:&=exp^{-1}_{F_{i_{k}+1}(p)}\circ F_{i_{k}+1}\circ
exp_{p}\\\ &=exp^{-1}_{F_{i_{k}+1}(p)}\circ\exp_{f(\gamma x_{i})}\circ
G_{i_{k},\gamma,\nu}\circ exp_{\gamma x_{i}}^{-1}\circ exp_{p}.\end{split}$
(1.37)

Now, we have that $exp^{-1}_{F_{i_{k}+1}(p)}\circ\exp_{f(\gamma x_{i})}$ and
$exp_{\gamma x_{i}}^{-1}\circ exp_{p}$ are just a change of normal
coordinates, and so they are $C^{k}_{bu}$-maps as consequence of Proposition
1.1.4. Then it is sufficient to study $G_{i_{k},\gamma,\nu}$. Applying the
Proposition 1.2.2, we have that

$|G_{i_{k},\gamma,\nu}|_{k}\leq C(\nu,k)|G_{i_{k-1},\gamma,\nu}|_{0}\leq
C(\nu,k)\sigma_{2}.$ (1.38)

  * •

$F$ is $\Gamma$-equivariant. In order to prove this we have to check that all
the $F_{i}$s are $\Gamma$-equivariant. Observe that $F_{0}=f$ is
$\Gamma$-equivariant. Moreover if $F_{i}$ is $\Gamma$-equivariant, then also
$F_{i+1}$ is $\Gamma$-equivariant. Then we conclude observing that

$\gamma F(p)=\gamma F_{i_{k}+1}(p)=F_{i_{k}+1}(\gamma p)=F(\gamma p).$ (1.39)

In order to conclude the proof we have to show that $F$ and $f$ are
$\Gamma$-lipschitz-homotopic. Let us define for each $\epsilon\in[0,1]$ the
map

$H_{i,\gamma,\epsilon}:=G_{i,\gamma,\epsilon\nu}$ (1.40)

where $G_{i,\gamma,\epsilon\nu}$ is the $C^{k}_{bu}$-approximation of
$g_{i,\gamma}=\exp_{f(\gamma x_{i})}^{-1}\circ F_{i}\circ exp_{\gamma x_{i}}$
using $\epsilon\nu$.
Let us define the map

$\begin{split}H_{0}:(M\times[0,1],g+g_{[0,1]})&\longrightarrow(N,h)\\\
(p,\epsilon)\longrightarrow f(p)\end{split}$ (1.41)

and

$H_{i+1}(p,\epsilon):=\begin{cases}\exp_{f(\gamma x_{i})}\circ
H_{i,\gamma,\epsilon}\circ exp_{\gamma x_{i}}^{-1}\mbox{ if }p\in
B_{r_{inj}(M)}(\bigcup_{\gamma}\\{\gamma x_{i}\\})\\\ H_{i}(p,\epsilon)\mbox{
otherwise}\end{cases}$ (1.42)

Then we define the map $H:(M\times[0,1],g+g_{[0,1]})\longrightarrow(N,h)$ as

$H(p,\epsilon)=\lim_{i\rightarrow+\infty}H_{i}(p,\epsilon).$ (1.43)

Let us suppose that if $H_{i}$ lipschitz implies $H_{i+1}$ is lipschitz with a
constant less or equal to $C\cdot C_{i}$ where $C_{i}$ is the costant of
$H_{i}$. Then we will have that the constant of $H$ will be $C^{K}C_{f}$.

To prove that if $H_{i-1}$ is lipschitz then $H_{i}$ is lipschitz it is
sufficient to prove that $H_{i}$ is lipschitz on $B_{\delta_{2}}(\gamma
x_{i})$ for all $\gamma$ in $\Gamma$. Moreover we also know that the
exponential map are lipschtiz with constant $C_{\sigma_{2}}$ and
$C_{\delta_{2}}$ on $B_{\delta_{2}}(\gamma x_{i})$ and on
$B_{\sigma_{2}}(f(\gamma x_{i}))$. This means that we can study
$H_{i,\gamma,\epsilon}$.
Let us consider

$d(H_{i,\gamma,\epsilon_{1}}(x_{1}),H_{i,\gamma,\epsilon_{2}}(x_{2}))\leq
d(H_{i,\gamma,\epsilon_{1}}(x_{1}),H_{i,\gamma,\epsilon_{1}}(x_{2}))+d(H_{i,\gamma,\epsilon_{1}}(x_{2}),H_{i,\gamma,\epsilon_{2}}(x_{2}))$
(1.44)

One have to observe that $g_{i,\gamma}$ is a lipschitz map with costant less
or equal to $C_{\delta_{2}}\cdot C_{\sigma_{2}}\cdot C_{F_{i}}$.
Then we have

$\begin{split}d(H_{i,\gamma,\epsilon_{1}}(x_{1}),H_{i,\gamma,\epsilon_{1}}(x_{2}))&\leq\int_{B^{m}}|g_{i,\gamma}(x_{2}-\epsilon_{1}y)-g_{i,\gamma}(x_{2}-\epsilon_{1}y)|\phi(y)dy\\\
&\leq C_{\delta_{2}}\cdot C_{\sigma_{2}}\cdot C_{F_{i}}|x_{1}-x_{2}|\\\ &\leq
C_{\delta_{2}}^{2}\cdot C_{\sigma_{2}}\cdot
C_{F_{i}}d((x_{1},\epsilon_{1}),(x_{2},\epsilon_{1})).\end{split}$ (1.45)

and, moreover

$\begin{split}d(H_{i,\gamma,\epsilon_{1}}(x_{1}),H_{i,\gamma,\epsilon_{2}}(x_{1}))&\leq\int_{B^{m}}|g_{i,\gamma}(x_{2}-\epsilon_{1}y)-g_{i,\gamma}(x_{2}-\epsilon_{2}y)|\phi(y)dy\\\
&\leq C_{\delta_{2}}\cdot C_{\sigma_{2}}\cdot
C_{F_{i}}||x_{2}-\epsilon_{1}y-x_{2}+\epsilon_{2}y|\\\ &\leq
C_{\delta_{2}}\cdot C_{\sigma_{2}}\cdot
C_{F_{i}}|\epsilon_{1}-\epsilon_{2}|\\\ &\leq C_{\delta_{2}}\cdot
C_{\sigma_{2}}\cdot
C_{F_{i}}d((x_{1},\epsilon_{1}),(x_{2},\epsilon_{2})).\end{split}$ (1.46)

Then, we have that $H$ is a lipschitz map and so $f$ and $F$ are lipschitz-
homotopic. The $\Gamma$-equivariance of $H$ follows exactly as for $F$. ∎

######  Corollary 1.2.4.

Consider a lipschitz map $f:(M,g)\longrightarrow(N,h)$ between manifolds of
bounded geometry. Let us suppose that there is a closed set $C$ such that
$f_{|_{M\setminus C}}$ is a $C^{k}_{bu}$-map. Then for all $\epsilon>0$ there
is a map $F:(M,g)\longrightarrow(N,h)$ such that $F$ is a $C^{k}_{bu}$-map and
if $C_{\epsilon}$ is the $\epsilon$-neighborhood of $C$, then

$F_{|_{M\setminus C_{\epsilon}}}=f_{|_{M\setminus C_{\epsilon}}}.$ (1.47)

Moreover if $C$ is $\Gamma$-invariant and $f$ is $\Gamma$-equivariant then
also $F$ is $\Gamma$-equivariant and they are $\Gamma$-lipschitz homotopic.

###### Proof.

The proof is exactly the same of the previous proposition. Indeed it is
sufficient to impose

$\epsilon=2\delta_{2},$ (1.48)

and to define the $F_{i}$s and the $H_{i}$s only on the
$\bigsqcup\limits_{\gamma\in\Gamma}B_{\delta_{2}}(\gamma x_{i})$ that
intersect $C$. ∎

######  Lemma 1.2.5.

Let $f,f^{\prime}:(M,g)\longrightarrow(N,h)$ be two $C^{k}_{b}$ lipschitz-
homotopy equivalences such that $f\sim_{\Gamma}f^{\prime}$, where $\Gamma$
acts FUPD on $M$ and $N$. Let us fix some normal coordinates $\\{x^{i},s\\}$
on $M\times[0,1]$ and $\\{y^{j}\\}$ on $N$. Then there is a lipschitz homotopy
$H:(M\times[0,1],g+g_{eucl})\longrightarrow(N,h)$ such that all its
derivatives in normal coordinates of order minus or equal of $k$ are uniformly
bounded.

###### Proof.

Let us denote by $h$ the homotopy between $f$ and $f^{\prime}$. Then we can
define the lipschitz map
$H_{0}:(M\times\mathbb{R},g+g_{0,1})\longrightarrow(N,h)$ as

$H_{0}(p,s):=\begin{cases}f(p)\mbox{ if }s\leq\frac{1}{4}\\\
h(p,\frac{1}{2}(t+\frac{1}{2}))\mbox{ if }s\in[\frac{1}{4},\frac{3}{4}]\\\
f^{\prime}(p)\mbox{ if }s\geq\frac{3}{4}.\end{cases}$ (1.49)

Let us consider the action of $\Gamma$ on $M\times\mathbb{R}$ as the action
$\gamma(p,t):=(\gamma(p),t)$. Then we can consider the $\Gamma$-equivariant
closed subset given by $M\times[\frac{3}{8},\frac{5}{8}]$. Since
$M\times[0,1]$ is a manifold of bounded geometry, we can apply the Corollary
1.2.4, and we obtain a map $\tilde{H}$ which is a $C^{k}_{b}$-map such that
$\tilde{H}(p,0)=f(p)$ and $\tilde{H}(p,1)=f^{\prime}(p)$. Then it is
sufficient to define

$H:=\tilde{H}_{|_{M\times[0,1]}}.$ (1.50)

∎

######  Corollary 1.2.6.

Let $H:M\times[0,1]\longrightarrow N$ be the homotopy in the Lemma 1.2.5. If
$p_{H}:f^{*}(T^{\delta}N)\times[0,1]\longrightarrow N$ is the submersion
related to $H$ then $p_{H}$ has, in normal coordinates, the derivatives of
order $k$ uniformly bounded.

###### Proof.

It is a consequence of Proposition 1.4.1. ∎

###  1.3 $L^{2}$-cohomology and pull-back

####  1.3.1 $\mathcal{L}^{2}$-forms

Let us consider an orientable Riemannian manifold $(M,g)$ and let us denote by
$\Omega^{k}_{c}(M)$ the space of complex differential forms with compact
support. The Riemannian metric $g$ induces for every $k\in\mathbb{N}$ a scalar
product on $\Omega^{k}_{c}(M)$ as follows: consider $\alpha$ and $\beta$ in
$\Omega^{*}_{c}(M)$ then

$<\alpha,\beta>_{\mathcal{L}^{2}(M)}:=\int\limits_{M}\alpha\wedge\star\bar{\beta},$
(1.51)

where $\star$ is the Hodge star operator given by $g$.
This hermitian product gives a norm on $\Omega^{k}_{c}(M)$:

$\begin{split}\|\alpha\|^{2}_{\mathcal{L}^{2}(M)}:&=<\alpha,\alpha>_{\mathcal{L}^{2}(M)}\\\
&=\int\limits_{M}\alpha\wedge\star\bar{\alpha}<+\infty.\end{split}$ (1.52)

######  Definition 1.3.1.

We will denote by $\mathcal{L}^{2}\Omega^{k}(M)$ the Hilbert space given by
the closure of $\Omega^{*}_{c}(M)$ respect the norm
$|\cdot|_{\mathcal{L}^{2}\Omega^{k}(M)}$. Moreover we can also define the
Hilbert space $\mathcal{L}^{2}(M)$ given by

$\mathcal{L}^{2}(M):=\bigoplus\limits_{k\in\mathbb{N}}\mathcal{L}^{2}\Omega^{k}(M).$
(1.53)

The norm of $\mathcal{L}^{2}(M)$ it will be denoted by
$|\cdot|_{\mathcal{L}^{2}}$ or $|\cdot|_{\mathcal{L}^{2}(M)}$.

######  Remark 8.

Since $\Omega_{c}^{k}(M)$ is dense in $\mathcal{L}^{2}\Omega^{k}(M)$, then
$\Omega_{c}^{*}(M)$ is dense in $\mathcal{L}^{2}(M)$.

######  Definition 1.3.2.

Consider an orientable Riemannian manifold $(M,g)$ with $dim(M)=m$. Then, the
chirality operator
$\tau_{M}:\mathcal{L}^{2}(M)\longrightarrow\mathcal{L}^{2}(M)$ is the operator
defined for each $\alpha$ in $\Omega_{c}^{*}(M)$ as

$\tau_{M}(\alpha):=i^{\frac{m}{2}}\star\alpha$ (1.54)

if $m$ is even and

$\tau_{M}(\alpha):=i^{\frac{m+1}{2}}\star\alpha$ (1.55)

if $m$ is odd.

######  Remark 9.

The chirality operator is a bounded operator. In particular, respect to
$||\cdot||_{\mathcal{L}^{2}}$ it is an isometry and an involution and so we
obtain that

$\tau_{M}^{2}=1$ (1.56)

and

$||\tau_{M}||=1.$ (1.57)

####  1.3.2 Exterior derivative

Let us consider two orientable Riemannian manifolds $(M,g)$ and $(N,h)$ and
let $F$ be a linear operator
$F:dom(F)\subseteq\mathcal{L}^{2}(M)\longrightarrow\mathcal{L}^{2}(N)$.

######  Definition 1.3.3.

The operator $F$ is $\mathcal{L}^{2}$-bounded if $F$ is a bounded operator
respect to the $\mathcal{L}^{2}$-norms.
Moreover we say that $F$ is a closed operator if for every $\alpha$ in
$dom(F)$ and for each sequence $\\{\alpha_{n}\\}$ converging to $\alpha$ there
is a $\beta$ in $\mathcal{L}^{2}(N)$ such that

$\lim\limits_{n\rightarrow+\infty}F(\alpha_{n})=\beta\in\mathcal{L}^{2}(N)$
(1.58)

and, moreover, $F(\alpha)=\beta$.

######  Remark 10.

Every bounded operator is closed. In particular we have that $F$ is a closed
operator if and only if its graph is a closed subset of
$\mathcal{L}^{2}(M)\times\mathcal{L}^{2}(N)$.

Consider now a linear operator
$F:\Omega^{*}_{c}(M)\longrightarrow\Omega^{*}_{c}(N)$. Since
$\Omega_{c}^{*}(M)$ is dense in $\mathcal{L}^{2}(M)$, if $F$ is
$\mathcal{L}^{2}$-bounded, then there is a unique extension
$\overline{F}:\mathcal{L}^{2}(M)\longrightarrow\mathcal{L}^{2}(N)$ of $F$ and
it is bounded.
On the other hand, if $F$ is unbounded, then there could be some different
ways to extend $F$.
Let us denote by $d:\Omega_{c}^{*}(M)\longrightarrow\Omega_{c}^{*}(M)$ the
operator given by the exterior derivative of compactly supported complex
differential forms. We introduce some closed extensions i.e. some extensions
$\overline{d}$ of $d$ such that $\overline{d}$ is a closed operator.

######  Definition 1.3.4.

The maximal extension of $d$ is the operator
$d_{max}:dom(d_{max})\subseteq\mathcal{L}^{2}(M)\longrightarrow\mathcal{L}^{2}(N)$
defined on

$\begin{split}dom(d_{max}):=\\{&\alpha\in\mathcal{L}^{2}(M)|\exists\alpha^{\prime}\in\mathcal{L}^{2}(N)s.t.\forall\beta\in\Omega_{c}^{*}(N)\\\
&\langle\alpha^{\prime},d(\beta)\rangle_{\mathcal{L}^{2}(\Omega^{*}(M))}=\langle\alpha,\beta\rangle_{\mathcal{L}^{2}(N)}\\}\end{split}$
(1.59)

posing

$d_{max}(\alpha)=\alpha^{\prime}.$ (1.60)

Moreover we also have

######  Definition 1.3.5.

The minimal extension of $d$ is the operator
$d_{min}:dom(d_{min})\subseteq\mathcal{L}^{2}(M)\longrightarrow\mathcal{L}^{2}(N)$
defined on

$\begin{split}dom(d_{min}):=\\{&\alpha\in\mathcal{L}^{2}(M)|\exists\\{\alpha_{n}\\}\in\Omega_{c}^{*}(N)\mbox{
and}\exists y\in\mathcal{L}^{2}(N)s.t.\\\
&\lim\limits_{n\rightarrow+\infty}\alpha_{n}=\alpha\mbox{
and}\lim\limits_{n\rightarrow+\infty}d\alpha_{n}=y\\}\end{split}$ (1.61)

as

$d_{min}(\alpha)=y.$ (1.62)

Given two operators
$F:dom(F)\subseteq\mathcal{L}^{2}(M)\longrightarrow\mathcal{L}^{2}(N)$ and
$G:dom(G)\subseteq\mathcal{L}^{2}(M)\longrightarrow\mathcal{L}^{2}(N)$ we will
say that

$F\subseteq G\iff Graph(F)\subseteq Graph(G)$ (1.63)

. We can observe that

$d\subseteq d_{min}\subseteq d_{max}.$ (1.64)

Moreover every closed extension $\overline{d}$ is such that

$d_{min}\subseteq\overline{d}\subseteq d_{max}.$ (1.65)

However if the Riemannian manifold $(M,g)$ is complete then the following
proposition of [6] holds.

######  Proposition 1.3.1.

Let $(M,g)$ a complete Riemannian manifold. Then the maximal and the minimal
domain of the exterior derivative coincide and so

$d_{max}=d_{min}.$ (1.66)

As consequence of this Proposition we also have that there is a unique closed
extension for $d^{*}:\Omega_{c}^{*}(M)\longrightarrow\Omega_{c}^{*}(M)$, the
adjoint of $d$ and also for the operator
$d+d^{*}:\Omega_{c}^{*}(M)\longrightarrow\Omega_{c}^{*}(M)$.
When $(M,g)$ is a manifold of bounded geometry, in particular is a complete
Riemannian manifold and so, using an abuse of notation, we denote the closed
extensions of these operators by $d$, $d^{*}$, $d+d^{*}$.

####  1.3.3 $L^{2}$-cohomology and reduced $L^{2}$-cohomology

Let us consider a closed extension $\overline{d}$ of the exterior derivative
on an orientable Riemannian manifold $(M,g)$. Let us suppose that
$\overline{d}(dom(\overline{d}))\subseteq ker(\overline{d})\subseteq
dom(\overline{d})$, and so $\overline{d}\circ\overline{d}$ is well defined and
equals to zero. In particular, this condition is satisfied for each closed
extension $\overline{d}$ if $(M,g)$ is a complete Riemannian manifold.
Let us consider now the sequence

$0\longrightarrow\mathcal{L}^{2}(\Omega^{1}(M))\xrightarrow{\overline{d}}\mathcal{L}^{2}(\Omega^{2}(M))\xrightarrow{\overline{d}}\mathcal{L}^{2}(\Omega^{3}(M))\xrightarrow{\overline{d}}...$
(1.67)

It is, following the definition given in [2], an Hilbert complex.

######  Definition 1.3.6.

We will define $i$-th group of $L^{2}$-cohomology  the group

$H^{i}_{2,\overline{d}}(M):=\frac{ker(\overline{d}_{i})}{im(\overline{d}_{i-1})}.$
(1.68)

######  Definition 1.3.7.

We will define $i$-th group of reduced $L^{2}$-cohomology  the group

$\overline{H}^{i}_{2,\overline{d}}(M):=\frac{ker(\overline{d}_{i})}{\overline{im(\overline{d}_{i-1})}}.$
(1.69)

######  Remark 11.

If $(M,g)$ is a complete manifold, since [6], the (reduced or not)
$L^{2}$-cohomology is uniquely defined. In this case we denote the groups as
$H^{i}_{2}(M)$ and $\overline{H}^{i}_{2}(M)$.

######  Proposition 1.3.2.

Let $(N,h)$ and $(M,g)$ be two orientable Riemannian manifolds. Let $B$ and
$K:\mathcal{L}^{2}(N)\longrightarrow\mathcal{L}^{2}(M)$ be two
$\mathcal{L}^{2}$-bounded operators. Let us suppose that
$B(\Omega^{*}_{c}(N))\subseteq\Omega^{*}_{c}(M)$ and

$dB=\pm Bd+K$ (1.70)

over $\Omega^{*}_{c}(N)$. Then we have that

$B(dom(d_{min}))\subseteq dom(d_{min})$ (1.71)

and $d_{min}B=\pm Bd_{min}+K$ on the minimal domain of $d$.

###### Proof.

Let $\alpha$ be an element in $dom(d_{min})$. This means that there is a
sequence $\\{\alpha_{n}\\}$ in $\Omega^{*}_{c}(N)$ such that

$\begin{cases}&\alpha=\lim\limits_{n\rightarrow+\infty}\alpha_{n}\\\
&d\alpha=\lim\limits_{n\rightarrow+\infty}d\alpha_{n}.\end{cases}$ (1.72)

Using that $B$ is continuous we have

$B\alpha=\lim\limits_{n\rightarrow+\infty}B\alpha_{n}$ (1.73)

where $\\{B\alpha_{n}\\}$ is a sequence in $\Omega^{*}_{c}(M)$. Moreover one
can see that the limit of $dB\alpha_{n}$ exists:

$\begin{split}\lim\limits_{n\rightarrow+\infty}dB\alpha_{n}&=\lim\limits_{n\rightarrow+\infty}\pm
Bd\alpha_{n}+K\alpha_{n}\\\ &=\pm
B\lim\limits_{n\rightarrow+\infty}d\alpha_{n}+K\lim\limits_{n\rightarrow+\infty}\alpha_{n}\\\
&=\pm Bd\alpha+K\alpha.\end{split}$ (1.74)

So $d_{min}$ is well defined in $B\alpha$ and

$d_{min}B\alpha=\pm Bd_{min}\alpha+K\alpha.$ (1.75)

∎

######  Remark 12.

Let us suppose $(M,g)$ and $(N,h)$ are complete, orientable, Riemannian
manifolds. In general an operator
$A:dom(A)\subseteq\mathcal{L}^{2}(N)\longrightarrow\mathcal{L}^{2}(M)$ induces
a map in $L^{2}$-cohomology if

  * •

$dom(d_{N})\subseteq dom(A)$,

  * •

$A(dom(d_{N}))\subseteq dom(d_{M})$,

  * •

$A$ and $d$ commute.

Moreover if we also want that $A$ induces a map in reduced $L^{2}$-cohomology,
we need that $A$ is $\mathcal{L}^{2}$-bounded on $\overline{im(d)}$. Indeed,
given $\alpha$ in $dom(d_{N})$, we have that

$\begin{split}A(\alpha+\lim\limits_{k\rightarrow+\infty}d\beta_{k})&=A(\alpha)+A(\lim\limits_{k\rightarrow+\infty}d\beta_{k})\\\
&=A(\alpha)+\lim\limits_{k\rightarrow+\infty}A(d\beta_{k})\\\
&=A(\alpha)+\lim\limits_{k\rightarrow+\infty}dA(\beta_{k})\end{split}$ (1.76)

and so, in reduced $L^{2}$-cohomology,

$[A(\alpha+\lim\limits_{k\rightarrow+\infty}d\beta_{k})]=[A(\alpha)].$ (1.77)

######  Corollary 1.3.3.

Consider two operators $B$ and $K$ as in Proposition 1.3.2. Let us suppose
that $K=0$ and that

$dB=Bd$ (1.78)

on $\Omega_{c}^{*}(M)$. Then $B$ induces a map in (un)-reduced
$L^{2}$-cohomology.444We are considering $\overline{d}=d_{min}$.

####  1.3.4 Problems with pullback

Let $f:(M,g)\longrightarrow(N,h)$ be a map between orientable Riemannian
manifolds. In general the pullback $f^{*}$ is not an $\mathcal{L}^{2}$-bounded
operator and it doesn’t induce an operator in $L^{2}$-cohomology (reduced or
not). In this section we will see some examples.

######  Example 1.3.1.

In this first example we will show that the pullback of a map
$p:(E,g)\longrightarrow(N,h)$ between complete Riemannian manifolds is not a
$\mathcal{L}^{2}$-bounded operator even if $p$ is a proper submersion. Let us
consider $(N,h)=(\mathbb{R},g_{eucl})$, $E=(\mathbb{R}\times S^{1},g)$ where

$g(x,\theta)=dx^{2}+e^{\frac{x^{2}}{2}}d\theta^{2}.$ (1.79)

Consider $p:(E,g)\longrightarrow(N,h)$ given by the projection
$(x,y)\longrightarrow x$. One can observe that $p$ is a proper submersion, but
$p^{*}$ is not $\mathcal{L}^{2}$-bounded.
Indeed if $\alpha=e^{-\frac{x^{2}}{2}}dx$ in $\mathcal{L}^{2}(N)$, then

$\begin{split}\|p^{*}(\alpha)\|_{\mathcal{L}^{2}(E)}&=\int\limits_{\mathbb{R}}\int\limits_{S^{1}}e^{-x^{2}}Vol_{\mathbb{R}\times
S^{1}}\\\ &=\int\limits_{\mathbb{R}}\int_{S^{1}}e^{-x^{2}}e^{x^{2}}d\theta
dx\\\ &=\int\limits_{\mathbb{R}}2\pi
e^{-x^{2}}e^{x^{2}}dx=+\infty,\end{split}$ (1.80)

and so $p^{*}(\mathcal{L}^{2}(N))$ is not in $\mathcal{L}^{2}(E)$.

The pullback is not a bounded operator between the $\mathcal{L}^{2}$-spaces
even if $(M,g)$ and $(N,h)$ are manifolds of bounded geometry and $f$ is a
smooth lipschitz-homotopy equivalence.

######  Example 1.3.2.

Let us consider $\alpha:\mathbb{R}\longrightarrow\mathbb{R}$ defined by

$\alpha(x)=\begin{cases}\alpha_{0}e^{-\frac{1}{\tan^{2}(x)}}\mbox{ if
$x\in[-1,1]$}\\\ |x|\mbox{ otherwise}\end{cases}$ (1.81)

where

$\alpha_{0}:=e^{-\frac{1}{\tan^{2}(1)}}.$ (1.82)

Consider the map $\psi:\mathbb{R}\longrightarrow\mathbb{R}$

$\beta(x)=\begin{cases}\alpha(x)\mbox{ if $x\geq 0$}\\\ -\alpha(x)\mbox{
otherwise}.\end{cases}$ (1.83)

Observe that for each $\epsilon>0$ we have that $\beta$ is a smooth map
outside
$C_{\epsilon}:=[-1-\epsilon,-1+\epsilon]\sqcup[1-\epsilon,1+\epsilon]$. Fix
$\epsilon\leq\frac{1}{2}$. Consider $\psi$ a smooth approximation of $\beta$
which is equal to $\beta$ outside $C_{\epsilon}$. We know that such a
approximation exists since Corollary 1.2.4.
We have that $\psi$ is also lipschitz-homotopic to $id_{\mathbb{R}}$, indeed
the homotopy $h:\mathbb{R}\longrightarrow\mathbb{R}$

$h(x,t):=\psi(x)\cdot t+(1-t)\cdot x$ (1.84)

is lipschitz. We have that $\psi$ is smooth, indeed it is clearly smooth if
$x\neq 0$ and

$\frac{\partial^{k}}{\partial^{k}x}\psi(0)=0,$ (1.85)

for each $k\geq 1$. Moreover $\psi$ is lipschitz-homotopic to the identity:
the homotopy is $h:\mathbb{R}\times[0,1]\longrightarrow\mathbb{R}$ defined as
follow

$h(x,s)=s\psi(x)+(1-s)x.$ (1.86)

Consider now

$f(x,y)=\begin{cases}(x,\psi(y-1))\mbox{ if $y\geq 1$}\\\ (x,\psi(y+1))\mbox{
if $y\leq-1$}\\\ (x,0)\mbox{ otherwise}.\end{cases}$ (1.87)

We have that $f$ is a uniformly proper and lipschitz map. In particular it is
smooth since

$\frac{\partial^{k}}{\partial^{k}x}\psi(0)=\frac{\partial^{k}}{\partial^{k}x}\beta(0)=0.$
(1.88)

Finally $f$ is lipschitz-homotopic to the identity: indeed the homotopy

$H_{1}(x,y,s)=\begin{cases}(x,\psi(y-s))\mbox{ if $y\geq s$}\\\
(x,\psi(y+s))\mbox{ if $y\leq-s$}\\\ (x,0)\mbox{ otherwise}.\end{cases}$
(1.89)

is a lipschitz homotopy between $f$ and $(id_{\mathbb{R}},\psi)$ and
$H_{2}(x,y,s):=(x,h(y,s))$ is a lipschitz homotopy between
$(id_{\mathbb{R}},\psi)$ and $id_{\mathbb{R}^{2}}$.

Let us consider the $0$-form given by $\chi_{A}$, where

$A:=\\{(x,y)\in R^{2}|y\in[-\frac{e^{-x^{2}}}{2},+\frac{e^{-x^{2}}}{2}]\\}.$
(1.90)

We have that $\chi_{A}$ is in $\mathcal{L}^{2}(\mathbb{R}^{2})$, but
$f^{*}(\chi_{A})$ is not squared-integrable.

Finally we will show that given a lipschitz-homotopy equivalence $\phi$
between manifolds of bounded geometry, then $\phi^{*}(dom(d))\nsubseteq
dom(d)$ and so $\phi^{*}$ doesn’t induce a morphism on $L^{2}$-cohomology
(reduced or not).

######  Example 1.3.3.

Let us consider
$\phi:(\mathbb{R}^{4},g_{eucl})\longrightarrow(\mathbb{R}^{4},g_{eucl})$ such
that $\phi(x,y,z,w)=(\sigma(x),\sigma(y),\sigma(z),w)$ where

$\sigma(t):=\begin{cases}0\mbox{ if }|t|\leq 1\\\ \psi(t-1)\mbox{ if
}t\leq-1\\\ \psi(t+1)\mbox{ if }t\geq 1.\end{cases}$ (1.91)

where $\psi$ is the map defined in Example 1.3.2. Since the derivatives of
each order of $\psi$ in $0$ are null, then $\sigma$ is a smooth map. So $\phi$
is a smooth map.
Consider a compactly supported $0$-form $\alpha$ such that around the origin
is $\frac{1}{|(x,y,z,w)|^{k}}$ where $\frac{1}{2}<k<2$, then $\alpha$ is in
$dom(d)$, indeed we know that $\alpha$ is in the Sobolev space $W^{1,2}$.
On the other hand $\phi^{*}\alpha$ is not an $\mathcal{L}^{2}$-form, indeed we
have that on $[-1,1]\times[-1,1]\times[-1,1]\times U$, where $U$ is a
sufficiently small neighborhood of $0$ in $\mathbb{R}$, we have that

$\phi^{*}\alpha(x,y,z,w)=\frac{1}{w^{k}}$ (1.92)

and so

$\begin{split}\int_{\mathbb{R}^{4}}(\phi^{*}\alpha)^{2}d\mu_{\mathbb{R}^{4}}&\geq\int_{[-1,1]^{3}}\int_{U}\frac{1}{w^{2k}}d\mu_{\mathbb{R}}d\mu_{\mathbb{R}^{3}}\\\
&\geq 2^{3}\cdot\int_{U}\frac{1}{w^{2k}}d\mu_{\mathbb{R}}\\\
&=+\infty,\end{split}$ (1.93)

since $2k>1$.
Observe that $\phi$ is a lipschitz-homotopy equivalence because there is a
lipschitz homotopy $H$ between $\phi$ and the identity. This homotopy is
defined as follow

$\begin{split}H:(\mathbb{R}^{4}\times[0,1],g_{eucl}\times
g_{[0,1]})&\longrightarrow(\mathbb{R}^{4},g_{eucl})\\\
(x,y,z,w,t)&\longrightarrow(h(x,t),h(y,t),h(z,t),w).\end{split}$ (1.94)

where $h$ is defined in the Example 1.3.2.

####  1.3.5 Radon-Nicodym derivative and Fiber Volume

In this section we will introduce some tools of Measure Theory, that we will
use to define a family of maps whose pullbacks are $\mathcal{L}^{2}$-bounded
operators.

######  Definition 1.3.8.

Let $X$ be a set with a $\sigma$-algebra $\Sigma$ (i.e. a measurable set). Let
$\mu$ and $\nu$ two measures respect $\Sigma$. We will say that $\nu$ is
abolutely continuous with respect to $\mu$ if and only if for all measurable
$A$

$\mu(A)=0\implies\nu(A)=0.$ (1.95)

We will denote it by $\nu<<\mu$.

######  Definition 1.3.9.

A measure $\mu$ over a measurable set $(X,\Sigma)$ is said $\sigma$-finite if
$X$ is the countable union of substsets with finite measure.

######  Remark 13.

The measure over a Riemannian manifold given by its metric is always
$\sigma$-finite. It follows from para-compactness.

######  Theorem 1.3.4 (Radon-Nyikodym).

Let $(X,\Sigma)$ be a meausurable set. Given two measures $\mu$ and $\nu$ such
that $\nu<<\mu$ and $\mu$ is $\sigma$-finite, then there is a non-negative
$\mu$-measurable function $f:X\longrightarrow\mathbb{R}$ such that

$\nu(A)=\int_{A}fd\mu.$ (1.96)

We will call $f$ the Radon-Nicodym derivative of $\nu$ respect to $\mu$ and we
will denote it by $\frac{\partial\nu}{\partial\mu}$.

######  Remark 14.

If $g$ is an integrable function respect to $\nu$, then

$\int_{M}gd\nu=\int_{M}gfd\mu.$ (1.97)

######  Remark 15.

The Radon-Nicodym derivative is unique (up to $0$-measure sets).

######  Remark 16.

Let us consider on a measurable space $(X,\Sigma)$ and consider three
$\sigma$-finite measure $\lambda$, $\mu$ and $\sigma$ such that
$\lambda<<\mu<<\sigma$. Then

$\frac{\partial\sigma}{\partial\lambda}=\frac{\partial\sigma}{\partial{\mu}}\frac{\partial\mu}{\partial\sigma}$
(1.98)

Let us consider now a measurable map
$f:(M,\Sigma_{M},\nu)\longrightarrow(N,\Sigma_{N},\mu)$

######  Definition 1.3.10.

The pushforward measure is the measure $f_{\star}(\nu)$ on $(N,\Sigma_{N})$
defined for all $A$ in $\Sigma_{N}$ as

$f_{\star}(\nu)(A):=\nu(f^{-1}(A)).$ (1.99)

######  Remark 17.

If $g$ is an integrable function on $N$, then

$\int_{M}f^{*}(g)d\nu=\int_{N}gd(f_{\star}\nu).$ (1.100)

######  Definition 1.3.11.

Let $(M,\Sigma_{M},\nu)$ and $(N,\Sigma_{N},\mu)$ be two measured spaces and
let $f:(M,\Sigma_{M},\nu)\longrightarrow(N,\Sigma_{N},\mu)$ be a function such
that $f_{\star}(\nu)$ is absolutely continuous with respect to $\nu$. Let
$(N,\mu)$ be $\sigma$-finite, then the Fiber Volume is the function

$Vol_{f,\nu,\mu}:=\frac{\partial f_{\star}\nu}{\partial\mu}.$ (1.101)

####  1.3.6 Radon-Nicodym-Lipschitz maps

######  Definition 1.3.12.

A measured metric space is a triplet $(M,d_{M},\nu_{M})$ where $d_{M}$ is a
metric space, $M$ is a topological space with the topology induced by $d_{M}$
and $\nu_{M}$ is a measure defined on the $\sigma$-algebra of Borellians of
$(M,d_{M})$.

######  Definition 1.3.13.

Let $(M,d_{M},\nu_{M})$ and $(N,d_{N},\mu_{N})$ be two measured and metric
spaces. A map $f:(M,\nu)\longrightarrow(N,\mu)$ is Radon-Nikodym-lipschitz or
R.-N.-lipschitz if

  * •

$f$ is lipschitz

  * •

$f$ has bounded Fiber Volume.

Consider $f:(M,d_{M},\nu_{M})\longrightarrow(N,d_{N},\mu_{N})$ an
R.-N.-lipschitz map and let $C$ be the supremum of $\frac{\partial
f_{\star}\mu_{M}}{\partial\mu_{N}}$. Then for all measurable set $A\subseteq
N$, we have that

$\begin{split}\mu_{M}(f^{-1}(A))&=\int_{A}\frac{\partial
f_{\star}\mu_{M}}{\partial\mu_{N}}d\mu_{N}\\\ &\leq
C\int_{A}d\mu_{N}=C\mu_{N}(A).\end{split}$ (1.102)

A measurable, lipschitz map which satisfies the above inequality is called a
volume bounded map or v.b.-map: these maps are defined by Thomas Schick in his
Ph.D. thesis.

######  Proposition 1.3.5.

Let us consider a map $f:(M,d_{M},\nu_{M})\longrightarrow(N,d_{N},\mu_{N})$
between measured metric spaces. Then we have that $f$ is a R.-N.-map if and
only if it is a v.b.-map.

###### Proof.

If $f$ is a R.-N.-lipschitz map, then it follows by 1.102 that it is a
v.b.-map. To prove the other implication we can start observing that the Fiber
Volume is well-defined since $\mu_{N}<<\nu_{M}$. Consider $C$ the constant
such that $\nu_{M}(f^{-1}A)\leq C\cdot\mu_{N}(A)$: then

$|\frac{\partial f_{\star}\nu_{M}}{\partial\mu_{N}}|<C.$ (1.103)

Indeed if the above inequality is not satisfied, then there is $A\subseteq N$
such that $\frac{\mu_{M}(f^{-1}A)}{\mu_{N}(A)}>C$. But this implies

$\begin{split}\mu_{M}(f^{-1}(A))&=\int_{A}\frac{\partial
f_{\star}\nu_{M}}{\partial\mu_{N}}\\\ &\geq C\cdot\int_{A}d\mu_{N}\\\
&=C\cdot\mu_{N}(A)\end{split}$ (1.104)

and so we have the contradiction. ∎

Let us prove some properties of R.-N.-lipschitz maps.

######  Proposition 1.3.6.

Consider $f:(M,d_{M},\mu_{M})\longrightarrow(N,d_{N},\mu_{N})$ and
$g:(N,d_{N},\mu_{N})\longrightarrow(W,d_{W},\mu_{W})$ two R.-N.-lipschitz
maps. Then $(g\circ f):(M,d_{M},\mu_{M})\longrightarrow(W,d_{W},\mu_{W})$ is a
R.-N.-lipschitz map.

###### Proof.

Since the equivalence of the definitions, we can check that the composition of
two v.b.-maps is a v.b.-map. We can start observing that the composition of
lipschitz map is lipschitz. Moreover we also have that

$\mu_{M}((g\circ f)^{-1}A)\leq C_{f}\mu_{N}(g^{-1}(A))\leq C_{f}\cdot
C_{g}\mu_{N}(A)$ (1.105)

and so it means that $g\circ f$ is a v.b.-map. ∎

######  Proposition 1.3.7.

Let us consider two R.-N.-lipschitz maps
$f:(M,d_{M},\mu_{M})\longrightarrow(X,d_{X},\mu_{X})$ and
$g:(N,d_{N},\mu_{N})\longrightarrow(Y,d_{Y},\mu_{Y})$. Then the map

$(f,g):(M\times N,d_{M}\times
d_{N},\mu_{M}\times\mu_{N})\longrightarrow(X\times Y,d_{X}\times
d_{Y},\mu_{X}\times\mu_{Y})$ (1.106)

is a R.-N.-lipschitz map.

###### Proof.

Again we will show that $(f,g)$ is a v.b.-map. We can observe that $(f,g)$ is
lipschitz. Moreover we can also consider a subset $A\times B$ of $X\times Y$.
Then we have that

$\begin{split}\mu_{M\times N}((f,g)^{-1}(A\times
B))&=\mu_{M}(f^{-1}(A))\cdot\mu_{N}(g^{-1}(B))\\\ &\leq C_{f}\cdot
C_{g}\mu_{X}(A)\cdot\mu_{Y}(B)\\\ &\leq C_{f}\cdot C_{g}\mu_{X\times
Y}(A\times B).\end{split}$ (1.107)

Since the sets $\\{A\times B\\}$ are generators of the Borel $\sigma$-algebra
of $X\times Y$ we can conclude. ∎

######  Remark 18.

We can also prove the Proposition 1.3.7 checking that the Fiber Volume of
$(f,g)$ in a point $(p,q)$ is given by $Vol_{f}(p)\cdot Vol_{g}(q)$.

######  Proposition 1.3.8.

Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. Let
$f:(M,g)\longrightarrow(N,h)$ be a R.-N.-lipschitz map. Then $f$ induces an
$\mathcal{L}^{2}$-bounded pullback
$f^{*}:\mathcal{L}^{2}(N)\longrightarrow\mathcal{L}^{2}(M)$. In particular the
norm of $f^{*}$ is less or equal to $K_{f}\cdot\sqrt{C_{Vol}}$, where
$C_{Vol}$ is the maximum of the Fiber Volume.555In order to prove this
Proposition we will use some inequalities between the norms of differential
forms that can be found in Appendix A.

###### Proof.

Let $\omega$ be a smooth form with compact support in $\mathcal{L}^{2}(N)$.
Then

$\begin{split}||f^{*}\omega||_{\mathcal{L}^{2}(M)}^{2}&=\int_{M}||f^{*}\omega||^{2}d\mu_{g}\\\
&\leq\int_{M}K_{f}^{2}f^{*}(||\omega||^{2})d\mu_{g}\\\
&=K_{f}^{2}\int_{N}||\omega||^{2}d(f_{\star}\mu_{g})\\\
&=K_{f}^{2}\int_{N}||\omega||^{2}Vol_{f,g,h}d\mu_{h}\\\ &\leq
K_{f}^{2}C_{Vol}\int_{N}||\omega||^{2}d\mu_{h}\\\
&=K_{f}^{2}C_{Vol}||\omega||_{\mathcal{L}^{2}(N)}.\end{split}$ (1.108)

∎

######  Definition 1.3.14.

Consider a map $f:(M,g)\longrightarrow(N,h)$ between Riemannian manifolds. We
say that $f$ is a $L^{2}$-map, if it is a smooth, uniformly proper,
R.-N.-lipschitz map.

######  Proposition 1.3.9.

Consider a $L^{2}$-map $f:(M,g)\longrightarrow(N,h)$ between complete
Riemannian manifolds. Then $f^{*}$ induces a morphism between the (un)-reduced
$L^{2}$-cohomology.

###### Proof.

It is a consequence of Remark 12. ∎

####  1.3.7 Quotients of differential forms

In next sections we will focus on submersions. In particular we will study
their Fiber Volumes. In order to do this we need the notion of quotient of
differential forms

######  Definition 1.3.15.

Let us consider a differentiable manifold $M$. Given two differential forms
$\alpha\in\Omega^{k}(M)$, $\beta\in\Omega^{n}(M)$ we define a quotient between
$\alpha$ and $\beta$, denoted by $\frac{\alpha}{\beta}$ as a section of
$\Lambda^{k-n}(M)$ such that for all $p$ in $M$

$\alpha(p)=\beta(p)\wedge\frac{\alpha}{\beta}(p).$ (1.109)

######  Remark 19.

There are no condition about the continuity, or smoothness, of
$\frac{\alpha}{\beta}$.

In general, given two differential forms, there isn’t a quotient between them:
for example we can consider $dx^{1}$ and $dx^{2}\wedge dx^{3}$ in
$\mathbb{R}^{3}$. Moreover if there is a quotient between $\alpha$ and $\beta$
it may not be unique: for example if we consider $\alpha=dx^{1}\wedge dx^{2}$
and $\beta=dx^{1}+dx^{2}$ in $\mathbb{R}^{3}$ then
$\frac{1}{2}(dx^{2}-dx^{1})$ and $-dx^{1}$ are both quotients.
However there are some useful formulas concerning the quotients.

######  Proposition 1.3.10.

Let us consider a differentiable manifold $M$ and let
$\alpha,\beta,\gamma,\delta\in\Omega^{*}(M)$. Then, if
$\frac{\alpha}{\gamma}$, $\frac{\alpha}{\delta}$, $\frac{\beta}{\gamma}$, and
$\frac{\beta}{\delta}$ are well-defined, then the following formulas hold666in
these formulas the $=$ have to be read as _”exists and one of the possible
quotient is given by”_

  1. 1.

$\frac{\alpha+\beta}{\gamma}=\frac{\alpha}{\gamma}+\frac{\beta}{\gamma}$,

  2. 2.

if $\gamma$ as a $g$-form, $\beta$ a $b$-form and $\delta$ a $d$-form. Then
$\frac{\alpha\wedge\beta}{\gamma\wedge\delta}=(-1)^{d(a-g)}\frac{\alpha}{\gamma}\wedge\frac{\beta}{\delta}.$

  3. 3.

If $\gamma$ is closed, and $\frac{\alpha}{\gamma}$ is a smooth form, then
$\frac{d\alpha}{\gamma}=d(\frac{\alpha}{\gamma}).$

  4. 4.

if $\alpha$ is an $a$-form and $\beta$ is a $b$-form then
$\frac{\alpha\wedge\beta}{\beta\wedge\gamma}=(-1)^{ab}\frac{\alpha}{\gamma}$.

  5. 5.

if $\frac{\frac{\alpha}{\delta}}{\gamma}$ exists then
$\frac{\alpha}{\delta\wedge\gamma}=\frac{\frac{\alpha}{\delta}}{\gamma}.$

  6. 6.

if $\frac{\alpha}{\delta\wedge\gamma}$ exists, then
$\frac{\frac{\alpha}{\delta}}{\gamma}=\frac{\alpha}{\delta\wedge\gamma}.$

###### Proof.

  1. 1.

It follows since

$\gamma\wedge(\frac{\alpha}{\gamma}+\frac{\beta}{\gamma})=\gamma\wedge\frac{\alpha}{\gamma}+\gamma\wedge\frac{\beta}{\gamma}=\alpha+\beta$
(1.110)

  2. 2.

We have that the degree of $\frac{\beta}{\delta}$ is $(b-d)$.
Then we have that

$\gamma\wedge\delta\wedge(-1)^{d(a-g)}\frac{\alpha}{\gamma}\wedge\frac{\beta}{\delta}=\gamma\wedge\frac{\alpha}{\gamma}\wedge\delta\wedge\frac{\beta}{\delta}=\alpha\wedge\beta.$
(1.111)

  3. 3.

Since $\alpha=\gamma\wedge\frac{\alpha}{\gamma}$, then

$d\alpha=\gamma\wedge
d(\frac{\alpha}{\gamma})+d\gamma\wedge\frac{\alpha}{\gamma}=d(\frac{\alpha}{\gamma})\wedge\gamma.$
(1.112)

  4. 4.

Observe that

$\beta\wedge\gamma\wedge(-1)^{ab}\frac{\alpha}{\gamma}=((-1)^{ab}\beta\wedge\alpha=\alpha\wedge\beta.$
(1.113)

  5. 5.

We have that

$\delta\wedge\gamma\wedge\frac{\frac{\alpha}{\delta}}{\gamma}=\delta\wedge\frac{\alpha}{\delta}=\alpha.$
(1.114)

  6. 6.

In oreder to prove this, observe that
$\frac{\alpha}{\delta}=\gamma\wedge\frac{\alpha}{\delta\wedge\gamma}$, indeed

$\delta\wedge(\gamma\wedge\frac{\alpha}{\delta\wedge\gamma})=(\delta\wedge\gamma)\wedge\frac{\alpha}{\delta\wedge\gamma}=\alpha,$
(1.115)

and then we have that
$\frac{\frac{\alpha}{\delta}}{\gamma}=\frac{\gamma\wedge\frac{\alpha}{\delta\wedge\gamma}}{\gamma}$
is a form $F$ such that
$\gamma\wedge\frac{\alpha}{\delta\wedge\gamma}=\gamma\wedge F$ and so, by the
definition of quotient we have that

$\frac{\frac{\alpha}{\delta}}{\gamma}=F=\frac{\alpha}{\delta\wedge\gamma}.$
(1.116)

∎

Consider two oriented differentiable manifolds $X$ and $Y$. As proved in [3],
in particular in Proposition 16.21.7, when we have a submersion
$f:X\longrightarrow Y$, then if $F_{q}$ is the fiber of $f$ in $q$ and
$i_{q}:F_{q}\longrightarrow X$ is the immersion of the fiber, then
$i_{q}^{*}(\frac{\beta}{f^{*}\alpha})$ doesn’t depend on the choice of the
quotient. Moreover in [3] it is proved that if $\beta$ is a smooth form in
$X$, then $i_{q}^{*}(\frac{\beta}{f^{*}\alpha})$ is a smooth form on the fiber
$F_{q}$. It means that for all $p$ in $Y$ we obtain the orientation of $F_{p}$
defined imposing

$\int_{F_{p}}i_{p}^{*}\frac{Vol_{X}}{f^{*}Vol_{Y}}>0.$ (1.117)

We can also observe that if $f:(X,g)\longrightarrow(Y,h)$ is a submersion
between Riemannian manifolds, then it’s possible to define a locally smooth
quotient between $Vol_{X}$ and $f^{*}Vol_{Y}$. Indeed it is sufficient to
consider local fibered coordinates ${y^{j},x^{i}}$ on $X$ and $\\{y^{j}\\}$ in
$Y$ and we obtain that locally

$\frac{Vol_{X}}{f^{*}Vol_{Y}}=\frac{det(g_{ij}(y,x))}{det(h_{rs})(y)}dx^{1}\wedge...\wedge
dx^{n},$ (1.118)

where $g_{ij}$ and $h_{rs}$ are the matrix related to the metrics $g$ and $h$.
As consequence of this fact we have that the Projection Formula holds also for
quotients of volume forms, i.e. given a differential form $\alpha$ in $Y$ we
have that

$\int_{F}f^{*}\alpha\wedge\frac{Vol_{X}}{f^{*}Vol_{Y}}=\alpha\wedge\int_{F}\frac{Vol_{X}}{f^{*}Vol_{Y}}.$
(1.119)

In order to prove this we have to consider a cover of coordinate charts and to
decompose the integral using a partition of unity. Remember that, since [3],
the integral over the fiber in $p$ of a quotient between a form $\alpha$ and a
form $f^{*}\beta$ doesn’t depend on the choice of the quotient. This means
that in each chart one can choose a smooth quotient and apply the usual
Projection Formula (see Appendix B).

######  Proposition 1.3.11.

Let $p$ be a submersion $p:X\longrightarrow Y$. Consider
$\alpha\in\Omega_{vc}^{*}(X)$ and $\beta\in\Omega^{*}(X)$. Let us suppose that
there is a well-defined $\frac{\int_{F}\alpha}{\beta}$ which is locally
smooth. Then we have that777again $=$ have to be read as _”exists and one of
the possible quotient is given by”: this property holds for all possible
choice of the quotient._ , unless $\alpha$ and $\beta$ are top-degree forms:
in that case the $=$ actually means the equality as differential forms.

$\frac{\int_{F}\alpha}{\beta}=\int_{F}\frac{\alpha}{p^{*}\beta}.$ (1.120)

###### Proof.

We have that

$\beta\wedge(\int_{F}\frac{\alpha}{p^{*}\beta})=\int_{F}p^{*}\beta\wedge\frac{\alpha}{p^{*}\beta}=\int_{F}\alpha.$
(1.121)

∎

####  1.3.8 Fiber Volume of a submersion

In this section we study the Fiber Volumes of lipschitz submersions between
orientable manifolds. We use the notions of oriented fiber bundle, integration
along the fibers and the Projection Formula: we introduce them in Appendix B.

######  Proposition 1.3.12.

Let $(M,g)$ and $(N,h)$ two orientable, Riemannian manifolds. Let
$p:(M,g)\longrightarrow(N,h)$ be a proper lipschitz submersion. Then we have
that

$Vol_{p,g,h}(q)=\int_{F}\frac{Vol_{M}}{p^{*}Vol_{N}}(q)$ (1.122)

if $q$ is in $p(M)$, $0$ otherwise.

###### Proof.

Let $A$ a be measurable set of $N$. Then we have that

$\begin{split}f_{\star}\mu_{M}(A)&=\int_{p^{-1}(A)}1d\mu_{M}\\\
&=\int_{p^{-1}(A)}Vol_{M}\\\
&=\int_{p^{-1}(A)}p^{*}(Vol_{N})\wedge\frac{Vol_{M}}{p^{*}Vol_{N}}\\\
&=\int_{p^{-1}(A\cap
p(M))}p^{*}(Vol_{N})\wedge\frac{Vol_{M}}{p^{*}Vol_{N}}+\int_{A\cap[p(M)]^{c}}0d\mu_{N}\\\
\\\ &=\int_{A\cap
p(M)}Vol_{N}(\int_{F}\frac{Vol_{M}}{p^{*}Vol_{N}})+\int_{A\cap[p(M)]^{c}}0d\mu_{N}\\\
&=\int_{A\cap
p(M)}(\int_{F}\frac{Vol_{M}}{p^{*}Vol_{N}})Vol_{N}+\int_{A\cap[p(M)]^{c}}0d\mu_{N}\\\
&=\int_{A\cap
p(M)}(\int_{F}\frac{Vol_{M}}{p^{*}Vol_{N}})d\mu_{N}+\int_{A\cap[p(M)]^{c}}0d\mu_{N}\end{split}$
(1.123)

∎

######  Remark 20.

The Fiber Volume doesn’t depend on the choice of
$\frac{Vol_{M}}{p^{*}Vol_{N}}$. It is coherent with the uniqueness of the
Radon-Nicodym derivative.

######  Remark 21.

If the submersion $f:X\rightarrow Y$ is in particular a diffeomorphism between
oriented Riemannian manifold, then we have that the integration along the
fibers of $f$ is the pullback operator $(f^{-1})^{*}$ with a $+$ if $f$
preserves the orientation and a $-$ otherwise. Indeed the orientation of the
fibers is given posing

$\int_{f^{-1}(p)}\frac{Vol_{X}}{f^{*}Vol_{X}}>0$ (1.124)

and we know that the integration over a $0$-chain is just the evaluation on
the oriented points. So

$\begin{split}f_{\star}(\alpha)_{p}(v_{1,p},...,v_{k,p})&=\pm\alpha_{f^{-1}(p)}(df_{f^{-1}(p)}^{-1}(v_{1,p}),...,df_{f^{-1}(p)}^{-1}(v_{k,p}))\\\
&=\pm(f^{-1})^{*}\alpha(v_{1,p},...,v_{k,p}),\end{split}$ (1.125)

where we have a $+$ if $f$ preserves the orientation and $-$ otherwise. Then,
this means that

$f_{\star}=\pm(f^{-1})^{*}.$ (1.126)

and so, in particular, the Fiber Volume of $f$ is given by

$\pm(f^{-1})^{*}\frac{Vol_{X}}{f^{*}(Vol_{Y})}=|(f^{-1})^{*}\frac{Vol_{X}}{f^{*}(Vol_{Y})}|.$
(1.127)

######  Proposition 1.3.13.

Given a fiber bundle $p:(M,g)\longrightarrow(N,h)$ where $M$ and $N$ are
orientable, we have that if $p^{*}$ induces a map between the
$\mathcal{L}^{2}$-space, then the same is true for $p_{\star}$, the operator
of integration along the fibers of $p$. Moreover, if we denote by
$(p_{\star})^{*}$ the adjoint of $p_{\star}$, we have that, if $\tau_{X}$ and
$\tau_{Y}$ are the chiral operators of $X$ and $Y$

$(p_{\star})^{*}=\tau_{X}\circ p^{*}\circ\tau_{Y}.$ (1.128)

###### Proof.

Let $\alpha$ be in $\Omega_{vc}^{*}(X)$ and let $\beta$ be in $\Omega^{*}(Y)$,
if $n=dim(Y)$ we have that

$\langle
p_{\star}\alpha,\beta\rangle_{Y}=\int\limits_{Y}[\int\limits_{F}\alpha]\wedge\star_{Y}\overline{\beta}$
(1.129)

Now, applying the Projection Formula, we have that

$\begin{split}\langle
p_{\star}\alpha,\beta\rangle_{Y}&=\int\limits_{X}\alpha\wedge
p^{*}(\overline{\star_{Y}\beta})\\\
&=i^{-|\beta|(n-|\beta|)}\int\limits_{X}\alpha\wedge(\overline{p^{*}\tau_{Y}\beta})\\\
&=i^{-|\beta|(n-|\beta|)}i^{|\beta|(n-|\beta|)}\int\limits_{X}\alpha\wedge\star_{X}(\overline{\tau_{M}p^{*}\tau_{Y}\beta})\\\
&=\langle\alpha,\tau_{X}\circ
p^{*}\circ\tau_{Y}(\beta)\rangle_{X}.\end{split}$ (1.130)

∎

######  Remark 22.

The norm of $p_{\star}$ as operator between $\mathcal{L}^{2}$-spacesis the
same of the norm of $p^{*}$.

######  Corollary 1.3.14.

Consider $p:(M,g)\longrightarrow(N,h)$ a R.-N.-lipschitz submersion. Then the
operator $p_{\star}$ is a $\mathcal{L}^{2}$-bounded operator.

We conclude this section giving a formula which allows to compute the Fiber
Volume of the composition of two submersions.

######  Proposition 1.3.15.

Let $f:(M,g)\longrightarrow(N,h)$ and $g:(N,h)\longrightarrow(W,l)$ be two
submersions between oriented Riemannian manifolds. Then we have that

$Vol_{g\circ
f,\mu_{M},\mu_{W}}(q)=\int_{g^{-1}(q)}(\int_{f^{-1}g^{-1}(q)}\frac{Vol_{M}}{f^{*}(Vol_{N})})\frac{Vol_{N}}{g^{*}Vol_{N}}$
(1.131)

###### Proof.

We can observe that, as quotients, we have that

$\frac{Vol_{M}}{(g\circ
f)^{*}Vol_{W}}=\frac{Vol_{M}}{f^{*}Vol_{N}}\wedge\frac{f^{*}Vol_{N}}{(g\circ
f)^{*}Vol_{W}}$ (1.132)

and, in particular one can observe that we can choose as quotient

$\frac{f^{*}Vol_{N}}{(g\circ
f)^{*}Vol_{W}}=f^{*}(\frac{Vol_{N}}{g^{*}Vol_{W}}),$ (1.133)

indeed

$(g\circ f)^{*}Vol_{W}\wedge
f^{*}(\frac{Vol_{N}}{g^{*}Vol_{W}})=f^{*}(g^{*}Vol_{W}\wedge\frac{Vol_{N}}{g^{*}Vol_{W}})=f^{*}Vol_{N}.$
(1.134)

Then we can conclude by applying the Projection Formula. Indeed

$\begin{split}Vol_{g\circ f,\mu_{M},\mu_{W}}(q)&=\int_{(g\circ
f)^{-1}(q)}\frac{Vol_{M}}{(g\circ f)^{*}Vol_{W}}\\\ &=\int_{(g\circ
f)^{-1}(q)}\frac{Vol_{M}}{f^{*}Vol_{N}}\wedge
f^{*}(\frac{Vol_{N}}{g^{*}Vol_{W}})\\\
&=\int_{g^{-1}(q)}(\int_{f^{-1}g^{-1}(q)}\frac{Vol_{M}}{f^{*}(Vol_{N})})\frac{Vol_{N}}{g^{*}Vol_{N}}.\end{split}$
(1.135)

∎

###  1.4 Submersion related to lipschitz maps

####  1.4.1 The Sasaki metric

Let us consider a Riemannian manifold $(N,h)$ of dimension $n$,
$\pi_{E}:E\longrightarrow N$ a vector bundle of rank $m$ endowed with a bundle
metric $H_{E}\in\Gamma(E^{*}\otimes E^{*})$ and a linear connection
$\nabla_{E}$ which preserves $H_{E}$. Fix $\\{s_{j}\\}$ a local frame of $E$:
we have that if $\\{x^{i}\\}$ is a system of local coordinate over $U\subseteq
N$, then we can define the system of coordinates $\\{x^{i},\mu^{j}\\}$ on
$\pi_{E}^{-1}(U)$, where the $\mu^{j}$ are the components respect to
$\\{s_{j}\\}$.
Then we can denote by $K$ the map $K:TE\longrightarrow E$ defined as

$K(b^{i}\frac{\partial}{\partial
x^{i}}|_{(x_{0},\mu_{0})}+z^{j}\frac{\partial}{\partial\mu^{j}}|_{(x_{0},\mu_{0})}):=(z^{l}+b^{i}z^{j}\Gamma_{ij}^{l}(x_{0}))s_{l}(x_{0}),$
(1.136)

where the $\Gamma_{ij}^{l}$ are the Christoffel symbols of $\nabla_{E}$.

######  Definition 1.4.1.

The Sasaki metric on $E$ is the Riemannian metric $h^{E}$ defined for all
$A,B$ in $T_{(p,v_{p})}E$ as

$h^{E}(A,B):=h(d\pi_{E,v_{p}}(A),d\pi_{E,v_{p}}(B))+H_{E}(K(A),K(B)).$ (1.137)

######  Remark 23.

Let us consider the system of coordinate $\\{x^{i}\\}$ on $N$ and
$\\{x^{i},\mu^{j}\\}$ on $E$. We have that the components of $h^{E}$ are given
by

$\begin{cases}h^{E}_{ij}(x,\mu)=h_{ij}(x)+H_{\alpha\gamma}(x)\Gamma^{\alpha}_{\beta
i}(x)\Gamma^{\gamma}_{\eta j}(x)\mu^{\beta}\mu^{\eta}\\\
h^{E}_{i\sigma}(x,\mu)=H_{\sigma\alpha}(x)\Gamma^{\alpha}_{\beta
i}(x)\mu^{\beta}\\\ h^{E}_{\sigma\tau}(x,\mu)=H_{\sigma,\tau}(x),\end{cases}$
(1.138)

where $i,j=1,...,n$ and $\sigma,\tau=n+1,...,n+m$. Consider a point
$x_{0}=(x^{1}_{0},...,x^{n}_{0})$ in $M$. We have that all the Christoffel
symbols are in $x_{0}$ are zero, then, in local coordinates the matrix of
$h^{E}$ in a point $(x_{0},\mu)$ is given by

$\begin{bmatrix}h_{i,j}(x)&&0\\\ 0&&H_{\sigma,\tau}(x)\end{bmatrix}.$ (1.139)

Consider a fiber bundle $E$ over a Riemannian manifold $(M,g)$. Fix a bundle
metric $H$ and a connection $\nabla$ on $E$. Then the projection
$p:(E,g_{S})\longrightarrow(M,g)$, where $g_{S}$ is a Sasaki metric related to
$g$, $H$ and $\nabla$ is a Riemannian submersion.

######  Example 1.4.1.

Let us consider a Riemannian manifold $(N,h)$. We can consider as $E$ the
tangent bundle $TM$ and as $h_{E}$ the metric $g$ itself. In this case the
connection $\nabla_{E}$ is the Levi-Civita connection $\nabla_{g}^{LC}$.

####  1.4.2 Pull-back bundle and pull-back connection

Let us consider a smooth map $f:(M,g)\longrightarrow(N,h)$ between Riemannian
manifolds.

######  Definition 1.4.2.

The pullback bundle $f^{*}E$ is a bundle over $(M,g)$ given by

$f^{*}(E):=\\{(p,v)\in M\times E|\pi_{E}(t)=f(p)\\}.$ (1.140)

######  Remark 24.

Given a smooth map $f:(M,g)\longrightarrow(N,h)$, then there is a bundle map
$F:f^{*}E\longrightarrow E$ induced by $f$. This map is defined as

$F(p,v)=(f(p),v).$ (1.141)

Using this map it’s possible to define a metric bundle on $f^{*}(E)$ as
follows

$F^{*}(H)(A_{p},B_{p}):=H(F(A_{p}),F(B_{p}))$ (1.142)

Given a section $\sigma$ of $E$, one can define the section

$f^{*}\sigma:=f^{*}\sigma(p)=\sigma\circ f(p).$ (1.143)

A consequence of the existence of the pullback of a section of $E$ is that it
is possible to _pullback_ also a connection $\nabla_{E}$.

######  Definition 1.4.3.

The pullback connection $f^{*}\nabla_{E}$ on $f^{*}(E)$ is uniquely defined
imposing that

$(f^{*}\nabla_{E})f^{*}\sigma=f^{*}(\nabla_{E}\sigma).$ (1.144)

######  Remark 25.

The condition (1.144) is sufficient to uniquely define a connection on
$f^{*}(E)$. An equivalent condition is to impose that the local Christoffel
symbols $\tilde{\Gamma}^{\alpha}_{\beta,i}$ are given by

$\tilde{\Gamma}^{\alpha}_{\beta,i}:=\frac{\partial f^{l}}{\partial
x^{i}}f^{*}(\Gamma^{\alpha}_{\beta,l}).$ (1.145)

######  Remark 26.

Let us consider a map $f:(M,g)\longrightarrow(N,h)$ and consider, in
particular, the bundle $\pi:f^{*}(TN)\longrightarrow M$. Fix on $f^{*}(TN)$
the bundle metric $f^{*}h$ and $f^{*}\nabla^{LC}_{h}$. Consider the Sasaki
metric on $f^{*}(TN)$ induced by $g$, $f^{*}h$ and $f^{*}\nabla^{LC}_{h}$. Fix
some normal coordinates $x^{i}$ around a point $p$ on $M$ and fix a frame
$E_{j}$ of $f^{*}(TN)$ around $f(p)$. Then we can consider the fibered
coordinates $\\{x^{i},y^{j}\\}$ of $f^{*}(TN)$. Let us observe that in
$(p,w_{f(p)})$ we have the orthogonal decomposition

$\begin{split}T_{(p,w_{f(p)})}f^{*}(TN)&=Span\\{\frac{\partial}{\partial
x^{i}}|_{(p,w_{f(p)})}\\}\oplus ker(d\pi_{(p,w_{f(p)})})\\\
&=Span\\{\frac{\partial}{\partial x^{i}}|_{(p,w_{f(p)})}\\}\oplus
Span\\{\frac{\partial}{\partial y^{j}}|_{(p,w_{f}(p))}\\}\end{split}$ (1.146)

and, moreover, we have that $Span\\{\frac{\partial}{\partial
x^{i}}|_{(p,w_{f(p)})}\\}$ is isometric to $T_{p}M$, while
$Span\\{\frac{\partial}{\partial y^{j}}|_{(p,w_{f}(p))}\\}$ is isometric to
$T_{f(p)}N$. This fact follows by (1.138) and (1.145) and observing that the
Christoffel symbols of $\nabla^{LC}_{N}$ respect to normal coordinates around
$f(p)$ are null in $f(p)$.
This fact, in particular, implies that the projection
$\pi:f^{*}TN\longrightarrow M$ is a Riemannian submersion.

######  Remark 27.

Consider $(M,g)$ and $(N,h)$ two manifolds of bounded geometry and let
$f:(M,g)\longrightarrow(N,h)$ be a $C^{k}_{bu}$-map for each $k$ in
$\mathbb{N}$. Fix on $f^{*}(TN)$ the Sasaki metric $g_{S}$ induced by $g$,
$f^{*}h$ and $f^{*}\nabla^{LC}_{g}$. Then, if we denote by
$\nabla:=\nabla^{LC}_{g_{S}}$ and by $R$ the Riemann tensor on $f^{*}(TN)$, we
have that for each $i$ in $\mathbb{N}$ there is a continuous function
$C_{i}:\mathbb{R}\longrightarrow\mathbb{R}$ such that

$|\nabla^{i}R(v_{p})|\leq C_{i}(||v_{p}||).$ (1.147)

####  1.4.3 The map $p_{f}$

In this subsection we will define a submersion
$p_{f}:f^{*}(T^{\delta}N)\longrightarrow N$ where

$f^{*}(T^{\delta}N):=\\{(p,w_{f(p)})\in f^{*}(TN)||w_{f(p)}|\leq\delta\\}.$
(1.148)

######  Lemma 1.4.1.

Let us consider $f:(M,g)\longrightarrow(N,h)$ a smooth lipschitz map between
two oriented Riemannian manifolds. Suppose that $(N,h)$ has bounded geometry.
Let us denote by $F:f^{*}TN\longrightarrow TN$ the bundle morphism induced by
$f$ between $f^{*}(TN)$ and $TN$. Fix on $f^{*}TN$ the Sasaki metric $g_{s}$
defined using $f^{*}\nabla^{L}C_{h}$, $f^{*}h$ and $g$. Then there is a map
$p_{f}:(f^{*}(T^{\delta}N),g_{s})\longrightarrow(N,h)$ such that:

  1. 1.

$p_{f}$ is a submersion,

  2. 2.

${p}_{f}(x,0)=f(x)$,

  3. 3.

${p}_{f}$ is $\Gamma$-equivariant,

  4. 4.

${p}_{f}={p}_{id_{N}}\circ F$,

  5. 5.

Fix some local normal coordinates $\\{x^{i}\\}$ around a point $p$ in $M$ and
some local normal coordinates $\\{y^{j}\\}$ around $f(p)$ on $N$. Let us
suppose that for each $k$ in $\mathbb{N}$

$\sup\limits_{s=0,...,k}|\frac{\partial^{s}y^{j}\circ f}{\partial
x^{i}_{1}...\partial x^{i_{s}}}(x)|\leq L_{k}$ (1.149)

for some $L_{k}$. Consider the frame $\\{\frac{\partial}{\partial y^{j}}\\}$
around $f(0)$ and define the fibered coordinates $\\{x^{i},\mu^{j}\\}$ related
to $\\{\frac{\partial}{\partial y^{j}}\\}$ on $f^{*}T^{\delta}N$. Then for
each $k$ in $\mathbb{N}$ there is a constant $C_{k}$ such that

$\sup\limits_{s+t=0,...,k}|\frac{\partial^{s}y^{j}\circ p_{f}}{\partial
x^{i}_{1}...\partial
x^{i_{s}}\partial\mu^{j_{1}}...\partial\mu^{j_{t}}}(x,\mu)|\leq C_{k}$ (1.150)

where $C_{k}$ doesn’t depend on $x$ or $j$.

###### Proof.

We can define

$\begin{split}p_{f}:(f^{*}(T^{\delta}N),g_{s})&\longrightarrow(N,h)\\\
(p,w_{f(p)})&\longrightarrow exp_{f(p)}(w_{f(p)}).\end{split}$ (1.151)

Then we have that

  1. 1.

$p_{f}$ is a submersion, indeed for each fixed $p$ in $M$ we have that
${p}_{f}(p,\cdot):f^{*}(T^{\delta}N)_{p}=T^{\delta}_{f(p)}N\longrightarrow N$
is the exponential map in $f(p)$. We know that the exponential map is a local
diffeomorphism and so $\tilde{p}_{f}$ is a submersion,

  2. 2.

$p_{f}(p,0_{f(p)})=f(p)$: this follows immediately by the definition of
exponential map,

  3. 3.

$p_{f}$ is $\Gamma$-equivariant, indeed, since $\Gamma$ acts by isometries,

$\begin{split}p_{f}(\gamma p,d\gamma w_{f(p)})&=exp_{f(\gamma p)}d\gamma
w_{f(p)}\\\ &=exp_{\gamma f(p)}d\gamma w_{f(p)}\\\ &=\gamma
exp_{f(p)}w_{f(p)}=\gamma{p}_{f}(p,w_{f(p)}).\end{split}$ (1.152)

  4. 4.

It’s obvious: we have that $F(p,w_{f(p)})=w_{f(p)}$,

  5. 5.

Because of the last point it is sufficient to show the assertion just for
$p_{id}$.
Consider $V\subseteq N$ a normal coordinate chart. Let $\\{x^{i}\\}$ be the
coordinates system on $V$. Consider $\pi:TN\longrightarrow N$ and let
$\\{\frac{\partial}{\partial x^{i}}\\}$ be an orthonormal frame on $V$. Then
we have the fibered coordinate system $\\{x^{i},\mu^{j}\\}$ on $\pi^{-1}(V)$.
Let us study now $p_{id}$ restricted to $\pi^{-1}(V)$. We have that it can be
seen as $\pi\circ\phi(x,\mu)$ where $\pi(x,\mu)=x$ and $\phi$ is the flow of
the system of differential equations given by

$\begin{cases}\dot{x}^{k}=c^{k}\\\
\dot{c}^{k}=-\Gamma^{k}_{ij}(x)x^{i}x^{j}\end{cases}$ (1.153)

Then, applying the Lemma 3.4 of [20] we have that the partial derivatives of
$\phi$ are uniformly bounded. Then we conclude that the derivatives of
$p_{id}$ are uniformly bounded.

∎

######  Corollary 1.4.2.

We have that $\tilde{p}_{f}$ is a lipschitz map.

###### Proof.

It’s a direct computation. Observe that $||\frac{\partial}{\partial y^{j}}||$,
$||\frac{\partial}{\partial x^{i}}||$ and
$||\frac{\partial}{\partial\mu^{l}}||$ are uniformly bounded. Moreover we also
have that

$dp_{f}(\frac{\partial}{\partial x^{i}})=\frac{\partial p_{f}^{j}}{\partial
x^{i}}\frac{\partial}{\partial y^{j}}$ (1.154)

and

$dp_{f}(\frac{\partial}{\partial\mu^{l}})=\frac{\partial
p_{f}^{j}}{\partial\mu^{l}}\frac{\partial}{\partial y^{j}}.$ (1.155)

∎

###  1.5 The pull-back functor

####  1.5.1 The Fiber Volume of $p_{f}$

######  Lemma 1.5.1.

Consider $f:(M,g)\longrightarrow(N,h)$ a smooth lipschitz map between
Riemannian manifolds of bounded geometry. Then the map
${t}_{f}:f^{*}(T^{\delta}N)\longrightarrow M\times N$ defined as

${t}_{f}(p,w_{f(p)})=(p,p_{f}(p,w_{f(p)}))$ (1.156)

is an R.-N.-lipschitz diffeomorphism with its image.

###### Proof.

We start proving that ${t}_{f}$ is a diffeomorphism with its image. Observe
that

$dim(f^{*}(T^{\delta}N))=m+n=dim(M)+dim(N)=dim(M\times N).$ (1.157)

Fix some normal coordinates $\\{x^{i}\\}$ around a point $p$ in $M$ and let
$\\{y^{j}\\}$ be some normal coordinates around $f(p)$ in $N$. Consider the
frame $\\{\frac{\partial}{\partial y^{j}}\\}$ and define the fibered
coordinates $\\{x^{i},\mu^{j}\\}$ related to $\\{\frac{\partial}{\partial
y^{j}}\\}$ on $f^{*}(T^{\delta}N)$. On $M\times N$ we consider the normal
coordinates $\\{x^{i},y^{j}\\}$. Then the Jacobian of $\tilde{t}$ is given by

$Jt_{f}(x,\mu)=\begin{bmatrix}1&&\star\\\ 0&&Jexp_{x}(\mu)\end{bmatrix}$
(1.158)

Then, since the exponential map is a diffeomorphism for each $x_{0}$, we have
that $Jt_{f}$ is invertible. Moreover ${t}_{f}$ is also injective, indeed if
$(p,w_{f(p)})$ and $(q,v_{f(q)})$ have the same image, then $p=q$ and

$exp_{f(p)}w_{f(p)}=exp_{f(p)}v_{p}\implies w_{p}=v_{p},$ (1.159)

since their norm is less than $\delta$ and $\delta\leq inj_{N}$. We proved
that ${t}_{f}$ is a diffeomorphism with its image.

We know that ${t}_{f}$ is a lipschitz map because $p_{f}$ is a lipschitz map.
So, in order to prove that ${t}_{f}$ is a R.-N.-lipschitz map, we have to show
that it has bounded Fiber Volume.
Consider a point $(p,q)$ in $M\times N$. Then its fiber is empty or it is a
singleton $\\{(p,w_{f(p)})\\}$. This means, following the Remark 21 that the
Fiber Volume of ${t}_{f}$ is given by

$|{t_{f}}^{-1^{*}}\frac{Vol_{T^{\delta}N}}{{t_{f}}^{*}Vol_{M\times N}}|$
(1.160)

on the image of $t_{f}$ and it null otherwise.
Then if $\frac{Vol_{T^{\delta}N}}{{t}^{*}Vol_{M\times N}}$ is a bounded
function, which is a-priori not clear, then we can conclude that ${t}_{f}$ is
a R.-N.-lipschitz map.
Consider the fibered coordinates $\\{x^{i},\mu^{j}\\}$ on $f^{*}(TN)$ and the
coordinates $\\{x^{i},y^{j}\\}$ on $M\times N$. Since the definition of
exponential map we have that the image of ${t}_{f}$ is contained in a
$\delta$-neighborhood of the $Graph(f)\in M\times N$. This means that we can
cover all the image of ${t}_{f}$ using the normal coordinates
$\\{x^{i},y^{j}\\}$ around $(p,f(p))$. Out of the image of ${t}_{f}$ we
already know since Proposition 1.3.12 that the Fiber Volume is null.
Observe that, in this coordinates, we have

${t}_{f}(0,\mu^{j})=(0,\mu^{j}).$ (1.161)

Consider

$Vol_{f^{*}T^{\delta}N}(x,\mu)=\sqrt{det(G_{ij})}(x,\mu)dx^{1}\wedge...\wedge
d\mu^{n}$ (1.162)

and

$Vol_{M\times N}(x,y)=\sqrt{det(H_{ij})}(x,y)dx^{1}\wedge...\wedge dy^{n},$
(1.163)

where $G_{ij}$ is the matrix of $g_{S}$ on $f^{*}(T^{\delta}N)$ with respect
to $\\{x^{i},\mu^{j}\\}$ and $H_{ij}$ is the matrix of the metric on $M\times
N$. Then we have that

$\frac{Vol_{T^{\delta}N}}{t^{*}Vol_{M\times
N}}(x,\mu)=\frac{\sqrt{det(G_{ij})}}{t_{f}^{*}(\sqrt{det(H_{ij})})}(x,\mu)\cdot\frac{1}{det(Jexp_{f(x)}(\mu))}.$
(1.164)

Observe that in $(0,\mu)$ we have that $Jexp_{0}(\mu)$ is the identity matrix.
Moreover we also have that

$G_{ij}(0,\mu)=\begin{bmatrix}1&&0\\\ 0&&1\end{bmatrix}$ (1.165)

and so $\sqrt{det(G_{ij})}(0,y)=1$. Finally we have that

$H_{ij}(0,\mu)=\begin{bmatrix}1&&0\\\ 0&&h_{ij}(0,\mu)\end{bmatrix}$ (1.166)

where $h_{ij}$ is the matrix related to the Riemannian metric $h$ in normal
coordinates. Then we have that

$det(H_{ij})^{-1}(0,y)=det(h_{ij})^{-1}(0,y)\leq C$ (1.167)

because $N$ is a manifold of bounded geometry [20]. This means that

$\frac{Vol_{T^{\delta}N}}{t^{*}Vol_{M\times
N}}(0,\mu)=\sqrt{\frac{det(G_{ij})}{t_{f}^{*}det(H_{ij})}}(0,\mu)\leq C$
(1.168)

and so the Fiber Volume of $t_{f}$ is bounded. ∎

######  Corollary 1.5.2.

Let $f:(M,g)\longrightarrow(N,h)$ be a smooth uniformly proper lipschitz map
between Riemannian manifolds of bounded geometry. Then $p_{f}$ is a
R.-N.-lipschitz map and $p_{f}^{*}$ is $\mathcal{L}^{2}$-bounded.

###### Proof.

We already know, since Lemma 1.5.1, that $t_{f}$ is a R.-N.-lipschitz map.
Observe that $p_{f}=pr_{N}\circ t_{f}$.
Consider $\overline{f}:f^{*}T^{\delta}N\longrightarrow N$ defined as

$\overline{f}(w_{f(p)}):=f(p).$ (1.169)

We have that $p_{f}\sim_{\Gamma}\overline{f}$. Then, using Remark 3, in
particular (1.11), we have that there is a $C>0$ such that

$p_{f}^{-1}(q)\subset
A_{q}:=\overline{f}^{-1}(B_{C}(q))=\pi^{-1}f^{-1}(B_{C}(q)),$ (1.170)

where $\pi:f^{*}T^{\delta}N\longrightarrow M$ is the projection of the bundle.
This means that if we fix a $q$ in $N$, then the Fiber Volume of $t_{f}$ in a
point $(p,q)$ can be different from zero only if $p\in f^{-1}(B_{C}(q))$.
Then, using the Proposition 1.3.15, we have that the Fiber Volume of $p_{f}$
in a point $q$ is given by

$\begin{split}Vol_{p_{f}}(q)&=\int_{M}Vol_{t_{f}}(p,q)d\mu_{M}\\\
&=\int_{f^{-1}(B_{C}(q))}Vol_{t_{f}}(p,q)d\mu_{M}\\\ &\leq
K\cdot\mu_{M}(f^{-1}(B_{C}(q))).\end{split}$ (1.171)

where $K$ is the maximum of the Fiber Volume of $t_{f}$. Let us observe that
since $f$ is uniformly proper, then the diameter of $f^{-1}(B_{C}(q))$ is
uniformly bounded and so there are a point $x_{0}$ in $M$ and a radius $R$
such that

$f^{-1}(B_{C}(q))\subseteq B_{R}(x_{0}).$ (1.172)

Then, since the Ricci curvature of $M$ is bounded, we also have that (applying
Remark 4) there is a constant $V$ such that

$\mu_{M}(f^{-1}(B_{C}(q)))\leq\mu_{M}(B_{R}(x_{0}))\leq V$ (1.173)

and so

$Vol_{p_{f}}(q)\leq K\cdot V$ (1.174)

and $p_{f}$ is a R.-N.-lipschitz map. ∎

######  Remark 28.

Consider a smooth map $f:(M,g)\longrightarrow(N,h)$ which is lipschitz and
uniformly proper. Let us suppose, moreover, that $(N,h)$ is a manifold of
bounded geometry and $(M,g)$ has bounded Ricci curvature. Eventually suppose
$M$ incomplete or $\partial M\neq\emptyset$. Then we can observe that the map
$p_{f}:(f^{*}(T^{\delta}N),g_{S})\longrightarrow(N,h)$ is well-defined and,
moreover, using the same arguments we used in this section, it is also
R.-N.-lipschitz.

####  1.5.2 A Thom form for $f^{*}(TN)$

Let $(M,g)$ and $(N,h)$ be two manifolds of bounded geometry, let
$f:(M,g)\longrightarrow(N,h)$ be a smooth lipschitz map and consider
$(f^{*}(TN),g_{S})$ the pullback bundle $f^{*}(TN)$ endowed of the Sasaki
metric defined in the subsection 1.4.1. Let us introduce the notion of Thom
form.

######  Definition 1.5.1.

Let $\pi:E\longrightarrow M$ be a vector bundle. We say that $\omega$ in
$\Omega_{cv}^{*}(E)$ is a Thom form if it is closed and its integral along the
fibers of $\pi$ is equal to the constant function $1$.

It’s a well known fact that for each oriented vector bundle $(E,\pi,M)$ there
is at least a Thom form $\omega$ (see for example [1]). This, in particular,
implies that the tangent bundle $(TN,\pi,N)$ always admits a Thom form. Then,
if we consider a smooth map $f:M\longrightarrow N$, the same holds for
$(f^{*}TN,\pi_{1},M)$.
Given a Thom form $\omega$ of $f^{*}TN$ such that $supp(\omega)$ is contained
in a $\delta_{0}<\delta$ neighborhood of the null section, let us define an
operator
$e_{\omega}:\Omega^{*}(f^{*}(T^{\delta}N))\longrightarrow\Omega^{*}(f^{*}(T^{\delta}N))$,
for every smooth form $\alpha$ as

$e_{\omega}(\alpha):=\alpha\wedge\omega.$ (1.175)

Our goal, in this subsection, is to find a Thom form $\omega$ such that the
operator $e_{\omega}$ is $L^{2}$-bounded. To this end we follow the work of
Mathai and Quillen [16]. In their work, indeed, they compute a Thom form for a
vector bundle endowed with a connection and a metric bundle.

First we construct the Thom form of $f^{*}(TN)$ when $f=id_{N}$.
Consider the bundle $\pi:(TN,h_{S})\longrightarrow(N,h)$ and let $\pi^{*}(TN)$
be the pullback bundle over $TN$. Consider the bundle metric on $\pi^{*}(TN)$
given by $\pi^{*}h$ and we can also consider the connection
$\pi^{*}\nabla^{LC}_{h}$. Then we can denote by $\Omega^{i,j}$ the algebra

$\Omega^{i,j}:=\Omega^{i}(TN,\Lambda^{j}\pi^{*}TN)=\Gamma(TN,\Lambda^{i}T^{*}(TN)\otimes\Lambda^{j}\pi^{*}TN).$
(1.176)

Let us consider the following section of $TN$

$\begin{split}X:TN&\longrightarrow\pi^{*}TN\\\
v_{p}&\longrightarrow(v_{p},v_{p}).\end{split}$ (1.177)

Observe that, identifying $\pi^{*}TN$ and $\Lambda^{1}(\pi^{*}TN)$ we obtain
that $X$ is in $\Omega^{0,1}$.
Let us fix some normal coordinates $\\{x^{i}\\}$ on $N$ and the coordinates
$\\{x^{i},\mu^{j}\\}$ on $TN$, where $\\{\mu^{j}\\}$ are referred to the frame
$\\{\frac{\partial}{\partial x^{j}}\\}$. Then we have that

$X(x,\mu)=\mu^{i}\frac{\partial}{\partial x^{i}}.$ (1.178)

Let us consider

$\begin{split}\pi^{*}g(X,X)=|X|^{2}:TN&\longrightarrow\mathbb{R}\\\
v_{p}&\longrightarrow h_{p}(v_{p},v_{p}).\end{split}$ (1.179)

This map can be see as a differential form in $\Omega^{0,0}$. In coordinates
it can be expressed as follow

$|X|^{2}(x,\mu)=h_{i,j}(x)\mu^{i}\mu^{j}.$ (1.180)

Consider $\pi^{*}\nabla^{LC}_{h}(X)$: this is a form in $\Omega^{1,1}$ and, in
local coordinates, it is given by

$\begin{split}\pi^{*}\nabla^{LC}_{h}(X)&=d\mu^{i}\otimes\frac{\partial}{\partial
x^{i}}+\mu^{i}\nabla\frac{\partial}{\partial x^{i}}\\\
&=d\mu^{i}\otimes\frac{\partial}{\partial
x^{i}}+\mu^{i}\Gamma^{k}_{ij}(x)dx^{j}\otimes\frac{\partial}{\partial
x^{k}}\\\
&=(\delta^{k}_{j}+\mu^{i}\Gamma^{k}_{ij}(x))dx^{j}\otimes\frac{\partial}{\partial
x^{k}}\end{split}$ (1.181)

Finally let us consider $\Omega$ the curvature form of $TN$: this is a
$2$-form on $N$ with values in $\Lambda^{2}(TN)$. We know that $\Omega$, in
coordinates is given by

$\Omega(x)=R^{ij}_{kl}(x)dx^{k}\wedge dx^{l}\otimes(\frac{\partial}{\partial
x^{i}}\wedge\frac{\partial}{\partial x^{k}})$ (1.182)

where $R^{ij}_{kl}=g^{is}R^{j}_{kls}$ and $R^{j}_{kls}$ are the components of
the Riemann tensor of $N$. Pulling back $\Omega$ along $\pi$, we obtain
$\pi^{*}\Omega$ which is a differential form in $\Omega^{2,2}$.
Let $\phi:\mathbb{R}\longrightarrow\mathbb{R}$ be a smooth function which
support is contained in $[\delta_{0},\delta_{0}]$. Then we define

$\overline{\omega}:=\sum_{k=0}^{n}\frac{\phi^{(k)}(\frac{|X^{2}|}{2})}{k!}(\pi^{*}\nabla
X+\pi^{*}\Omega)^{k},$ (1.183)

where $(\pi^{*}\nabla X+\pi^{*}\Omega)^{k}$ is the $k$-times wedge of
$\pi^{*}\nabla X+\pi^{*}\Omega$.
Observe that the support of $\overline{\omega}$ is strictly contained in a
$\delta_{0}$-nighborhood of the zero section since
$supp(f)\subseteq[-\delta_{0},\delta_{0}]$.
Let us consider $I,J,K$ some multi-index and let us denote
$dx^{I}:=dx^{i}_{1}\wedge...\wedge dx^{i}_{|I|}$,
$d\mu^{J}:=d\mu^{i}_{1}\wedge...\wedge d\mu^{i}_{|J|}$ and
$\frac{\partial}{\partial x^{K}}:=\frac{\partial}{\partial
x^{i_{1}}}\wedge...\wedge\frac{\partial}{\partial x^{i_{|K|}}}$. Since $N$ is
a manifold of bounded geometry, we have that $\overline{\omega}$ in local
fibered coordinate $\\{U,x^{i},\mu^{j}\\}$ is given by

$\overline{\omega}(x,\mu):=\alpha_{IJ}^{K}(x,\mu)dx^{I}\wedge
d\mu^{J}\otimes\frac{\partial}{\partial x^{K}}$ (1.184)

where $\alpha_{IJ}$

$|\alpha_{IJ}^{K}(x,\mu)|\leq C$ (1.185)

and $C$ doesn’t depend on the choice of $U$. Indeed the bounded geometry of
$N$ implies that the components of the metric $h$, the Christoffel symbols of
$\nabla$ and also the components of $h_{S}$ are uniformly bounded in the
fibered coordinates $\\{x^{i},\mu^{j}\\}$.
We have to introduce an important tool: the Berezin integral B. This is the
isometry

$\begin{split}\textbf{B}:\Lambda^{n}(TN)&\longrightarrow N\times\mathbb{R}\\\
\alpha_{p}&\longrightarrow(p,Vol_{p}(\alpha_{p}))\end{split}$ (1.186)

where $n=dim(N)$ and $Vol_{p}$ is the volume form of $N$ in a point $p$
888Actually the definition of Berezin integral is much more general: this is
the definition of Berezin integral for the fiber bundle $TN$.. We can observe
that the Berezin integral can be extended to $\Omega^{i,j}$

$\mathcal{B}:\Omega^{i,j}\longrightarrow\Omega^{i}(TN)\\\ $ (1.187)

posing $\mathcal{B}(\alpha\otimes\beta):=\textbf{B}(\beta)\alpha$ if $j=n$,
$\textbf{B}(\alpha\otimes\beta):=0$ otherwise.
As showed in [7] and in [16], we can define our Thom form as the differential
form given by

$\omega:=\mathcal{B}(\overline{\omega}).$ (1.188)

We can observe that, in coordinates $\\{x,\mu\\}$ on $TN$,

$\mathcal{B}(\alpha^{I}_{J,L}(x,\mu)\frac{\partial}{\partial x^{I}}\otimes
dx^{J}\wedge d\mu^{L})=\alpha^{0}_{J,L}(x,\mu)dx^{J}\wedge d\mu^{L},$ (1.189)

where $I$, $J$, $L$ are multi-index and $0$ is the multi-index $(1,...,n)$.
This fact implies that, in local coordinates $(x,\mu)$ we can express

$\omega(x,\mu)=\alpha_{IJ}(x,\mu)dx^{I}\wedge d\mu^{J},$ (1.190)

where $|\alpha_{IJ}|$ are uniformly bounded. Moreover the support of $\omega$
is contained in a $\delta_{0}$-neighborhood of the null section. The
differential form $\omega$ is a Thom form for $TN$: a proof of this fact can
be find in [16] or in [7].

######  Proposition 1.5.3.

Consider $(N,h)$ a manifold of bounded geometry. Let $\alpha$ be a
differential form on $(T^{\delta}N,h_{S})$. If in fibered coordinates
$\\{x^{i},\mu^{j}\\}$, where $\mu^{j}$ are referred to
$\\{\frac{\partial}{\partial x^{i}}\\}$ the coefficients of $\alpha$ are
uniformly bounded, then the punctual norm $|\alpha_{p}|$ is uniformly bounded.

###### Proof.

It is a direct computation. We know that the norm of $dx^{i}$ and $d\mu^{j}$
are uniformly bounded in $f^{*}T^{\delta}N$. Moreover, applying the Hadamard-
Schwartz inequality ([19] or Lemma A.0.2 in Appendix A) we have that there is
a number $K$ such that $|dx^{I}\wedge d\mu^{J}|_{p}\leq K$. Then, since

$|\alpha_{p}|\leq\sqrt{|\alpha_{IJ}(x,\mu)|^{2}\cdot|\alpha_{RS}(x,\mu)|^{2}\cdot|dx^{I}\wedge
d\mu^{J}|_{p}^{2}\cdot|dx^{R}\wedge d\mu^{S}|_{p}^{2}}\leq C.$ (1.191)

∎

Let $f:(M,g)\longrightarrow(N,h)$ be a smooth lipschitz map and consider
$(f^{*}(TN),g_{S})$ where $g_{S}$ is the Sasaki metric defined as in the
previous section. Observe that if $\omega$ is the Thom form on $TN$ defined in
(1.188) and if we consider the map $F:f^{*}(TN)\longrightarrow TN$ given by
$F(p,w_{f(p)})=(f(p),w_{f(p)})$, then it’s easy to check that $F^{*}\omega$ is
a Thom form for $f^{*}(TN)$ and moreover, applying Lemma A.0.3 in Appendix A,
we also have a uniform bound on the norm of $|F^{*}\omega|_{p}$.

######  Proposition 1.5.4.

Let us consider a form $\alpha$ on a Riemannian manifold $(M,g)$. Suppose that
there is a number $C$ such that

$|\alpha_{p}|\leq C$ (1.192)

for each $p$ in $(M,g)$. Then the operator
$e_{\alpha}(\beta):=\beta\wedge\alpha$ defines an $\mathcal{L}^{2}$-bounded
operator.

###### Proof.

It’s again a direct consequence of Hadamard-Schwartz inequality. Indeed

$\begin{split}||e_{\alpha}(\beta)||_{\mathcal{L}^{2}(M)}^{2}&=\int_{M}|\alpha\wedge\beta|_{p}^{2}d\mu_{M}\\\
&\leq\int_{M}C\cdot|\alpha|_{p}^{2}\cdot|\beta|_{p}^{2}d\mu_{M}\\\ &\leq
C_{\alpha}\int_{M}|\beta|_{p}^{2}d\mu_{M}=C_{\alpha}||\beta||_{\mathcal{L}^{2}(M)}^{2}.\end{split}$
(1.193)

∎

####  1.5.3 The $T_{f}$ operator

Let $(M,g)$ and $(N,h)$ be two manifolds of bounded geometry and let
$f:(M,g)\longrightarrow(N,h)$ be a uniformly proper, smooth, lipschitz map.
Let us denote by $pr_{M}$ the projection
$pr_{M}:(f^{*}(T^{\delta}N),g_{S})\longrightarrow(M,g)$, let $\omega$ be a
Thom form defined as in the previous subsection and consider
$p_{f}:(f^{*}(T^{\delta}N),g_{S})\longrightarrow(M,g)$ the submersion related
to $f$ that we introduced in the previous section. If $f$ is differentiable,
then we can define the operator $T_{f}$ for each smooth $\mathcal{L}^{2}$-form
as

$\begin{split}T_{f}(\alpha)&=pr_{M,\star}\circ e_{\omega}\circ
p_{f}^{*}(\alpha)\\\
&=\int_{B^{\delta}}p_{f}^{*}(\alpha)\wedge\omega.\end{split}$ (1.194)

where $B^{\delta}$ denotes the fibers of $pr_{M}$. If $f$ is not
differentiable, then we can consider a differentiable lipschitz map
$f^{\prime}$ which is lipschitz-homotopic to $f$ 999we know that such a
$f^{\prime}$ exists as consequence of Proposition 1.2.3 and we impose

$T_{f}:=T_{f^{\prime}}.$ (1.195)

Actually, if $f$ is not differentiable, the definition of $T_{f}$ depends on
the choice of $f^{\prime}$. We will not denote the choice of the $f^{\prime}$
because, as we will see later, $T_{f}$ induces an operator in (un)-reduced
$L^{2}$-cohomology which doesn’t depend on the choice of $f^{\prime}$.

######  Proposition 1.5.5.

Let us consider a uniformly proper lipschitz map $f$ between Riemannian
manifolds of bounded geometry. Then the operator $T_{f}$ is a bounded
operator.

###### Proof.

Let us suppose $f$ is smooth. We know, since Corollary 1.5.2, that $p_{f}^{*}$
is $\mathcal{L}^{2}$-bounded.
Because of since Corollary 1.5.4, we also know that $e_{\omega}$ is a
$\mathcal{L}^{2}$-bounded operator.
We just have to prove that $pr_{M,\star}$ is $\mathcal{L}^{2}$-bounded.
Observe that $pr_{M}$ is a Riemannian submersion (Remark 23) and so, in
particular, it is a lipschitz map. Moreover it’s easy to check that its Fiber
Volume is the Volume of an euclidean ball of radius $\delta$ in each point of
$M$. Then, applying Proposition 1.3.12, we have that $pr_{M}$ is a
R.-N.-lipschitz map and, in particular $pr_{M}^{*}$ is a
$\mathcal{L}^{2}$-bounded operator. Then, beacause of Remark 22,

$||pr_{M,\star}||=||pr_{M}^{*}||=L<+\infty.$ (1.196)

Then we can conclude since $T_{f}$ is composition of bounded operators. ∎

######  Remark 29.

In particular the previous Proposition holds if $f$ is a lipschitz-homotopy
equivalence.

######  Corollary 1.5.6.

Given a uniformly proper lipschitz map $f:(M,g)\longrightarrow(N,h)$ between
two Riemannian manifolds of bounded geometry, then we have

$T_{f}(dom(d_{min}))\subseteq dom(d_{min})$ (1.197)

and $T_{f}d=dT_{f}$. This, in particular, means that $T_{f}$ induces a
morphism in $L^{2}$-cohomology. Moreover, since $T_{f}$ is
$\mathcal{L}^{2}$-bounded, it also induces a morphism between the reduced
$L^{2}$-cohomology groups.

###### Proof.

Let us suppose, again, that $f$ is smooth. In order to conclude the proof we
will prove that the operator $T_{f}$ satisfies the hypothesis of Proposition
1.3.3. We already know that $T_{f}$ is $\mathcal{L}^{2}$-bounded. Then, in
order to conclude the proof, we just have to prove that
$T_{f}(\Omega^{*}_{c}(N))\subseteq\Omega^{*}_{c}(M)$ and
$dT_{f}\alpha=T_{f}d\alpha$ for each $\alpha$ in $\Omega^{*}_{c}(N)$
Since $p_{f}$ is uniformly proper we have that for each smooth form $\alpha$
in $L^{2}(N)$

$diam(supp(e_{\omega}\circ p_{f}^{*}(\alpha))\leq C_{\alpha}.$ (1.198)

Moreover, since $supp(pr_{M,\star}\beta)\subset pr_{M}(supp(\beta))$ for each
smooth $\beta$ in $L^{2}(f^{*}T^{\delta}N)$, we have that $supp(T_{f}\alpha)$
is bounded in $M$. This means that $supp(T_{f}\alpha)$ is compact since
$(M,g)$ is complete and $T_{f}(\Omega_{c}(N))\subseteq\Omega_{c}(M)$.

We have that $e_{\omega}\circ
p_{f}^{*}(\Omega^{*}_{c}(N))\subseteq\Omega^{*}_{vc}(f^{*}(T^{\delta}N))$,
where in $\Omega^{*}_{vc}(f^{*}(T^{\delta}N))$ we are considering the
vertically compactly supported smooth forms related to the projection on the
first component $pr_{M}:f^{*}(T^{\delta}(M))\longrightarrow M$.
Then we can apply the Proposition B.0.3 of [9], which allow us to say
$pr_{M,\star}(\Omega^{*}_{vc}(f^{*}T^{\delta}N)\subseteq\Omega_{c}(M)$ and
that

$\int_{B^{\delta}}d\eta=d\int_{B^{\delta}}\eta,$ (1.199)

if $\eta$ is in $\Omega_{vc}^{*}(f^{*}T^{\delta}N)$. Then, using that $T_{f}$
is a $\mathcal{L}^{2}$-bounded operator we conclude applying Corollary 1.3.3.
∎

######  Remark 30.

Since $p_{f}$ is a $\Gamma$-equivariant map, then $p_{f}^{*}$ is
$\Gamma$-equivariant and, in particular $T_{f}$ is $\Gamma$-equivariant.

####  1.5.4 R.-N.-lipschitz equivalences of vector bundles

Let us consider two smooth lipschitz maps
$f_{0},f_{1}:(M,g)\longrightarrow(N,h)$ between Riemannian manifolds and
consider a vector bundle $E$ over $N$. Fix a group $\Gamma$ acting on $M$ and
on $N$. Then it’s a known fact (see for example [22]) that if
$f_{0}\sim_{\Gamma}f_{1}$ with a lipschitz homotopy $H$, then

$f_{0}^{*}(E)\times[0,1]\cong f_{1}^{*}(E)\times[0,1]\cong H^{*}(E)$ (1.200)

and

$f_{0}^{*}(E)\cong f_{1}^{*}(E)$ (1.201)

where with $\cong$ we mean that they are isomorphic as fiber bundles. As
consequence of this fact we also have that they are homeomorphic as manifolds.

######  Definition 1.5.2.

Consider two vector bundles $E$, $F$ over a Riemannian manifold $(N,h)$. Let
us suppose that $e$ is a Riemannian metric on $E$ and $f$ a Riemannian metric
on $F$. Let us denote by $E^{\delta}$ and $F^{\delta}$ the disk bundles of
radius $\delta$. We say that $(E,e)$ and $(F,f)$ are R.-N.-lipschitz
equivalent if there is a number $\delta>0$ and there is a bundle isomorphism
$\phi:E\longrightarrow F$ such that $\phi(E^{\delta})=F^{\delta}$ and $\phi$
restricted to $E^{\delta}$ is a R.-N.-lipschitz map.

######  Remark 31.

If $E$ and $F$ are R.-N.-lipschitz equivalent, then if
$A:(F^{\delta},f)\longrightarrow(M,g)$ is an R.-N.-lipschitz map, then
$\phi^{*}A$ is again an R.-N.-lipschitz map. Moreover if
$B:(E^{\delta},e)\longrightarrow(M,g)$ is an R.-N.-lipschitz map it is also
true that $(\phi^{-1})^{*}B$ is an R.-N.-lipschitz map, indeed, since $\phi$
is a diffeomorphism,

$(\phi^{-1})^{*}B=\phi_{\star}B,$ (1.202)

and the $\mathcal{L}^{2}$-norm of $\phi_{\star}$ is the same of the norm of
$\phi^{*}$.

######  Lemma 1.5.7.

Let us consider two Riemannian manifold $(M,g)$ and $(N,l)$, where $(N,l)$ is
a manifold of bounded geometry. Consider
$f_{0},f_{1}:(M,g)\longrightarrow(N,l)$ two smooth lipschitz maps and suppose
that there is a lipschitz-homotopy $H:(M\times[0,1],g\times
g_{[0,1]})\longrightarrow(N,l)$ such that $H(p,i):=f_{i}(p)$ for $i=0,1$.
Let us suppose, moreover, that

$d(f_{0}(p),f_{1}(p))\leq\frac{inj_{N}}{2}$ (1.203)

where $\delta$ is the radius of injectivity of $N$. Let us denote by
$g_{S,f^{*}_{i}}$ the Sasaki metric on $f_{i}^{*}(TN)$ defined using $g$,
$f^{*}_{i}(l)$ and $f^{*}_{i}(\nabla^{LC}_{l})$ where $i=0,1$.
Then $(f^{*}_{0}(TN),g_{S,f^{*}_{0}})$ and $(f^{*}_{1}(TN),g_{S,f^{*}_{1}})$
are R.-N.-lipschitz equivalent. In particular $\delta$ can be choose less or
equal to $inj_{N}$.

###### Proof.

We already know that $f^{*}_{0}(TN)$ and $f^{*}_{1}(TN)$ are homeomorphic.
Then we will consider $f^{*}_{0}(TN)$ and $f^{*}_{1}(TN)$ as the same bundle,
but with different metrics. Then we have to check that the identity map
$Id:f^{*}_{1}(T^{\delta}N)\longrightarrow f^{*}_{0}(T^{\delta}N)$ is lipschitz
and that its Fiber Volume is bounded. Let us start with its Fiber Volume.
Fix some normal coordinates $\\{U,x^{i}\\}$ on $M$. Fix, moreover, some normal
coordinates $\\{V,y^{j}\\}$ on $N$ such that $f_{0}(U)\subseteq V$. We also
suppose that $f_{0}(0)=0$ in this coordinates. Then we can consider
$\\{x^{i},\mu^{j}\\}$ the fibered coordinates on $f_{0}^{*}(TN)$ related to
the frame $\\{\frac{\partial}{\partial y^{j}}\\}$. Then we have that
$Id(x,\mu)=(x,\mu)$ and so

$Vol_{Id}(x,\mu)=\sqrt{\frac{det(G_{ij})}{det(L_{jk})}(x,\mu)}$ (1.204)

where $G_{ij}$ is the matrix related to the metric $g_{S,f^{*}_{1}}$ and
$L_{jk}$ the matrix related to $g_{S,f^{*}_{0}}$. Observe that in $(0,\mu)$ we
have that

$det(L_{jk}(0,\mu))=1.$ (1.205)

On the other hand we have that $det(G_{ij}(0,\mu))$ is an algebraic
combination of pullback of Christoffell symbols and terms of the metric on $N$
along the lipschitz map $f_{1}$, first derivatives of $f_{1}$ and
$\mu^{j}\mu^{i}$. We obtain a uniform bound on $|h_{E,ij}|$ and
$|\Gamma^{k}_{E,ij}|$ on a ball of radius less or equal to $inj_{r}N$.
Moreover we have that $|\mu^{i}|\leq\delta$. Then we can conclude that the
function

$\sqrt{\frac{det(G_{ij})}{det(L_{jk})}}:U\times[0,1]\longrightarrow\mathbb{R}$
(1.206)

is bounded and its bound doesn’t depend on the choice of $U$.

Let us verify that $Id$ is lipschitz. This time we fix the coordinates
$\\{x^{i}\\}$ on $M$ and we consider some normal coordinates $\\{y^{j}\\}$ on
$N$ such that $f_{1}(0)=0$. Then, exactly as we did in the previous point, we
choose the coordinates $\\{x^{i},\mu^{j}\\}$ related to the frame
$\\{\frac{\partial}{\partial y^{j}}\\}$.
Then we have to check that

$\frac{|\frac{\partial}{\partial
x^{i}}|_{f_{0}^{*}(TN),(0,\mu)}}{|\frac{\partial}{\partial
x^{i}}|_{f_{1}^{*}(TN),(0,\mu)}}$ (1.207)

and

$\frac{|\frac{\partial}{\partial\mu^{j}}|_{f_{0}^{*}(TN),(0,\mu)}}{|\frac{\partial}{\partial\mu^{j}}|_{f_{1}^{*}(TN),(0,\mu)}}$
(1.208)

are uniformly bounded. Observe that $\frac{\partial}{\partial
x^{i}}|_{f_{1}^{*}(TN),(0,\mu)}$ and
$|\frac{\partial}{\partial\mu^{j}}|_{f_{1}^{*}(TN),(0,\mu)}$ are equal to $1$.
Let us concentrate on $|\frac{\partial}{\partial
x^{i}}|_{f_{0}^{*}(TN),(0,\mu)}$ and on
$|\frac{\partial}{\partial\mu^{j}}|_{f_{0}^{*}(TN),(0,\mu)}$.
Then, similarly to what we did proving the boundedness of the Fiber Volume, we
can observe that the components of metric $g_{S,f^{*}_{0}}$, in these
coordinates, are algebraic combinations of pullback of Christoffell symbols
and terms of the metric on $N$ along the lipschitz map $f_{0}$, first
derivatives of $f_{0}$ and $\mu^{j}\mu^{i}$. Then, since on a ball of radius
less or equal to $\delta=inj_{r}N$ we have a bound on $|h_{E,ij}|$ and
$|\Gamma^{k}_{E,ij}|$, we conclude that the lipschitz constant is bounded. ∎

######  Proposition 1.5.8.

Let us consider $f_{0},f_{1}:(M,g)\longrightarrow(N,h)$ two lipschitz maps
such that $f_{0}$ and $f_{1}$ are lipschitz-homotopic. Then
$(f^{*}_{0}(TN),g_{S,0})$ and $(f^{*}_{1}(TN),g_{S,1})$ are R.-N.-lipschitz
equivalent. In particular $\delta$ can be choose less or equal to $inj_{N}$.

###### Proof.

Let us consider the homotopy $H$ between $f_{0}$ and $f_{1}$. Consider a
finite partition $\\{[s_{i},s_{i+1}]\\}$ of $[0,1]$ such that
$s_{i+1}-s_{i}\leq\frac{inj_{N}}{2C_{H}}$, where $C_{h}$ is the lipschitz
constant of $h$. Let us define the maps $H_{s}:(M,g)\longrightarrow(N,h)$ as
$H_{i}(p):=H(p,s_{i})$ then we can observe that

$d(H_{i}(p),H_{i+1}(p))\leq d(H(p,s_{i}),(p,s_{i+1}))\leq
C_{H}\cdot(s_{i+1}-s_{i})=\frac{inj_{N}}{2}.$ (1.209)

Then it is sufficient to apply the previous lemma to $H_{i}$ and $H_{i+1}$ and
observe that composition of R.-N.-lipschitz map is a R.-N.-lipschitz map. ∎

######  Corollary 1.5.9.

Let us consider a lipschitz map
$H:(M\times[0,1],g+g_{[0,1]}])\longrightarrow(N,h)$. For each $i$ in $[0,1]$,
let us denote by $f_{i}:(M,g)\longrightarrow(N,h)$ the map defined as
$f_{i}(p):=H(p,i)$. Then we have that the vector bundles $(H^{*}(TN),g_{S,H})$
and $(f^{*}_{i}(TN)\times[0,1],g_{S}\times g_{[0,1]})$ are R.-N.-lipschitz
equivalent and $\delta$ can choose less or equal to $inj_{N}$.

###### Proof.

Let us define the map $\overline{f}_{i}:M\times[0,1]\longrightarrow N$ as
$\overline{f}_{i}(p,s):=f_{i}(p)$. Observe that the Sasaki metric on
$\overline{f}_{i}^{*}(E)$ defined using $g$, $\overline{f}^{*}_{i}(h)$ and
$\overline{f}^{*}_{i}(\nabla^{LC}_{h})$ is the product metric between the
Sasaki metric on $f^{*}_{i}(E)$ and the metric $g_{[0,1]}$ on $[0,1]$.
Then we can conclude observing that the map

$\begin{split}\mathcal{H}:(M\times[0,1]\times[0,1],g\times g_{[0,1]}\times
g_{[0,1]})&\longrightarrow(N,h)\\\ (p,s,t)&\longrightarrow H(p,i\cdot
t+s\cdot(1-t))\end{split}$ (1.210)

is a lipschitz-homotopy between $H$ and $\overline{f}_{i}$. ∎

####  1.5.5 Lemmas about homotopy

In this section we study the $\mathcal{L}^{2}$-boundedness of the pullback of
some homotopies.

######  Lemma 1.5.10.

Let $f:(M,g)\longrightarrow(N,h)$ be a smooth lipschitz map and let
$\delta\leq inj_{N}$. Consider the homotopy
$H:(f^{*}(T^{\delta}N)\times[0,1],g_{S}\times g_{[0,1]})\longrightarrow(N,l)$
defined as

$H(p,w,t):=p_{f}(t\cdot w_{f(p)}).$ (1.211)

Then, if $f$ is a smooth R.-N.-lipschitz map, then also $H$ and
$(H,id_{[0,1]}):(f^{*}(T^{\delta}N)\times[0,1],g_{S}\times
g_{[0,1]})\longrightarrow(N\times[0,1],l\times g_{[0,1]})$ are R.-N.-lipschitz
maps.

######  Remark 32.

The map $H$ above is important for us because it is a $\Gamma$-equivariant
lipschitz-homotopy between $p_{f}$ and
$\overline{f}:f^{*}(T^{\delta}N)\longrightarrow N$. In particular, proving
this Lemma, we have that the maps $p_{id}:TN\longrightarrow N$ and the
projection $pr_{N}:TN\longrightarrow N$ are $\Gamma$-lipschitz-homotopic with
an homotopy $H$ which is a R.-N.-lipschitz map, and so $H$ induces a morphism
in (un)-reduced $L^{2}$-cohomology.

###### Proof.

Observe that $H=pr_{N}\circ(H,id_{[0,1]})$, where
$pr_{N}:N\times[0,1]\longrightarrow N$. We have that $pr_{N}$ is a
R.-N.-lipschitz map. This means that if $(H,id_{[0,1]})$ is a R.-N.-lipschitz
map, then also $H$ is a R.-N.-lipschitz map.
Moreover we can observe that $f^{*}(T^{\delta}N)\times\\{0,1\\}$ is a set of
null measure an so we can consider the interval $[0,1]$ as open.

Let us define the map

$\begin{split}(pr_{M},H,id_{(0,1)}):(f^{*}T^{\delta}N\times(0,1),g_{S}\times
g_{(0,1)})&\longrightarrow(M\times N\times(0,1),g\times h\times g_{(0,1)})\\\
(v_{f(p)},s)&\longrightarrow(p,p_{f}(s\cdot v_{p}),s).\end{split}$ (1.212)

This map is a submersion since we are considering $s\neq 0$. In particular we
have that $(pr_{M},H,id_{(0,1)})$ is lipschitz since it is composition of
lipschitz maps.
Moreover we have that
$(H,id_{(0,1)})=pr_{N\times(0,1)}\circ(pr_{M},H,id_{(0,1)})$ and so, applying
Proposition 1.3.15, we have

$Vol_{(H,id_{(0,1)})}(q,t)=\int_{M}Vol_{(pr_{M},H,id_{(0,1)})}(p,q,t)d\mu_{M}.$
(1.213)

Let us calculate the Fiber Volume of $(pr_{M},H,id_{(0,1)})$.
Similarly to the Lemma 1.5.1, we can consider for each $p$ in $M$ some normal
coordinates $\\{x^{i}\\}$ around $p$, some normal coordinates $\\{y^{j}\\}$
around $f(p)$ in $N$ and, considering the frame $\\{\frac{\partial}{\partial
x^{i}}\\}$, we have the coordinates $\\{x^{i},\mu^{j},t\\}$ on
$f^{*}(TN)\times[0,1]$ around $(p,0,0)$.
Moreover we can also consider the coordinates $\\{x^{i},y^{j},t\\}$ on
$M\times N\times[0,1]$.
Observe that, the coordinates $\\{x^{i},y^{j},t\\}$ are enough to cover the
image of $(id_{M},H,id_{[0,1]})$ which is contained in a $\delta$-neighborhood
of $Graph(f)\times[0,1]$.
Finally, respect to these coordinates, we have that

$(pr_{M},H,id_{(0,1)})(0,\mu^{j},t)=(0,t\cdot y^{j},t).$ (1.214)

Similarly as we did in Lemma 1.5.1 one can show that

$Vol_{f^{*}(TN)\times[0,1]}(0,\mu,t)=dx^{1}\wedge...\wedge dx^{m}\wedge
d\mu^{1}\wedge...\wedge d\mu^{n}\wedge dt$ (1.215)

and that

$Vol_{M\times
N\times[0,1]}(0,y,t)=\sqrt{det(H_{ij}(0,y))}dx^{1}\wedge...\wedge dx^{m}\wedge
dy^{1}\wedge...\wedge dy^{n}\wedge dt,$ (1.216)

where

$H_{ij}=\begin{bmatrix}1&&0&&0\\\ 0&&h_{l,s}(y)&&0\\\ 0&&0&&1\end{bmatrix}$
(1.217)

and $h_{l,s}$ are the components of the metric $h$ on $N$. Then we have that
$(pr_{M},H,id_{[0,1]})$ is a diffeomorphism with its image and so applying
Remark 21, its Fiber Volume on $im(pr_{M},H,id_{[0,1]})$ is given by

$|[(pr_{M},\tilde{h},id_{[0,1]})^{-1}]^{*}\frac{Vol_{f^{*}TN\times[0,1]}}{(pr_{M},\tilde{h},id_{[0,1]})^{*}Vol_{M\times
N\times[0,1]}}|$ (1.218)

and it is null otherwise.
Observe that, similarly to Lemma 1.5.1, in a point $(0,y^{j},t)$ we have that
the Fiber Volume of $(pr_{M},\tilde{h},id_{[0,1]})$ is

$\frac{Vol_{f^{*}TN\times[0,1]}}{(pr_{M},\tilde{h},id_{[0,1]})^{*}Vol_{M\times
N\times[0,1]}}=\frac{1}{t^{n}}(1+C(t,y)),$ (1.219)

where $C$ is a bounded function.

Let us consider now the projection $pr_{N\times[0,1]}:M\times
N\times(0,1)\longrightarrow N\times(0,1)$. Let us recall that, since the
Proposition 1.3.15, we have

$Vol_{(H,id_{(0,1)})}(q,t)=\int_{M}Vol_{(id_{M},H,id_{[0,1]})}(p,q,t)d\mu_{M}.$
(1.220)

We know that outside the image of $(id_{M},H,id_{[0,1]})$ the Fiber Volume
$Vol_{(id_{M},H,id_{[0,1]})}$ is null. Let us denote by

$\begin{split}H_{t}:M\times N&\longrightarrow N\\\ (p,q)&\longrightarrow
H(p,q,t).\end{split}$ (1.221)

we have that $Vol_{(id_{M},H,id_{[0,1]})}$ on $M\times\\{q\\}\times\\{t\\}$ is
null outside

$pr_{M}(H^{-1}_{t}(q))\times\\{q\\}\times\\{t\\}\supseteq[im(id_{M},H,id_{[0,1]})\cap
M\times\\{q\\}\times\\{t\\}]$ (1.222)

Since $H$ is lipschitz, applying Remark 3, we have

$H^{-1}_{t}(q)\subseteq\pi^{-1}(f^{-1}(B_{C_{H}\cdot t}(q)))\times\\{t\\}$
(1.223)

where $\pi:f^{*}TN\longrightarrow M$ is the projection of the bundle. So

$pr_{M}(H^{-1}_{t}(q))\times\\{q\\}\times\\{t\\}\subset f^{-1}(B_{C_{H}\cdot
t}(q))\times\\{q\\}\times\\{t\\}.$ (1.224)

Then using that $f$ is a R.-N.-lipschitz map, we have that

$\mu(f^{-1}(B_{C_{H}\cdot t}(q)))\leq K_{0}\mu_{N}(B_{C_{H}\cdot t}(q))\leq
K_{0}C_{H}^{n}t^{n}(1+L(t)),$ (1.225)

and so, since $N$ has bounded geometry,

$\mu_{N}(B_{C_{H}\cdot t}(q))\leq C_{H}^{n}t^{n}(1+L(t))$ (1.226)

where $L$ is a bounded function which depends by $t$. Then we have that the
Fiber Volume of $(H,id_{(0,1)})$ is given by

$\begin{split}Vol_{(H,id_{(0,1)})}(q,t)&=\int_{f^{-1}(B_{C_{H}\cdot
t}(q))}Vol_{(id_{M},H,id_{[0,1]})}(p,q,t)d\mu_{M}\\\
&\leq\frac{1}{t^{n}}(1+C(t,\tilde{y}))\cdot(K_{0}C_{H}^{n}t^{n}(1+L(t)))\\\
&\leq K\cdot\frac{1}{t^{n}}\cdot t^{n}\leq K.\end{split}$ (1.227)

and so $(H,id_{(0,1)})$ is a R.-N.-lipschitz map and so also

$H=pr_{N}\circ(H,id_{(0,1)})$ (1.228)

is an R.-N.-lipschitz map. ∎

Let us consider two uniformly proper lipschitz maps
$g:(S,v)\longrightarrow(M,m)$ and $f:(M,m)\longrightarrow(N,l)$ between
manifolds of bounded geometry. Let us define the maps

$\begin{split}\overline{g}:g^{*}T^{\sigma}M&\longrightarrow M\\\
v_{g(p)}&\longrightarrow g(p).\end{split}$ (1.229)

and

$\begin{split}p_{g}:g^{*}T^{\sigma}M&\longrightarrow M\\\
v_{g(p)}&\longrightarrow p_{g}(v_{g(p)}).\end{split}$ (1.230)

Consider the compositions $f\circ\overline{g}$ and $f\circ
p_{g}:g^{*}T^{\sigma}M\longrightarrow M$. Observe that these maps are
lipschitz-homotopy equivalent. Because of Proposition 1.5.8, we have that the
pullback bundles

$(f\circ\overline{g})^{*}TN\longrightarrow S$ (1.231)

and

$(f\circ p_{g})^{*}TN\longrightarrow S$ (1.232)

are R.-N.-lipschitz homotopy equivalent101010for $\delta\leq inj_{N}$.. Denote
by $h$ the homotopy

$\begin{split}h:g^{*}(T^{\sigma}M)\times[0,1]&\longrightarrow N\\\
(v_{g(p)},t)&\longrightarrow f\circ p_{g}(t\cdot v_{g(p)})\end{split}$ (1.233)

between $f\circ p_{g}$ and $f\circ\overline{g}$. Because of Corollary 1.5.9 we
have that $(h^{*}(TN),g_{S,h})$ and

$(f\circ\overline{g}^{*}(T^{\delta}N)\times[0,1],g_{S}\times g_{[0,1]})$
(1.234)

are R.-N.-lipschitz equivalent. Then we have that

$p_{h}:((f\circ\overline{g})^{*}(T^{\delta}N)\times[0,1],g_{S}\times
g_{[0,1]})\longrightarrow(N,l)$ (1.235)

is a R.-N.-lipschitz map because $h$ is uniformly proper and lipschitz map and
because of Remark 28.
Observe that $p_{h}$ is a lipschitz homotopy between the maps

$p_{f\circ\overline{g}}:f\circ\overline{g}^{*}(T^{\delta}N)\longrightarrow(N,l)$
(1.236)

and

$p_{f\circ p_{g}}:f\circ\overline{g}^{*}(T^{\delta}N)\longrightarrow(N,l).$
(1.237)

Applying the point 4. of Proposition 1.4.1, we can obtain that

$\begin{split}p_{f\circ p_{g}}&=p_{id_{M}}\circ F\circ P_{g},\\\ &=p_{f}\circ
P_{g}\end{split}$ (1.238)

where $F:f^{*}(T^{\delta}N)\longrightarrow T^{\delta}N$ is the bundle map
induced by $f$ and $P_{g}:(f\circ\overline{g})^{*}T^{\delta}N\longrightarrow
f^{*}T^{\delta}N$ is the bundle map induced by $p_{g}$.

######  Proposition 1.5.11.

The map $H$

$\begin{split}H:((f\circ\overline{g})^{*}T^{\delta}N\times[0,1],g_{S}\times
g_{[0,1]})&\longrightarrow(N,l)\\\ (v_{f\circ\overline{g}},t)&\longrightarrow
p_{f}(P_{g}(t\cdot v_{f\circ\overline{g}}))\end{split}$ (1.239)

is a R.-N.-lipschitz map.

###### Proof.

Observe that $H=p_{h}$ defined above. Observe that since Remark 27 we have
that $(f\circ\overline{g})^{*}T^{\delta}N\times[0,1]$ has bounded Ricci
curvature. Then, since $h$ is a smooth, uniformly proper and lipschitz map and
since Remark 28, we have that $H=p_{H}$ is a R.-N.-lipschitz map. ∎

####  1.5.6 Lipschitz-homotopy invariance of (un)-reduced $L^{2}$-cohomology

Consider the category $\mathcal{C}$ which has manifolds of bounded geometry as
objects and uniformly proper lipschitz maps as arrows. Consider, moreover, the
category Vec which has complex vector spaces as objects and linear maps as
arrows. In this section we will show that, for every $z$ in $\mathbb{N}$, the
association $\mathcal{F}_{z}:\mathcal{C}\longrightarrow\textbf{Vec}$ defined
as

$\begin{cases}\mathcal{F}_{z}(M,g)=H^{z}_{2}(M)\\\
\mathcal{F}_{z}((M,g)\xrightarrow{f}(N,h))=H^{z}_{2}(N)\xrightarrow{T_{f}}H^{z}_{2}(M)\end{cases}$
(1.240)

is a controvariant functor.
Moreover, we will show that if two maps $f_{1}$ and $f_{2}$ are uniformly
proper lipschitz-homotopic then $T_{f_{1}}=T_{f_{2}}$ in (un)-reduced
$L^{2}$-cohomology and that if $f$ is a $L^{2}$-map111111i.e. a smooth
R.-N.-lipschitz map which is also uniformly proper, then $f^{*}=T_{f}$ in
(un)-reduced $L^{2}$-cohomology.
This fact will imply that if $f$ is a lipschitz-homotopy equivalence between
manifolds of bounded geometry, then the $L^{2}$-cohomology groups are
isomorphic. In other words, the $L^{2}$-cohomology is a lipschitz-homotopy
invariant for manifolds of bounded geometry.

Let us introduce, the operator $\int_{0\mathcal{L}}^{1}$: it will be the main
tool that we will use to study homotopies.

######  Lemma 1.5.12.