2021

[1,2]Dr Stefano Spessato ORCID: 0000-0003-0069-5853

[1]Università degli Studi ”Niccolò Cusano”, via don Carlo Gnocchi 3, Rome,
00166, Italy

# Pullback functors for reduced and unreduced $L^{q,p}$-cohomology

<EMAIL_ADDRESS>*

###### Abstract

In this paper we study the reduced and unreduced $L^{q,p}$-cohomology groups
of oriented manifolds of bounded geometry and their behaviour under uniform
maps. A uniform map is a uniformly continuous map such that the diameter of
the preimage of a subset is bounded in terms of the diameter of the subset
itself. In general, for each $p,q\in[1,+\infty)$, the pullback map along a
uniform map does not induce a morphism between the between the spaces of
$p$-integrable forms or even in $L^{q,p}$-cohomology. Then our goal is to
introduce, for each $p$ in $[1,+\infty)$ and for each uniform map $f$ between
manifolds of bounded geometry, an $\mathcal{L}^{p}$-bounded operator $T_{f}$,
such that it does induce in a functorial way the appropriate morphism in
reduced and unreduced $L^{q,p}$-cohomology.

###### keywords:

$L^{q,p}$-cohomology, bounded geometry, pullback, Fiber Volume

### Acknowledgments

I am grateful to Paolo Piazza, my advisor, and to Vito Felice Zenobi for the
several discussions that we had and for the competences they shared with me. I
would also like to thank Francesco Bei and Thomas Schick for their advice.
Finally my thanks go to the reviewers for their patience in correcting my
manuscript.
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## Introduction

In this paper our goal is to define, for each $p\in[1,+\infty)$ and for each
uniform map $f$ between manifolds of bounded geometry, a pullback operator
$T_{f}$ between the spaces of $p$-integrable forms. It is a well-known fact,
indeed, that the classical pullback of an $\mathcal{L}^{p}$-form is not, in
general, an $\mathcal{L}^{p}$-form. This means that the pullback is not
inducing a well-defined morphism between the reduced and unreduced
$L^{q,p}$-cohomology. On the other hand we prove that $T_{f}$ does induce some
morphisms in (un)-reduced $L^{q,p}$-cohomology and we also prove that the
functorial properties hold.
More formally, fix the category $\mathcal{C}$ which has oriented manifolds of
bounded geometry as objects and uniform maps as arrows. Let Vec be the
category which has complex vector spaces as objects and linear maps as arrows.
Then we prove that, for every $z$ in $\mathbb{N}$,

$\begin{cases}\mathcal{F}_{z}(M,g)=H^{z}_{q,p}(M)\\\
\mathcal{F}_{z}((M,g)\xrightarrow{f}(N,h))=H^{z}_{q,p}(N)\xrightarrow{T_{f}}H^{z}_{q,p}(M)\end{cases}$
(1)

is a controvariant functor. The same happens if we define
$\overline{\mathcal{F}}_{z}(M,g):=\overline{H}^{z}_{q,p}(M)$. Moreover, we
will show that if two maps $f_{1}$ and $f_{2}$ are uniformly homotopic then
$T_{f_{1}}=T_{f_{2}}$ in (un)-reduced $L^{q,p}$-cohomology and in
$L^{q,p}$-quotient cohomology. Finally we will show that if the naive pullback
$f^{*}$ does induce a morphism in reduced and unreduced $L^{q,p}$-cohomology,
then $f^{*}=T_{f}$.
As a consequence of the existence of these functors, we obtain that the
reduced and unreduced $L^{q,p}$-cohomology of a manifold of bounded geometry
is invariant under uniform homotopy equivalence. This result is stated in
Corollary 4.18. Finally, as a consequence of the existence of the functors
$\overline{\mathcal{F}}_{z}$, the invariance of the $L^{2}$-index of the
signature operator defined by Bei at page 20 of Bei3  under uniform homotopy
equivalence for manifolds of bounded geometry is proved.
The idea of the operator $T_{f}$ comes from the work of Hilsum and Skandalis
hils : in their paper, the authors define an $\mathcal{L}^{2}$-bounded
operator $T_{f}$ for compact manifolds. Our operator $T_{f}$ is a bounded
geometry version of their one.

The structure of the paper is the following: in the first section we introduce
the objects and the arrows of the category $\mathcal{C}$. In particular, we
show that every uniform map can be approximated by a smooth map with uniformly
bounded derivatives. In the second one we introduce the reduced and unreduced
$L^{q,p}$-cohomology. Moreover we also introduce the Fiber Volume of a
Lipschitz submersion $\pi:(M,g)\longrightarrow(N,h)$. This is a real function
defined on $(N,h)$ such that its boundedness implies the
$\mathcal{L}^{p}$-boundedness of $\pi^{*}$. In the third Section, given a
smooth uniform map $f:(M,g)\longrightarrow(N,h)$, we introduce a Lipschitz
submersion $p_{f}:f^{*}TN\longrightarrow N$ such that $p_{f}(0_{x})=f(x)$. In
the last section, we introduce a specific Thom form $\omega$ for the bundle
$f^{*}TN$, we define $T_{f}$ and, in conclusion, we prove the functorial
properties. Finally we prove, as a consequence of the main result, the uniform
homotopy invariance of the $L^{2}$-index of the signature operator defined by
Bei in Bei3 .

##  1 Maps between manifolds of bounded geometry

###  1.1 Uniform maps and uniform homotopy

Let us consider two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$.

######  Definition 1.1.

A map $f:(X,d_{X})\longrightarrow(Y,d_{Y})$ is uniformly continuous if for
each $\epsilon>0$ there is a $\delta(\epsilon)>0$ such that for each $x_{1}$,
$x_{2}$ in $X$

$d_{X}(x_{1},x_{2})\leq\delta(\epsilon)\implies
d_{Y}(f(x_{1}),f(x_{2}))\leq\epsilon.$ (2)

Moreover $f$ is uniformly (metrically) proper if for each $R\geq 0$ there is a
number $S(R)>0$ such that for each subset $A$ of $(Y,d_{Y})$

$diam(A)\leq R\implies diam(f^{-1}(A))\leq S(R).$ (3)

A map $f:(X,d_{X})\longrightarrow(Y,d_{Y})$ is a uniform map if it is
uniformly continuous and uniformly proper.

######  Remark 1.

Compositions of uniform maps are also uniform.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be two metric spaces and fix two actions of a
group $\Gamma$ on $X$ and $Y$. Assume that $\Gamma$ acts by isometries on $X$
and on $Y$.

######  Definition 1.2.

Two maps $f_{0}$ and $f_{1}:(X,d_{X})\longrightarrow(Y,d_{Y})$ are
$\Gamma$-uniformly-homotopic if they are $\Gamma$-equivariant maps and they
are homotopic with a uniformly continuous homotopy
$H:(X\times[0,1],d_{X}\times d_{[0,1]})\longrightarrow(Y,d_{Y})$ which is
$\Gamma$-equivariant111The action of $\Gamma$ on $X\times[0,1]$ is defined as
$\gamma(x,t):=(\gamma x,t)$ for each $\gamma$ in $\Gamma$.
We will denote it by

$f_{1}\sim_{\Gamma}f_{2}.$ (4)

Moreover $f_{1}$ and $f_{2}$ are $\Gamma$-Lipschitz-homotopic if
$f_{1}\sim_{\Gamma}f_{2}$ and $H$ is a Lipschitz map.

######  Remark 2.

Consider two uniformly continuous maps $F,f:(X,d_{X})\longrightarrow(Y,d_{Y})$
such that $f\sim_{\Gamma}F$. Then, if $f$ is a uniformly proper map, also the
homotopy $h$ is uniformly proper. As a consequence of this, in particular,
also $F$ is uniformly proper.
In order to prove it fix $\epsilon>0$. Since $h$ is uniformly continuous,
there is a $\delta(\epsilon)$ such that

$d(t,s)\leq\delta(\epsilon)\implies d(h(x,t),h(x,s))\leq\epsilon.$ (5)

Fix $t$ in $[0,1]$. If we divide $[0,t]$ in $N_{t}$ intervals of length less
or equal to $\delta(\epsilon)$, we obtain that

$\begin{split}d(f(x),h(x,t))&\leq d(h(x,0),h(x,t))\\\ &\leq
d(h(x,0),h(x,t_{1}))+...+d(h(x,t_{N_{s}-1}),h(x,t))\\\ &\leq
N_{t}\epsilon=:R(t).\end{split}$ (6)

Observe that $R(t_{1})\leq R(t_{2})$ if $t_{1}\leq t_{2}$ and
$R(1)=\frac{\epsilon}{\delta(\epsilon)}$.
If $A$ is a subset of $Y$ and $h_{t}:X\longrightarrow Y$ is defined as
$h_{t}(p):=h(p,t)$, then

$\begin{split}h_{t}^{-1}(A)&=\\{p\in X|h(p,t)\in A\\}\\\ &\subseteq\\{p\in
X|f(p)\in B_{R(t)}(A)\\}=f^{-1}(B_{R(t)}(A))\end{split}$ (7)

where $B_{R(t)}(A)$ are the points $y$ of $Y$ such that $d(y,A)\leq R(t)$.
Then

$h^{-1}(A)\subseteq f^{-1}(B_{R(1)}(A))\times[0,1].$ (8)

and consequently, since $f$ is uniformly proper, and since
$diam(B_{R(1)}(A))\leq diam(A)+2R(1)$,

$\begin{split}diam(h^{-1}(A))&\leq diam(f^{-1}(B_{R(1)}(A)))+diam([0,1])\\\
&\leq S(diam(A)+2R(1))+1.\end{split}$ (9)

######  Definition 1.3.

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

  * •

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

  * •

$g$ is uniformly continuous,

  * •

$f\circ g\sim_{\Gamma}id_{N}$ and $g\circ f\sim_{\Gamma}id_{M}$.

###  1.2 Manifolds of bounded geometry

In this section we introduce the notion of manifolds of bounded geometry. All
the definitions and propositions below can be found in Chapter 2 of the thesis
of Eldering bound .

######  Definition 1.4.

A 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. $\forall i=0,...,k$ there is a constant $V_{i}$ such that
$|\nabla^{i}K(x)|\leq V_{i}$ for all $x$ in $M$.

  * •

there is a number $C>0$ such that for all $p$ in $M$ the injectivity radius
$inj_{M}(p)$ in $p$ satisfies $inj_{M}(p)\geq C.$ The maximal number which
satisfies this inequality will be denoted by $inj_{M}$.

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

######  Definition 1.5.

Let $(M,g)$ be a manifold of bounded geometry. We define $\delta>0$ to be
$M$-small if, for each $k>0$ in $\mathbb{N}$,

  * •

there is a constant $C_{k}>0$ such that for all $x_{1},x_{2}\in M$ with
$d(x_{1},x_{2})<\delta$ the coordinate transition map
$\phi_{2,1}=exp^{-1}_{x_{2}}\circ exp_{x_{1}}:U\longrightarrow T_{x_{2}}M$
where $U=exp_{x_{1}}^{-1}(B_{\delta}(x_{1})\cap B_{\delta}(x_{2}))\subset
T_{x_{1}}M$ is $C^{k-1}$-bounded222This means that all the derivatives of
degree less or equal to $k-1$ are bounded. with $|\phi_{2,1}|_{k-1}\leq
C_{k}$.

  * •

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 $x$.

As reported in page 45 of bound , every manifold $(M,g)$ of bounded geometry
admits an $M$-small $\delta$.

######  Remark 3.

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

$Ric(M)\geq(n-1)C$ (10)

where $C$ is a constant. In particular, if $M$ has $k_{\geq 0}$-bounded
geometry, then (10) is satisfied using $C=-V_{0}$ where $V_{0}$ bounds the
norm of sectional curvature. Let us denote the measure $\mu_{M}$ induced by
$g$ on $M$. Because of the Bishop-Gromov inequality, if $(M,g)$ satisfies
(10), then for each $p\in M$ and for each $r\geq 0$ we obtain
$\mu_{M}(B_{r}(p))\leq Q(r)$ for some function $Q:\mathbb{R}_{\geq
0}\longrightarrow\mathbb{R}_{\geq 0}$. So, if $A\subset M$ has $diam(A)=r$,

$\mu_{M}(A)\leq\mu_{M}(B_{diam(A)}(p))\leq C(diam(A)),$ (11)

where $p$ is a point of $A$.

###  1.3 Uniformly proper and discontinuous actions

######  Definition 1.6.

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

  * •

the action of $\Gamma$ is free,

  * •

there is a number $\delta>0$ such that $d_{X}(x,\gamma x)\leq\delta\implies
x=\gamma x.$

######  Remark 4.

If the action of $\Gamma$ is FUPD, then it is properly discontinuous and free.

######  Proposition 1.1.

Let $(M,g)$ be a manifold of bounded geometry. Consider $\Gamma$ a group
acting on $M$ by isometries. Suppose the action of $\Gamma$ is free and
properly discontinuous. Then the following statements are equivalent:

  1. 1.

a group $\Gamma$ acting FUPD on $M$ and $N$,

  2. 2.

the quotient $M/\Gamma$ has bounded geometry333We are considering on
$M/\Gamma$ the Riemannian metric induced by $g$.

Proof: $1\implies 2$. Let $\delta$ be the constant of the FUPD action of
$\Gamma$. Then $B_{\delta}(p)$ is a trivializing open of $M/\Gamma$ for each
$p$ in $M$. So $inj_{M/\Gamma}\geq\min\\{inj_{M},\delta\\}$ and the curvature
of $M/\Gamma$ 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 $d_{M}(p,\gamma p)\leq\delta$. Suppose
$\delta<\min\\{inj_{M},inj_{M/\Gamma}\\}$. Then there is a vector $v$ in
$T_{p}M$ whose norm equals $\delta$ and $exp_{p}(v)=\gamma p$. Consider the
Riemannian covering $s:M\longrightarrow M/\Gamma$. Observe that

$s\circ exp_{p}=exp_{s(p)}\circ ds$ (12)

Then

$s(\gamma p)=s\circ exp_{p}(v)=exp_{s(p)}\circ ds(v).$ (13)

Since $ds$ is an isometry, the norm of $ds(v)$ is less or equal to
$inj_{M/\Gamma}$. This means that $ds(v)=0$ and, in particular, $v=0$. Then

$\gamma p=exp_{p}(v)=exp_{p}(0)=p.$ (14)

Because the action of $\Gamma$ is free, $\gamma=id_{\Gamma}$

###  1.4 Lipschitz approximation of a uniformly continuous map

Let $(M,g)$ and $(N,h)$ be two manifolds of bounded geometry and $f\in
C^{k}(M,N)$.

######  Definition 1.7.

A map $f$ is of class $C^{k}_{b}$ if there exist $M$-small $\delta_{M}$ and an
$N$-small $\delta_{N}>0$ such that for each $x\in X$ we have
$f(B_{\delta_{M}}(x))\subset B_{\delta_{N}}(f(x))$ and the composition

$F_{p}=exp^{-1}_{f(p)}\circ f\circ exp_{p}:B_{\delta_{M}}(0)\subset
T_{p}M\longrightarrow T_{f(p)}N$ (15)

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

######  Remark 5.

There are some remarks we wish to make:

  * •

The assumption of bounded geometry is necessary in order to define
$C^{k}_{b}$-maps. More details can be found in pages 44-45 of bound .

  * •

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

  * •

Uniformly continuous maps are $C^{0}_{b}$-maps.

  * •

A $C^{1}_{b}$-map is a Lipschitz map.

Consider a uniform map $f:(M,g)\longrightarrow(N,h)$ between manifolds of
bounded geometry. Let us suppose, moreover, that there is a group $\Gamma$ on
$M$ and $N$ acting FUPD and assume that $f$ is $\Gamma$-equivariant. We want
to show that there is a $\Gamma$-equivariant uniform map
$F:(M,g)\longrightarrow(N,h)$ which is $C^{k}_{b}$ for all $k$ and that
$F\sim_{\Gamma}f$.
In order to prove this fact we need the Lemma 2.34 of the thesis of Eldering
bound .

######  Lemma 1.2.

Let $r,h>0$, $g\in C^{k}(B_{r+2h}(0)\subset\mathbb{R}^{m},\mathbb{R}^{n})$.
Then for each $\epsilon>0$ there is a $\nu_{0}$ such that $g$ can be
approximated by a function $G_{\nu}$ for each $0<\nu\leq\nu_{0}$ such that

  * •

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

  * •

$G_{\nu}$ is a smooth function in $B_{r}(0)$;

  * •

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

  * •

$|G_{\nu}|_{l}\leq C(\nu,l)|g|_{0}$ 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. Consider a mollifier,
i.e. a compactly supported, smooth, positive function
$\phi:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ such that its integral over
$\mathbb{R}^{n}$ equals to 1. Suppose that the support of $\phi$ is contained
in the Euclidean ball of radius $1$ in $\mathbb{R}^{m}$. Let us consider a
non-increasing smooth map $\chi:[0,r+2h]\longrightarrow[0,1]$ which is $1$ if
$x\leq r$ and $0$ if $x\geq r+h$. Then we define

$G_{\nu}(x)=(1-\chi(||x||))g(x)+\chi(||x||)\int_{B^{m}_{1}(0)}g(x-\nu
y)\phi(y)dy.$ (16)

Observe that if $\nu=0$ then $G_{\nu}=g$.

######  Remark 7.

Fix $\epsilon>0$. By the proof of Lemma 2.34 of bound , if
$\nu\leq\delta(\epsilon)$ where $\delta(\epsilon)$ satisfies

$d(p,q)\leq\delta(\epsilon)\implies d(g(p),g(q))\leq\epsilon$ (17)

then

$d(g(p),G_{\nu}(p))\leq\epsilon.$ (18)

Let us define $\delta_{\chi}:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0}$ a
function such that for each $\epsilon>0$

$|x_{1}-x_{0}|\leq\delta_{\chi}(\epsilon)\implies|\chi(||x_{1}||)-\chi(||x_{0}||)|\leq\epsilon.$
(19)

Observe that, without loss of generality, we can always suppose that
$\delta:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0}$ is non-decreasing and,
at the same time,

$\delta(\epsilon)\leq\delta_{\chi}(\frac{\epsilon}{|g|_{0}})$ (20)

where $|g|_{0}:=\max\limits_{p\in B_{1}^{m}(0)}\\{|g(p)|\\}$. Under this
assumption on $\delta$, if we define
$\tilde{\delta}(\epsilon):=\delta(\frac{\epsilon}{4})$, we obtain that

$d(p,q)\leq\tilde{\delta}(\epsilon)\implies
d(G_{\nu}(p),G_{\nu}(q))\leq\epsilon.$ (21)

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

######  Proposition 1.3.

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

  * •

$d(F(p),f(p))\leq\epsilon$ for all $p$ in $M$,

  * •

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

  * •

Consider $\Gamma$ a group that acts FUPD by isometries on $M$ and $N$. Assume
that $f$ is $\Gamma$-equivariant. Then $F$ can be chosen to be
$\Gamma$-equivariant,

  * •

$f\sim_{\Gamma}F$.

This implies that if $f$ is a uniform map, then, for each $\epsilon>0$, there
is a smooth, uniformly proper and Lipschitz approximation $f_{\epsilon}$ of
$f$.

Proof: Since $f$ is uniformly continuous, there are two positive numbers
$\sigma_{1}<\sigma_{2}$ both $N$-small and there is an $M$-small number
$\delta_{f}$ such that $f(B_{\delta_{f}}(q))\subseteq B_{\sigma_{1}}(f(q))$.
Let us consider the Riemannian manifold $X:=M/\Gamma$. Let us define
$R_{0}:=\frac{1}{4}min\\{\delta_{f},inj_{X}\\}$.
Following Proposition 1.1, observe 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.
By Lemma 2.16 of bound , there is a number $K$ and a countable cover of $X$
given by $\\{B_{\delta_{1}}(x_{i})\\}$ such that for all $x$ in $X$ the ball
$B_{\delta_{2}}(x)$ intersects at most $K$ balls of
$\\{B_{\delta_{2}}(x_{i})\\}_{i}$.
Consider the preimage of the cover $\\{B_{\delta_{2}}(x_{i})\\}_{i}$ on $M$.
Since $\delta_{2}<\delta_{0}$, it has the form
$\\{\sqcup_{\gamma\in\Gamma}B_{\delta_{2}}(\gamma\tilde{x}_{i})\\}_{i}$, where
$\tilde{x}_{i}$ is an element of the fiber of $x_{i}$.
Moreover $f(B_{\delta_{2}}(q))\subseteq B_{\sigma_{1}}(f(q))$ for all $q$ in
$M$.

Fix $\epsilon<\sigma_{2}-\sigma_{1}$ and let $C$ be a number which is greater
then the Lipschitz constants of $\exp_{f(\gamma\tilde{x}_{i})}$,
$exp_{\gamma\tilde{x}_{i}}$ and their inverses. Fix $F_{0}:=f$. Then define
for all $i$ in $\mathbb{N}$

$F_{i+1}(p):=\begin{cases}\exp_{f(\gamma\tilde{x}_{i})}\circ
G_{i,\gamma,\nu}\circ exp_{\gamma\tilde{x}_{i}}^{-1}\mbox{ if }p\in
B_{R_{0}}(\bigcup_{\gamma}\\{\gamma\tilde{x}_{i}\\})\\\ F_{i}(p)\mbox{
otherwise}\end{cases}$ (22)

where $G_{i,\gamma,\nu}:B_{\delta_{2}}(0)\subset
T_{\gamma\tilde{x}_{i}}M\cong\mathbb{R}^{m}\longrightarrow
B_{\sigma_{2}}(0)\subset T_{f(\gamma\tilde{x}_{i})}N\subset\mathbb{R}^{n}$ is
a $C^{k}_{b}$-approximation of
$g_{i,\gamma}=\exp_{f(\gamma\tilde{x}_{i})}^{-1}\circ F_{i}\circ
exp_{\gamma\tilde{x}_{i}}$ defined by using a function $\chi$ as in Remark 6
and $\nu$ such that

$d(p,q)\leq\nu\implies d(f(p),f(q))\leq\frac{\epsilon}{CK}.$ (23)

In particular, in order to obtain the $\Gamma$-equivariance of $F_{i+1}$, we
choose for each $i$ a $\tilde{x}_{i}$ in the fiber of $x_{i}$. Fix an
orthonormal basis of $T_{\tilde{x}_{i}}M$ and an orthonormal basis of
$T_{f(\tilde{x}_{i})}N$. We define $G_{i,e,\nu}$ with respect to the
coordinates on the tangent spaces induced by these basis. Then we define for
each $\gamma$ in $\Gamma$

$G_{i,\gamma,\nu}:=d\gamma\circ G_{i,e,\nu}\circ d\gamma^{-1}.$ (24)

Observe that, by the Gluing lemma (Theorem III.9.4, p. 83. of the book of
Dugundji Dug ), $F_{i}$ is well-defined. Finally we define

$F(p):=\lim_{i\rightarrow+\infty}F_{i}(p).$ (25)

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 $s(B_{\delta_{2}}(p))$ intersects at most $K$
balls $B_{\delta_{2}}(x_{i})$. Moreover, since
$\delta_{2}<\frac{1}{4}\delta_{0}$, then for all of these $x_{i}$ there exists
only one $\gamma$ in $\Gamma$ such that $B_{\delta_{2}}(p)$ intersects
$B_{\delta_{2}}(\gamma\tilde{x}_{i})$.
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.

  * •

Let us denote by $x:=exp^{-1}_{\gamma\tilde{x}_{i}}(p)$. We obtain
$d(F(p),f(p))\leq\epsilon.$ Indeed, by applying Proposition 1.2 and Remark 7,

$\begin{split}d(f(p),F(p))&\leq
d(f(p),F_{i_{1}}(p))+...+d(F_{i_{K}}(p),F_{i_{K+1}}(p))\\\ &\leq
C(|g_{0,\gamma,\nu}(x)-G_{0,\gamma,\nu}(x)|+...+|g_{i_{K},\gamma,\nu}(x)-G_{i_{K},\gamma,\nu}(x)|)\\\
&\leq C\cdot K\cdot\frac{\epsilon}{CK}=\epsilon\end{split}$ (26)

  * •

$F$ is a $C^{k}_{b}$-map for each $k$ in $\mathbb{N}$. Consider a point $p$.
As a consequence of Lemma 2.16 of bound , there is a finite sequence
$i_{1},...,i_{K}$ of the indexes $i$ such that $F_{i_{j}+1}\neq F_{i_{j}}$ on
$B_{\delta_{2}}(p)$. So, we have to study the boundedness of the derivatives
of

$\begin{split}F_{p}:&=exp^{-1}_{F_{i_{K}+1}(p)}\circ F_{i_{K}+1}\circ
exp_{p}\\\ &=exp^{-1}_{F_{i_{K}+1}(p)}\circ\exp_{f(\gamma x_{K})}\circ
G_{i_{K},\gamma,\nu}\circ exp_{\gamma x_{i_{K}}}^{-1}\circ
exp_{p}.\end{split}$ (27)

Observe that $exp^{-1}_{F_{i_{K}+1}(p)}\circ\exp_{f(\gamma\tilde{x}_{i})}$ and
$exp_{\gamma x_{i_{K}}}^{-1}\circ exp_{p}$ are changes of normal coordinates,
and so, since we are in a bounded geometry setting, they are $C^{k}_{b}$-maps
with uniformly bounded norm. Consider $G_{i_{K},\gamma,\nu}$. By applying
Proposition 1.2, we obtain

$|G_{i_{K},\gamma,\nu}|_{k}\leq C(\nu,k)|g_{i_{K},\gamma,\nu}|_{0}\leq
C(\nu,k)\sigma_{2}.$ (28)

  * •

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

$\gamma F(p)=\gamma F_{i_{K}+1}(p)=F_{i_{K}+1}(\gamma p)=F(\gamma p).$ (29)

In order to conclude the proof we have to show that $F$ and $f$ are
$\Gamma$-uniformly-homotopic.
Let us define the map $H_{0}:(M\times[0,1],g+dt^{2})\longrightarrow(N,h)$ as
$H_{0}(p,t):=f(p)$. Fix for each $i$ a $\tilde{x}_{i}$ in the fiber of
$x_{i}$. Fix an orthonormal basis of $T_{\tilde{x}_{i}}M$ and an orthonormal
basis of $T_{f(\tilde{x}_{i})}N$. We denote by $H_{i,e,\nu}$ the
$C^{k}_{b}$-approximation444With respect to the coordinates on the tangent
spaces induced by these basis.  of

$h_{i,\gamma,t}=\exp_{f(\gamma\tilde{x}_{i})}^{-1}\circ H_{i}(\cdot,t)\circ
exp_{\gamma\tilde{x}_{i}}$ (30)

defined by using $t\cdot\nu$ and the function $\chi$.
Then, for each $\gamma$ in $\Gamma$ we have
$H_{i,\gamma,t}:B_{\delta_{2}}(0)\subset
T_{\gamma\tilde{x}_{i}}M\cong\mathbb{R}^{m}\longrightarrow
B_{\sigma_{2}}(0)\subset T_{f(\gamma\tilde{x}_{i})}N\subset\mathbb{R}^{n}$
which is the map

$H_{i,\gamma,t}:=d\gamma\circ H_{i,e,t\cdot\nu}\circ d\gamma^{-1}.$ (31)

So, for each $i\in\mathbb{N}$, we define

$H_{i+1}(p,t):=\begin{cases}\exp_{f(\gamma\tilde{x}_{i})}\circ
H_{i,\gamma,t}\circ exp_{\gamma\tilde{x}_{i}}^{-1}\mbox{ if }p\in
B_{inj_{M}}(\cup_{\gamma}\\{\gamma\tilde{x}_{i}\\})\\\ H_{i}(p,t)\mbox{
otherwise}\end{cases}$ (32)

Finally we define the map $H:(M\times[0,1],g+dt^{2})\longrightarrow(N,h)$ as

$H(p,t)=\lim_{i\rightarrow+\infty}H_{i}(p,t).$ (33)

As well as for $F$, also $H$ is well-defined and $\Gamma$-equivariant.
We will check the uniformly continuity of $H$ in the following way. First we
observe that, by definition, $H_{0}$ is uniformly continuous. Then for each
$\varepsilon$ there is a $\delta_{0}(\epsilon)$ such that

$d((p_{0},t),(q,s))\leq\delta_{0}(\varepsilon)\implies
d(H_{0}(p_{0},t),H_{0}(q,s))\leq\varepsilon.$ (34)

Observe that, since $H_{0}$ is uniformly continuous, then $\delta_{0}$ does
not depends on $(p_{0},t)$. Moreover, without loss of generality we can also
suppose that
$\delta_{0}(\varepsilon)\leq\delta_{\chi}(\frac{\varepsilon}{\sigma_{2}})$ and
that $\delta_{0}$ is non-decreasing with respect to $\varepsilon$.
Let us assume now that $H_{i}$ is continuous. Fix an $\varepsilon$ and a point
$(p,t)$ and let us denote by $\delta_{i}(\varepsilon,p,t)$ a number such that,
for each $q$ in $M$

$d((p,t),(q,s))\leq\delta_{i}(\varepsilon,p,t)\implies
d(H_{i}(p,t),H_{i}(q,s))\leq\varepsilon.$ (35)

Assume that
$\delta_{i}(\varepsilon,p,t)\leq\delta_{\chi}(\frac{\varepsilon}{\sigma_{2}})$
and that $\delta_{i}(\cdot,p,t)$ is non-decreasing with respect to
$\varepsilon$ for each $p$ and $t$.
Choose $p_{0}$ in $M$: if
$B_{\delta_{2}}(p_{0})\cap[\sqcup_{\gamma\in\Gamma}B_{\delta_{2}}(\gamma\tilde{x}_{i})]=\emptyset$,
then

$d((p_{0},t),(q,s))\leq\delta_{i}(\varepsilon,p_{0},t)\implies
d(H_{i+1}(p_{0},t),H_{i+1}(q,s))\leq\varepsilon.$ (36)

So we can set

$\delta_{i+1}(\varepsilon,p_{0},t):=\delta_{i}(\varepsilon,p_{0},t).$ (37)

Let us suppose that
$B_{\delta_{2}}(p_{0})\cap[\sqcup_{\gamma\in\Gamma}B_{\delta_{2}}(\gamma\tilde{x}_{i})]\neq\emptyset$.
In this case we have to study the uniform continuity of

$\exp_{f(\gamma\tilde{x}_{i})}\circ H_{i,\gamma,t}\circ
exp_{\gamma\tilde{x}_{i}}^{-1}.$ (38)

In particular, we start studying the distance

$d(H_{i+1}(p_{0},t),H_{i+1}(p_{0},s)).$ (39)

Recall that $h_{i,\gamma,t}:T_{\gamma\tilde{x}_{i}}M\longrightarrow
T_{f(\gamma\tilde{x}_{i})}N$ is defined as

$h_{i,\gamma,t}=\exp_{f(\gamma\tilde{x}_{i})}^{-1}\circ H_{i}(\cdot,t)\circ
exp_{\gamma\tilde{x}_{i}}$ (40)

and that $H_{i,\gamma,t}$ is its approximation defined by using $t\cdot\nu$
and $\chi$.
Observe that the exponential maps and their inverses are Lipschitz maps with
uniformly bounded constant $C$. Assume $C>1$.
Fix a $\varepsilon>0$ and let us denote by
$x_{0}:=exp^{-1}_{\gamma\tilde{x}_{i}}(p_{0})$. Define

$\delta_{h,i}(\varepsilon,x_{0},t):=\frac{1}{C}\delta_{i}(\frac{\varepsilon}{C},p_{0},t).$
(41)

Observe that
$\delta_{h,i}(\varepsilon,x_{0},t)\leq\delta_{i}(\varepsilon,p_{0},t)\leq\delta_{\chi}(\frac{\varepsilon}{\sigma_{2}})$.
Moreover, $h_{i,\gamma,t}$ is also continuous in $t$, indeed,

$|(x_{0},t)-(x_{2},s)|\leq\delta_{h,i}(\varepsilon,x_{0},t)\implies
d(h_{i,\gamma,t}(x_{0}),h_{i,\gamma,s}(x_{2}))\leq\varepsilon.$ (42)

Then for any $s$ in $[0,1]$ such that
$|s-t|\leq\delta_{h,i}(\frac{\varepsilon}{2},x_{0},t)$, then

$\begin{split}&|H_{i,\gamma,s}(x_{0})-H_{i,\gamma,t}(x_{0})|\leq(1-\chi(||x_{0}||))|h_{i,\gamma,s}(x_{0})-h_{i,\gamma,t}(x_{0})|\\\
&+\chi(||x_{0}||)\int_{B^{m}}|h_{i,\gamma,s}(x_{0}-s\cdot\nu
y)-h_{i,\gamma,s}(x_{0}-t\cdot\nu y)|\phi(y)dy\\\
&+\chi(||x_{0}||)\int_{B^{m}}|h_{i,\gamma,s}(x_{0}-t\cdot\nu
y)-h_{i,\gamma,t}(x_{0}-t\cdot\nu y)|\phi(y)dy\\\
&\leq(1-\chi(||x_{0}||))\frac{\varepsilon}{2}+\chi(||x_{0}||)\frac{\varepsilon}{2}+\frac{\varepsilon}{2}\leq\varepsilon.\end{split}$
(43)

This means that, given

$\tilde{\delta_{i}}(\varepsilon,p_{0},t):=\frac{1}{C}\delta_{h,i}(\frac{\varepsilon}{2C},p_{0},t)=\frac{1}{C^{2}}\delta_{i}(\frac{\varepsilon}{2C^{2}},p_{0},t),$
(44)

then, since $H_{i+1}=\exp_{f(\gamma\tilde{x}_{i})}\circ H_{i,\gamma,t}\circ
exp_{\gamma\tilde{x}_{i}}^{-1}$,

$d((p_{0},t),(p_{0},s))\leq\tilde{\delta_{i}}(\varepsilon)\implies
d(H_{i+1}(p_{0},t),H_{i+1}(p_{0},s))\leq\varepsilon.$ (45)

Let us consider, now, the distance

$d(H_{i+1}(p_{0},t),H_{i+1}(q,t)).$ (46)

Let us denote by
$\Delta_{i}(\varepsilon):=\delta_{h,i}(\frac{\varepsilon}{4})$. Then, by
Remark 7 and by

$\delta_{h,i}(\frac{\varepsilon}{4})\leq\delta_{i}(\frac{\varepsilon}{4})\leq\delta_{\chi}(\frac{\varepsilon}{4\sigma_{2}})\leq\delta_{\chi}(\frac{\varepsilon}{4|h_{i,\gamma,t}|_{0}})$
(47)

we obtain that

$d(x_{1},x_{2})\leq\Delta_{i}(\varepsilon)\implies
d(H_{i,\gamma,t}(x_{1}),H_{i,\gamma,t}(x_{2}))\leq\varepsilon.$ (48)

Moreover, if we define

$\hat{\delta_{i}}(\varepsilon):=\frac{1}{C}\Delta_{i}(\frac{\varepsilon}{C},p_{0},t),$
(49)

then

$d(p_{0},q)\leq\hat{\delta_{i}}(\varepsilon,p_{0},t)\implies
d(H_{i+1}(p_{0},t),H_{i+1}(q,t))\leq\varepsilon.$ (50)

So given a point $p_{0}$ such that $B_{\delta_{2}}(p_{0})$ intersects
$\bigsqcup\limits_{\gamma\in\Gamma}B_{\delta_{2}}(\gamma\tilde{x}_{i})$, then

$d((p_{0},t),(q,s))\leq\delta_{i+1}(\varepsilon,p_{0},t)\implies
d(H_{i+1}(p_{0},t),H_{i+1}(q,s))\leq\varepsilon.$ (51)

where

$\begin{split}\delta_{i+1}(\varepsilon,p_{0},t):&=\min\\{\hat{\delta_{i}}(\frac{\varepsilon}{2},p_{0},t),\tilde{\delta_{i}}(\frac{\varepsilon}{2},p_{0},t)\\}\\\
&=\min\\{\frac{1}{C^{2}}\delta_{i}(\frac{\varepsilon}{4C^{2}},p_{0},t),\frac{1}{C^{2}}\delta_{i}(\frac{\varepsilon}{8C^{2}},p_{0},t)\\}\\\
&=\frac{1}{C^{2}}\delta_{i}(\frac{\varepsilon}{8C^{2}},p_{0},t),\end{split}$
(52)

indeed

$\begin{split}d(H_{i+1}(p_{0},t),H_{i+1}(q,s))&\leq
d(H_{i+1}(p_{0},t),H_{i+1}(p_{0},s))+d(H_{i+1}(p_{0},s),H_{i+1}(q,s))\\\
&\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.\end{split}$ (53)

Then, since a ball $B_{\delta_{2}}(p_{0})$ intersects
$\bigsqcup\limits_{\gamma\in\Gamma}B_{\delta_{2}}(\gamma\tilde{x}_{i})$ at
most $K$ times, we obtain that there is a $\delta(\varepsilon)$ which is given
by

$\delta(\varepsilon):=\frac{1}{C^{2K}}\delta_{0}(\frac{\varepsilon}{8^{K}C^{2K}})$
(54)

such that for each $(p,t)$ and $(q,s)$ in $M\times[0,1]$, we have

$d((p,t),(q,s))\leq\delta(\varepsilon)\implies
d(H(p,t),H(q,s))\leq\varepsilon.$ (55)

######  Corollary 1.4.

Let $f:(M,g)\longrightarrow(N,h)$ be a uniformly continuous map between
manifolds of bounded geometry. Assume the existence of a closed set $C$ such
that $f_{|_{M\setminus C}}$ is a $C^{k}_{b}$-map. Then for all $\epsilon>0$
there is a map $F:(M,g)\longrightarrow(N,h)$ such that $F$ is a
$C^{k}_{b}$-map. Moreover if $C_{\epsilon}$ is the $\epsilon$-neighborhood of
$C$, then

$F_{|_{M\setminus C_{\epsilon}}}=f_{|_{M\setminus C_{\epsilon}}}.$ (56)

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

######  Corollary 1.5.

Let $f,f^{\prime}:(M,g)\longrightarrow(N,h)$ be two smooth $C^{k}_{b}$-maps
such that $f\sim_{\Gamma}f^{\prime}$, where $\Gamma$ acts FUPD on $M$ and $N$
by isometries. Let us fix some normal coordinates $\\{x^{i},s\\}$ on
$M\times[0,1]$ and $\\{y^{j}\\}$ on $N$. Then there is a uniformly continuous
homotopy $H:(M\times[0,1],g+dt^{2})\longrightarrow(N,h)$ between $f$ and
$f^{\prime}$ such that all its derivatives in normal coordinates of order
minor or equal of $k$ are uniformly bounded.

######  Corollary 1.6.

Let $f:(M,g)\longrightarrow(N,h)$ be a uniform homotopy equivalence between
manifolds of bounded geometry. Then $f$ is uniformly proper and, in
particular, $f$ is a uniform map.

Proof: Let us denote by $g$ a uniform homotopy inverse of $f$ and by $H$ a
uniformly continuous homotopy between $g\circ f$ and $id_{X}$. Because of
Proposition 1.3 and Remark 2, we can assume that $f$, $g$ and $H$ are
Lipschitz maps. Then, if $x$ is a point in $M$,

$d_{X}(x,g\circ f(x))=d_{X}(H(x,0),H(x,1))\leq
C_{H}d_{X\times[0,1]}((x,0),(x,1))=C_{H}$ (57)

where $C_{H}$ is the Lipschitz constant of $H$. Fix a subset $A$ of $N$. Let
$x_{1}$ and $x_{2}$ be two points in $f^{-1}(A)$. Then

$\begin{split}d_{X}(x_{1},x_{2})&\leq d_{X}(x_{1},g\circ
f(x_{1}))+d_{X}(g\circ f(x_{1}),g\circ f(x_{2}))+d_{X}(x_{1},g\circ
f(x_{1}))\\\ &\leq 2C_{H}+C_{g}d_{Y}(f(x_{1}),f(x_{2}))\\\ &\leq
2C_{H}+C_{g}diam(A)\end{split}$ (58)

where $C_{g}$ is the Lipschitz constant of $g$. And so we obtain

$diam(f^{-1}(A))\leq 2C_{H}+C_{g}diam(A).$ (59)

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

###  2.1 $L^{p}$-forms

Let us consider a Riemannian manifold $(M,g)$ and let us denote by
$\Omega^{k}_{c}(M)$ the space of complex differential forms with compact
support. Fix a $p\in[1,+\infty)$. We can define a norm on $\Omega^{*}_{c}(M)$
as follows:

$||\alpha||_{p}:=[\int_{M}|\alpha|^{p}(q)d\mu_{M}(q)]^{\frac{1}{p}},$ (60)

where $\mu_{M}$ is the measure on $M$ induced by $g$.

######  Definition 2.1.

Fix $p\in[1,+\infty)$. We denote by $L^{p}\Omega^{k}(M)$ the Banach space
given by the closure of $\Omega^{*}_{c}(M)$ with respect to the norm
$||\cdot||_{p}$. Moreover we can also define the Banach space
$\mathcal{L}^{p}(M):=\bigoplus\limits_{k\in\mathbb{N}}L^{p}\Omega^{k}(M)$.

######  Definition 2.2.

Consider two Riemannian manifolds $(M,g)$ and $(N,h)$. Let us consider a
linear operator $A:\Omega^{*}(N)\longrightarrow\Omega^{*}(N)$. The operator
$A$ is $\mathcal{L}^{\star}$-bounded if for each $p$ in $[1,+\infty)$ the
operator $A_{|_{\Omega^{*}_{c}(M)}}$ is a bounded operator with respect to the
$\mathcal{L}^{p}$-norm.

###  2.2 $L^{q,p}$-cohomology and reduced $L^{q,p}$-cohomology

In the next section we use definitions and results from the work of
Gol’dshtein and Troyanov Gol  and from the work of Bei Bei2 . Consider a
complete Riemannian manifold $(M,g)$. Then, for each choice of
$q,p\in[1,+\infty)$ and $k$, we need a closed extension of the exterior
derivative operator

$d_{q,p}^{k-1}:\Omega^{k-1}_{c}(M)\subset\mathcal{L}^{q}\Omega^{k-1}(M,g)\longrightarrow\mathcal{L}^{p}\Omega^{k}(M,g).$
(61)

######  Definition 2.3.

The minimal extension $d_{min,q,p}^{k-1}$ of the exterior derivative
$d_{q,p}^{k-1}$ has domain given by the $\mathcal{L}^{q}$-forms $\alpha$ such
that there is a sequence of compactly supported differential forms
$\\{\alpha_{k}\\}$ and an $\mathcal{L}^{p}$-form $\beta$ such that555For each
$p$ in $[1,+\infty)$ we consider on $L^{p}\Omega^{k}(M)$ the topology induced
by $||\cdot||_{p}$ $\alpha=\lim\limits_{k\rightarrow+\infty}\alpha_{k}$ and
$\beta=\lim\limits_{k\rightarrow+\infty}d\alpha_{k}$.
Then we obtain

$d_{min,q,p}^{k-1}\alpha:=\lim\limits_{k\rightarrow+\infty}d\alpha_{k}.$ (62)

If there is no need to specify $q,p$ and $k$ we will just denote it by $d$.

######  Remark 8.

Observe that $d_{min,p,s}^{k-1}(dom(d_{min,p,s}^{k-1}))\subseteq
dom(d_{min,s,q}^{k})$ and $d^{2}:=d_{min,s,q}^{k}\circ d_{min,p,s}^{k-1}=0$
for each choice of $p,s,q$ and $k$.

######  Remark 9.

Because of the lower bound on the injectivity radius, every manifold of
bounded geometry is a complete Riemannian manifold. As a consequence of this
fact, $d_{min,p,q}$ is the unique closed extension of the exterior derivative
for each choice of $p,q\in[1,+\infty)$. The proof is exactly the same of
Proposition 3.1 of Bei2 , indeed the arguments used by the author and Theorem
12.5 of Gol  also hold if $q\neq p$.

Consider $p,q$ in $[1,+\infty)$.

######  Definition 2.4.

The $i$-th group of $L^{q,p}$-cohomology  is the group

$H^{i}_{q,p}(M):=\frac{ker(d^{i}_{p,p})}{im(d^{i-1}_{q,p})}.$ (63)

The $i$-th group of reduced $L^{q,p}$-cohomology  is the group

$\overline{H}^{i}_{q,p}(M):=\frac{ker(d^{i}_{p,p})}{\overline{im(d^{i-1}_{q,p})}}.$
(64)

Finally, the $i$-th group of $L^{q,p}$-quotient cohomology is the group

$T^{i}_{q,p}(M):=\frac{\overline{im(d^{i-1}_{q,p})}}{im(d^{i-1}_{q,p})}.$ (65)

######  Remark 10.

The $i$-th group of $L^{q,p}$-quotient group is also called $i$-th group of
$L^{q,p}$-torsion by Gol’dshtein and Kopylov in page 4 of Gol .

######  Proposition 2.1.

Let $(N,h)$ and $(M,g)$ be two oriented Riemannian manifolds and let
$p\in[1,+\infty)$. Fix $b\in\mathbb{Z}$ and consider
$B:\Omega^{*}(N)\longrightarrow\Omega^{*+b}(M)$ and
$K:\Omega^{*}(N)\longrightarrow\Omega^{*+b-1}(M)$ two
$\mathcal{L}^{*}$-bounded operators. Let us suppose that
$B(\Omega^{*}_{c}(N))\subseteq\Omega^{*+b}_{c}(M)$ and

$dB=\pm Bd+K$ (66)

over $\Omega^{*}_{c}(N)$. Then $B(dom(d_{min,q,p}))\subseteq dom(d_{min,q,p})$
and $d_{min,q,p}B=\pm Bd_{min}+K$ on the minimal domain of $d$.

Proof: Let $\alpha$ be an element in $dom(d_{min,q,p})$. This means that there
is a sequence $\\{\alpha_{n}\\}$ in $\Omega^{*}_{c}(N)$ such that
$\alpha=\lim\limits_{n\rightarrow+\infty}\alpha_{n}$ and
$d\alpha=\lim\limits_{n\rightarrow+\infty}d\alpha_{n}$. Since $B$ is
continuous we obtain

$B\alpha=\lim\limits_{n\rightarrow+\infty}B\alpha_{n}$ (67)

where $\\{B\alpha_{n}\\}$ is a sequence in $\Omega^{*}_{c}(M)$. Moreover the
limit of $dB\alpha_{n}$ exists, indeed

$\begin{split}\lim\limits_{n\rightarrow+\infty}dB\alpha_{n}&=\lim\limits_{n\rightarrow+\infty}\pm
Bd\alpha_{n}+K\alpha_{n}\\\ &=\pm
B\lim\limits_{n\rightarrow+\infty}d\alpha_{n}+K\lim\limits_{n\rightarrow+\infty}\alpha_{n}=\pm
Bd\alpha+K\alpha.\end{split}$ (68)

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

$d_{min,q,p}B\alpha=\pm Bd_{min,q,p}\alpha+K\alpha.$ (69)

######  Remark 11.

Consider an $\mathcal{L}^{*}$-bounded operator $A$ such that
$A(dom(d_{min,N}))\subseteq dom(d_{min,M})$. Suppose that $Ad=dA$. Then $A$
induces a map in reduced $L^{p,q}$-cohomology and in $L^{q,p}$-quotient
cohomology. Indeed, given $\alpha$ in $dom(d_{N})$, we have

$\begin{split}&A(\alpha+\lim\limits_{k\rightarrow+\infty}d\beta_{k})=A(\alpha)+A(\lim\limits_{k\rightarrow+\infty}d\beta_{k})\\\
&=A(\alpha)+\lim\limits_{k\rightarrow+\infty}A(d\beta_{k})=A(\alpha)+\lim\limits_{k\rightarrow+\infty}dA(\beta_{k}).\end{split}$
(70)

So, in reduced $L^{q,p}$-cohomology and in $L^{q,p}$-quotient cohomology, we
obtain

$[A(\alpha+\lim\limits_{k\rightarrow+\infty}d\beta_{k})]=[A(\alpha)].$ (71)

######  Corollary 2.2.

Consider two operators $B$ and $K$ as in Proposition 2.1.
Then $K$ induces the null operator in (un)-reduced $L^{q,p}$-cohomology and in
$L^{q,p}$-quotient cohomology.
Moreover, if $K=0$ as operator between the $\mathcal{L}^{p}$-spaces, then
$dB=Bd$ on $\Omega_{c}^{*}(M)$. Then $B$ induces a map in (un)-reduced
$L^{q,p}$-cohomology and in $L^{q,p}$-quotient cohomology.666We are
considering $\overline{d}=d_{min}$.

###  2.3 Fiber Volume and Radon-Nikodym-Lipschitz maps

Let $(M,\nu)$ and $(N,\mu)$ be two measured spaces and let
$f:(M,\nu)\longrightarrow(N,\mu)$ be a function such that the pushforward
measure $f_{\star}(\nu)$ is absolutely continuous with respect to $\mu$.

######  Definition 2.5.

Let $(N,\mu)$ be $\sigma$-finite, then the Fiber Volume is the Radon-Nikodym
derivative

$Vol_{f,\nu,\mu}:=\frac{\partial f_{\star}\nu}{\partial\mu}.$ (72)

Consider $(M,d_{M},\mu_{M})$ and $(N,d_{N},\mu_{N})$ two measured and metric
spaces.

######  Definition 2.6.

A map $f:(M,d_{M},\mu_{M})\longrightarrow(N,d_{N},\mu_{N})$ is Radon-Nikodym-
Lipschitz or R.-N.-Lipschitz if

  * •

$f$ is Lipschitz

  * •

$f$ has a well-defined and bounded Fiber Volume.

######  Remark 12.

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 $A\subseteq
N$,

$\mu_{M}(f^{-1}(A))=\int_{A}\frac{\partial
f_{\star}\mu_{M}}{\partial\mu_{N}}d\mu_{N}\leq C\int_{A}d\mu_{N}=C\mu_{N}(A).$
(73)

The vice-versa also holds: if $f$ satisfies (73), then it is a R.N.-Lipschitz
map. This implication can be proved using a reductio ad absurdum argument.

######  Remark 13.

Composition of R.-N.-Lipschitz maps is a R.-N.-Lipschitz map.

######  Lemma 2.3.

Let $f:(M,g)\longrightarrow(N,h)$ be a Lipschitz map between Riemannian
manifolds. Let $x$ be a point on $M$ such that $f$ is differentiable on $x$.
Then there is a number $C$, which does not depend on $x$, such that for all
$\alpha_{f(x)}$ in $T^{*}_{f(x)}(N)$ and for all $k\in[1,+\infty)$

$|f^{*}\alpha_{f(x)}|^{k}_{x}\leq C^{k}\cdot|\alpha_{f(x)}|^{k}_{f(x)},$ (74)

where $|\cdot|_{x}$ and $|\cdot|_{f(x)}$ are the norms induced by the metrics
$g$ and $h$ on $\Lambda^{*}_{x}M$ and $\Lambda^{*}_{f(x)}N$.

Proof: It follows since the norms on $\Lambda^{*}_{p}N$ and $\Lambda^{*}_{q}M$
can be seen as operatorial norms and since the $df_{q}$ has bounded norm.

######  Remark 14.

By the Rademacher Theorem, we know that a Lipschitz map is differentiable
almost everywhere and so the previous lemma holds for almost all $x$ in $M$.

######  Proposition 2.4.

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
$\mathcal{L}^{*}$-bounded pullback. This means that if $f$ is a R.N.-Lipschitz
map, then $f^{*}$ is an $\mathcal{L}^{*}$-bounded operator.

Proof: Let $\omega$ be a smooth form with compact support in
$\mathcal{L}^{p}(N)$ and let $K_{f}:=\max\\{1,C_{f}^{n}\\}$, where $n=dim(N)$.
Then

$\begin{split}||f^{*}\omega||_{p}^{p}&=\int_{M}|f^{*}\omega|^{p}d\mu_{M}\leq\int_{M}K_{f}^{p}f^{*}(|\omega|^{p})d\mu_{M}\\\
&=K_{f}^{p}\int_{N}|\omega|^{p}d(f_{\star}\mu_{M})=K_{f}^{p}\int_{N}|\omega|^{p}Vol_{f,\mu_{M},\mu_{N}}d\mu_{N}\\\
&\leq
K_{f}^{p}C_{Vol}\int_{N}|\omega|^{p}d\mu_{N}=K_{f}^{p}C_{Vol}||\omega||_{p}^{p}.\end{split}$
(75)

In the 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 2.7.

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 (possibly not
continuous) section of $\Lambda^{k-n}(M)$ such that for all $p$ in $M$

$\alpha(p)=\beta(p)\wedge\frac{\alpha}{\beta}(p)..$ (76)

Consider two oriented differentiable manifolds $X$ and $Y$. Given a submersion
$f:X\longrightarrow Y$, let us denote by $F_{q}$ the fiber of $f$ in $q$ and
consider $i_{q}:F_{q}\longrightarrow X$ the immersion of the fiber in $X$.
Then $i_{q}^{*}(\frac{\beta}{f^{*}\alpha})$ does not depend on the choice of
the quotient. This is proved in Proposition 16.21.7 of the book of Dieudonné
Dieu . Moreover in Dieu  it is also proved that if $\beta$ is a smooth form in
$X$, then $i_{q}^{*}(\frac{\beta}{f^{*}\alpha})$ is a smooth form on the fiber
$F_{q}$. This means that for all $p$ in $Y$ we obtain an orientation of
$F_{p}$ defined setting

$\int_{F_{p}}i_{p}^{*}\frac{Vol_{X}}{f^{*}Vol_{Y}}>0.$ (77)

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

$\frac{Vol_{X}}{f^{*}Vol_{Y}}=\frac{det(g_{ij}(y,x))}{det(h_{rs})(y)}dx^{1}\wedge...\wedge
dx^{n},$ (78)

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

$\int_{F}f^{*}\alpha\wedge\frac{Vol_{X}}{f^{*}Vol_{Y}}=\alpha\wedge\int_{F}\frac{Vol_{X}}{f^{*}Vol_{Y}}.$
(79)

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, as a
consequence of Proposition 16.21.7 of Dieu , the integral over the fiber in
$p$ of a quotient between a form $\alpha$ and a form $f^{*}\beta$ does not
depend on the choice of the quotient. This means that in each chart we can
choose a smooth quotient and apply the usual Projection Formula.

###  2.4 Fiber Volume of a submersion

In this section we will study the Fiber Volumes of Lipschitz submersions
between orientable manifolds.

######  Proposition 2.5.

Let $(M,g)$ and $(N,h)$ two oriented, Riemannian manifolds possibly with
boundary. Let $\pi:(M,g)\longrightarrow(N,h)$ be a submersion. Then

$Vol_{\pi,\mu_{M},\mu_{N}}(q)=\int_{F}\frac{Vol_{M}}{\pi^{*}Vol_{N}}(q).$ (80)

Proof: Let $A$ a measurable set of $N$. Then

$\begin{split}f_{\star}\mu_{M}(A)&=\int_{\pi^{-1}(A)}1d\mu_{M}=\int_{\pi^{-1}(A)}Vol_{M}\\\
&=\int_{\pi^{-1}(A)}\pi^{*}(Vol_{N})\wedge\frac{Vol_{M}}{\pi^{*}(Vol_{N})}\\\
&=\int_{\pi^{-1}(A\cap\pi(M))}\pi^{*}(Vol_{N})\wedge\frac{Vol_{M}}{\pi^{*}(Vol_{N})}\\\
&=\int_{A\cap\pi(M)}Vol_{N}(\int_{F}\frac{Vol_{M}}{\pi^{*}(Vol_{N})})\\\
&=\int_{A\cap\pi(M)}(\int_{F}\frac{Vol_{M}}{\pi^{*}(Vol_{N})})Vol_{N}\\\
&=\int_{A\cap\pi(M)}(\int_{F}\frac{Vol_{M}}{\pi^{*}(Vol_{N})})d\mu_{N}.\end{split}$
(81)

######  Remark 15.

If the submersion $f:X\rightarrow Y$ is a diffeomorphism between oriented
manifold which preserves the orientations, then the integration along the
fibers of $f$ is the pullback $(f^{-1})^{*}$. This means that the Fiber Volume
of $f$ is given by $|(f^{-1})^{*}\frac{Vol_{X}}{f^{*}(Vol_{Y})}|$.

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

######  Proposition 2.6.

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

$Vol_{g\circ
f,\mu_{M},\mu_{W}}(q)=\int_{g^{-1}(q)}(\int_{f^{-1}g^{-1}(q)}\frac{Vol_{M}}{f^{*}(Vol_{N})})\frac{Vol_{N}}{g^{*}Vol_{N}}$
(82)

Proof: Observe that, as quotients,

$\frac{Vol_{M}}{(g\circ
f)^{*}Vol_{W}}=\frac{Vol_{M}}{f^{*}Vol_{N}}\wedge\frac{f^{*}Vol_{N}}{(g\circ
f)^{*}Vol_{W}}$ (83)

and, in particular, we can choose as quotient

$\frac{f^{*}Vol_{N}}{(g\circ
f)^{*}Vol_{W}}=f^{*}(\frac{Vol_{N}}{g^{*}Vol_{W}}).$ (84)

Then we conclude by applying the Projection Formula:

$\begin{split}&Vol_{g\circ f,\mu_{M},\mu_{W}}(q)=\int_{(g\circ
f)^{-1}(q)}\frac{Vol_{M}}{(g\circ f)^{*}Vol_{W}}\\\ &=\int_{(g\circ
f)^{-1}(q)}\frac{Vol_{M}}{f^{*}Vol_{N}}\wedge
f^{*}(\frac{Vol_{N}}{g^{*}Vol_{W}})=\int_{g^{-1}(q)}(\int_{f^{-1}g^{-1}(q)}\frac{Vol_{M}}{f^{*}(Vol_{N})})\frac{Vol_{N}}{g^{*}Vol_{N}}.\end{split}$
(85)

##  3 Sasaki metric and submersions

###  3.1 The Sasaki metric

The definition of Sasaki metric on a vector bundle given in this paper is a
generalization of the metric defined by Sasaki in SSK  for the tangent bundle
of a Riemannian manifolds. The definition of this generalized Sasaki metric
can be found in page 2 of the paper of Boucetta and Essoufi Boucetta .
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 $\\{s_{\alpha}\\}$ a local frame of
$E$: if $\\{x^{i}\\}$ is a system of local coordinates over $U\subseteq N$,
then we can define the system of coordinates $\\{x^{i},\mu^{\alpha}\\}$ on
$\pi_{E}^{-1}(U)$, where the $\mu^{\alpha}$ are the components with respect to
$\\{s_{\alpha}\\}$.
Let us denote by $K$ the map $K:TE\longrightarrow E$ defined as

$K(b^{i}\frac{\partial}{\partial
x^{i}}|_{(x_{0},\mu_{0})}+z^{\alpha}\frac{\partial}{\partial\mu^{\alpha}}|_{(x_{0},\mu_{0})}):=(z^{\alpha}+b^{i}\mu^{j}\Gamma_{ij}^{\alpha}(x_{0}))s_{\alpha}(x_{0}),$
(86)

where the $\Gamma_{ij}^{l}$ are the Christoffel symbols of $\nabla_{E}$. The
Christoffel symbols $\Gamma^{\gamma}_{\eta j}(x)$ are defined by the formula

$\nabla^{E}_{\frac{\partial}{\partial
x^{j}}}s_{\eta}(x):=\Gamma^{\gamma}_{\eta j}(x)s_{\gamma}(x).$ (87)

######  Definition 3.1.

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

$h^{E}(A,B):=h(d\pi_{E,v_{p}}(A),d\pi_{E,v_{p}}(B))+H_{E}(K(A),K(B)).$ (88)

######  Remark 16.

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

$\begin{cases}h^{E}_{ij}(x,\mu)=h_{ij}(x)+H_{\alpha\gamma}(x)\Gamma^{\alpha}_{\beta
i}(x)\Gamma^{\gamma}_{\eta j}(x)\mu^{\beta}\mu^{\eta}\\\
h^{E}_{i\sigma}(x,\mu)=H_{\sigma\alpha}(x)\Gamma^{\alpha}_{\beta
i}(x)\mu^{\beta}\\\ h^{E}_{\sigma\tau}(x,\mu)=H_{\sigma,\tau}(x),\end{cases}$
(89)

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

$\begin{bmatrix}h_{i,j}(x_{0})&&0\\\ 0&&H_{\sigma,\tau}(x_{0})\end{bmatrix}.$
(90)

Moreover, with respect to the coordinates $(x^{i},\mu^{\sigma})$, the matrix
$h_{E}:=(h^{E})^{-1}$ is given by

$\begin{cases}h^{ij}_{E}(x,\mu)=h^{ij}(x)\\\
h^{i\sigma}_{E}=-\Gamma^{\sigma}_{\beta j}(x)h^{ij}(x)\mu^{\beta}\\\
h_{E}^{\sigma\tau}=H^{\sigma\tau}(x)+h^{ij}(x)\Gamma^{\sigma}_{\beta
i}(x)\Gamma^{\tau}_{\eta j}(x)\mu^{\beta}\mu^{\eta}\end{cases}$ (91)

where $H^{\sigma\tau}$ and $h^{ij}(x)$ are the components of the inverse
matrices of $h_{ij}$ and $H_{\sigma,\tau}$.

######  Example 3.1.

Let $(M,g)$ a Riemannian manifold. Consider as $E$ the tangent bundle $TM$ and
as $h_{E}$ the metric $g$ itself. Choose the connection $\nabla_{E}$ as the
Levi-Civita connection $\nabla_{g}^{LC}$. We denote by $g_{S}$ the Sasaki
metric induced by $g$ and $\nabla^{LC}_{g}$.

######  Example 3.2.

Consider a smooth map $f:(M,g)\longrightarrow(N,h)$. Let
$\pi:f^{*}TN\longrightarrow M$ be the pullback bundle. Then the Riemannian
metric $h$ can be seen as a bundle metric on $TN$ and so we obtain a bundle
metric $f^{*}h$ on $f^{*}TN$. Fix the connection $f^{*}\nabla^{LC}_{h}$ on
$f^{*}TN$ which is the pullback of the Levi-Civita connection on $(N,g)$. Let
us denote by $g_{S,f}$ the Sasaki metric induced by $f^{*}\nabla^{LC}_{h}$,
$f^{*}h$ and $g$.

######  Remark 17.

Then the Christoffel symbols of the pullback connection $f^{*}\nabla^{E}$ with
respect to the pullback frame $\\{f^{*}e_{i}\\}$ and to the coordinates
$\\{x^{i}\\}$ are given by

$\tilde{\Gamma}^{\alpha}_{\beta,i}:=\frac{\partial f^{l}}{\partial
x^{i}}f^{*}(\Gamma^{\alpha}_{\beta,l}).$ (92)

A first consequence of this fact is that the map
$ID:(id^{*}TN,g_{S,id})\longrightarrow(TN,g_{S})$ which sends $(p,w_{p})$ to
$w_{p}$ is an isometry. So, from now on, we will identify
$(id^{*}TN,g_{S,id})$ and $(TN,g_{S})$.

######  Remark 18.

Let $f:(M,g)\longrightarrow(N,h)$ be a smooth map. Let us fix a chart
$\\{U,x^{i}\\}$ on $M$ and a chart $\\{V,y^{j}\\}$ on $N$ such that
$f(U)\subseteq V$. Fix a bundle $E$ on $N$ and let $\nabla^{E}$ be a
connection. Let us denote by $\Gamma^{\alpha}_{\beta l}$ the Christoffel
symbols of $\nabla^{E}$ with respect to a frame $\\{e_{i}\\}$ and the
coordinates $\\{y^{j}\\}$. If $f:(M,g)\longrightarrow(N,h)$ is a smooth
Lipschitz map, then also the induced bundle map

$\begin{split}F:(f^{*}T^{\delta}N,g_{S,f})&\longrightarrow(T^{\delta}N,g_{S})\\\
(p,w_{f(p)})&\longrightarrow w_{f(p)}\end{split}$ (93)

is a smooth Lipschitz map.

######  Proposition 3.1.

Consider a vector bundle $(E,\pi,M)$ over a Riemannian manifold $(M,g)$. Fix
on $E$ a bundle metric $h$, a connection $\nabla^{E}$ and let us denote by
$h_{E}$ the Sasaki metric induced by $g$, $h$ and $\nabla^{E}$. Let us suppose
that for each point $p$ on $M$ there is a system of normal coordinates
$\\{x^{i}\\}$ around $p$ and a local frame $\\{s_{\sigma}\\}$ such that the
Christoffel symbols of $\nabla^{E}$ vanishes at $x=0$. Moreover let us suppose
that $\frac{\partial h_{\sigma\tau}}{\partial x^{k}}(0)=0$, where
$h_{\sigma\tau}$ are the components of the Gram matrix of $h$ with respect to
the coordinates $\\{x^{i}\\}$. Then the fibers of $\pi$ are totally geodesic
submanifolds (in our case this means that the straight lines on the fibers are
geodesics) and $\pi$ is a Riemannian submersion.

######  Remark 19.

The vector bundles of the Examples 3.1 and 3.2 satisfy the assumptions of
Proposition 3.1.

Proof: Fix the coordinates $\\{x^{i},\mu^{\sigma}\\}$ where the coordinates
$\\{x^{i}\\}$ are normal and centered on a point $p$ and $\mu^{\sigma}$ refer
to the local frame $\\{s_{\sigma}\\}$. It follows from (90) that $\pi$ is a
Riemannian submersion. Let $v_{p}$ and $w_{p}$ be two vectors in $E_{p}$. Let
us suppose that $v_{p}=(0,v^{\sigma})$ and $w_{p}=(0,w^{\sigma})$ with respect
to the coordinates $\\{x^{i},\mu^{\sigma}\\}$. Consider the straight line
connecting $v_{p}$ and $v_{p}$ defined as
$\gamma(t)=t\cdot(v_{p}-w_{p})+w_{p}$. Consider the Christoffel symbols of the
Levi-Civita connection induced by $h^{E}$ with respect to the coordinates
$\\{x^{i},\mu^{\sigma}\\}$ and the local frame $\\{\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial\mu^{\sigma}}\\}$. Let us denote by
$\tilde{\Gamma}^{i}_{\sigma\tau}$ and by $\tilde{\Gamma}^{\eta}_{\sigma\tau}$
the Christoffel symbols defined by

$\nabla^{h^{E}}_{\frac{\partial}{\partial\mu^{\sigma}}}\frac{\partial}{\partial\mu^{\tau}}(x,\mu)=\tilde{\Gamma}^{i}_{\sigma\tau}(x,\mu)\frac{\partial}{\partial
x^{i}}+\tilde{\Gamma}^{\eta}_{\sigma\tau}(x,\mu)\frac{\partial}{\partial\mu^{\eta}}.$
(94)

Because the partial derivatives of $g_{ij}$ and of $h_{\sigma,\tau}$ in $x=0$
are zero, we obtain that
$\tilde{\Gamma}^{j}_{\sigma\tau}(0,\mu)=\tilde{\Gamma}^{\eta}_{\sigma\tau}(0,\mu)=0$.
Then, if we parametrize the curve
$\gamma(t):=(0,t(v^{\sigma}-w^{\sigma})+w^{\sigma})$, then $\gamma$ satisfies
the system of geodesics equations

$\begin{cases}\frac{d^{2}\gamma^{i}}{dt^{2}}+\tilde{\Gamma}^{i}_{\sigma\tau}(x,\mu)\frac{d\gamma^{\sigma}}{dt}\frac{d\gamma^{\tau}}{dt}+\tilde{\Gamma}^{i}_{j\tau}(x,\mu)\frac{d\gamma^{j}}{dt}\frac{d\gamma^{\tau}}{dt}+\tilde{\Gamma}^{i}_{jz}(x,\mu)\frac{d\gamma^{j}}{dt}\frac{d\gamma^{z}}{dt}=0\\\
\frac{d^{2}\gamma^{\alpha}}{dt^{2}}+\tilde{\Gamma}^{\alpha}_{\sigma\tau}(x,\mu)\frac{d\gamma^{\sigma}}{dt}\frac{d\gamma^{\tau}}{dt}+\tilde{\Gamma}^{\alpha}_{j\tau}(x,\mu)\frac{d\gamma^{j}}{dt}\frac{d\gamma^{\tau}}{dt}+\tilde{\Gamma}^{\alpha}_{jz}(x,\mu)\frac{d\gamma^{j}}{dt}\frac{d\gamma^{z}}{dt}=0.\end{cases}$
(95)

######  Corollary 3.2.

Consider $0_{E}$ the null section of a vector bundle $\pi:E\longrightarrow M$.
Under the same assumptions of the Proposition 3.1, the disk bundle
$E^{\delta}:=\\{v_{p}\in E|\sqrt{h(v_{p},v_{p})}\leq\delta\\}$ coincides with
$B_{\delta}(0_{E}):=\\{v_{p}\in E|d_{h^{E}}(v_{p},0_{E})\leq\delta\\}$ for
each $\delta>0$.

Proof: Fix $v_{p}$ a point on $E$. Because of Proposition 3.1 we already know
that the straight line $\gamma$ which connects $0_{p}$ and $v_{p}$ is a
geodesic and its length is equal to $\sqrt{h(v_{p},v_{p})}$. Observe that if
there is another point $0_{q}$ in $0_{E}$ such that
$d(0_{q},v_{p})<\sqrt{h(v_{p},v_{p})}$, then there is a geodesics $\sigma$
connecting $v_{p}$ to $0_{q}$ which is shorter than $\gamma$.
Let us suppose that $\dot{\sigma}(0_{q})$ is orthogonal to $T_{0_{q}}0_{E}$.
Because of Proposition 3.1, we know that $q=p$ and $\gamma=\sigma$. This is a
contradiction.
Let us prove that $\dot{\sigma}(0_{q})$ is orthogonal to $T_{0_{q}}0_{E}$. We
can assume, without loss of generality, that $d(v_{p},0_{E})$ is smaller than
the radius of injectivity of $E$ in $v_{p}$. Observe that the map
$d(v_{p},\cdot):0_{E}\longrightarrow\mathbb{R}$ is continuous and admits a
minimum.
Consider the ball $B_{d(v_{p},0_{E})}(v_{p})$. Then

$B_{d(v_{p},0_{E})}(v_{p})\cap 0_{E}\subseteq\partial
B_{d(v_{p},0_{E})}(v_{p}):=S_{d(v_{p},0_{E})}(v_{p})$ (96)

and the intersection is the set of the minima of $d(v_{p},\cdot)$. In
particular, if $q$ is a minimum for $d(v_{p},\cdot)$,

$T_{q}0_{E}\subseteq T_{q}S_{d(v_{p},0_{E})}(v_{p})$ (97)

as subspaces of $T_{q}E$. In order to prove (97) we use a simple reductio ad
absurdum argument. Let us suppose there is $v$ in $T_{q}0_{E}$ such that $v$
is not in $T_{q}S_{d(v_{p},0_{E})}(v_{p})$. Without loss of generality we can
assume that $v$ is an inward vector for $B_{d(v_{p},0_{E})}(v_{p})$. On a open
neighborhood of $B_{d(v_{p},0_{E})}(v_{p})$ there are some coordinates
$\\{x^{1},...,x^{n-1},y\\}$ such that the interior of
$B_{d(v_{p},0_{E})}(v_{p})$ is given by the points $y>0$. Let
$\tau:(-\varepsilon,\varepsilon)\longrightarrow U$ be a curve in $0_{E}$ such
that $\tau(0)=q$ and $\dot{\tau}(0)=v$. Observe that, in coordinates,
$v=v^{i}\frac{\partial}{\partial x^{i}}+v_{0}\frac{\partial}{\partial y}$
where $v_{0}\geq 0$. But this means that there is a $t_{0}$ in
$(-\varepsilon,\varepsilon)$ such that $y(\tau(t_{0}))>0$. But $\tau(t_{0})$
is a point of $0_{E}$ and so it cannot be a point on the interior of
$B_{d(v_{p},0_{E})}(v_{p})$. This proves (97).
Finally, thanks to the Gauss Lemma, the geodesic $\sigma:[0,1]\longrightarrow
E$ connecting $\sigma(0)=v_{p}$ to a minimum $\sigma(1)=q$ has
$\dot{\sigma}(1)$ orthogonal to $T_{q}S_{d(v_{p},0_{E})}(v_{p})\supseteq
T_{q}0_{E}$. In order to prove the following Proposition, we need a Lemma
which is a classical result of the Measure Theory. A proof of this result can
be found in page 29 of the book of Durrett Durr .

######  Lemma 3.3 (Multivariate Jensen’s Inequality).

Let $(X,\Sigma_{X},\mu_{X})$ be a probability space777This is a measured space
such that $\mu_{X}(X)=1$ and let $f:X\longrightarrow\mathbb{R}^{n}$. Consider
$\phi:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ a convex function. Then, for
each integrable function $f:X\longrightarrow\mathbb{R}^{n}$

$\phi(\int_{X}fd\mu_{X})\leq\int_{X}\phi(f)d\mu_{X}.$ (98)

######  Proposition 3.4.

Consider a Riemannian manifold $(M,g)$ and let $\pi:E\longrightarrow M$ be a
vector bundle. Fix on $E$ a metric bundle $h_{E}$ and a connection
$\nabla_{E}$. Let $h^{E}$ be the Sasaki metric on $E$ defined by using $g$,
$h_{E}$ and $\nabla_{E}$. Fix a $\delta>0$. Then, under the assumptions of
Proposition 3.1 $\pi:E^{\delta}\longrightarrow M$ is a R.N.-Lipschitz map. In
particular the Fiber Volume on a point $q$ is the Volume of an Euclidean ball
of radius $\delta$. Finally the integration along the fibers
$\pi_{\star}:\Omega_{c}^{*}(E^{\delta})\longrightarrow\Omega^{*}(M)$ is an
$\mathcal{L}^{*}$-bounded operator.

Proof: We know that $\pi$ is a Lipschitz map because it is a Riemannian
submersion (Proposition 3.1). In order to calculate the Fiber Volume of $\pi$,
we choose a point $q$ in $M$ and consider some normal coordinates
$\\{x^{i}\\}$ around $q$. Fix the frame $\\{e_{j}\\}$ of $E$ defined around
$q$ such that $\Gamma^{i}_{jk}(0)=0$ with respect to $\\{e_{j}\\}$ and to
$\\{x^{i}\\}$. Let $\\{x^{i},y^{j}\\}$ be the fibered coordinates on
$E^{\delta}$ where $\\{y^{j}\\}$ refer to $e_{j}$. The matrix related to
$h^{E}$ in a point of the fiber of $q$ is the identity.
This means that $Vol_{E}(x,y)=dx^{1}\wedge...\wedge dx^{n}\wedge
dy^{1}\wedge...\wedge dy^{m}$ and $\pi^{*}Vol_{M}(x)=dx^{1}\wedge...\wedge
dx^{n}$.
We obtain that

$\begin{split}Vol_{\pi,\mu_{E},\mu_{M}}&=\int_{F}\frac{Vol_{E}}{\pi^{*}Vol_{N}}=\int_{B_{\delta}(0_{q})}\frac{dx^{1}\wedge...\wedge
dx^{n}\wedge dy^{1}\wedge...\wedge dy^{m}}{dx^{1}\wedge...\wedge dx^{n}}\\\
&=\int_{B_{\delta}(0_{q})}dy^{1}\wedge...\wedge
dy^{m}=Vol(B_{\delta}(0)).\end{split}$ (99)

Let us study the $\mathcal{L}^{*}$-boundedness of $\pi_{\star}$. First we will
show that, given $\alpha$ in $\Omega^{*}_{c}(E)$, we have

$|\pi_{\star}\alpha|_{q}^{p}\leq
C^{p-1}\int_{E_{q}}|\alpha|_{y}^{p}d\mu_{E_{q}}(y),$ (100)

where $\mu_{E_{q}}$ is the measure on the fiber $E_{q}$ induced by
$\frac{Vol_{E}}{\pi^{*}Vol_{N}}$.
Fix the same coordinates $\\{x^{i},y^{j}\\}$ we used before. We obtain

$\alpha=\alpha_{I0}(x,y)dx^{I}\wedge dy^{1}\wedge
dy^{n}+\alpha_{IJ}(x,y)dx^{I}\wedge dy^{J},$ (101)

where $J\neq(1,2,...,m)$. Let us denote by
$\alpha_{0}=\alpha_{I0}(x,y)dx^{I}\wedge dy^{1}\wedge...\wedge dy^{n}$.
Observe that for each $r$ in the fiber of $q$ we have
$|\alpha_{0}|_{r}\leq|\alpha|_{r}$.
Moreover

$\int_{B_{\delta}(0_{q})}\alpha_{0}=\int_{B_{\delta}(0_{q})}f(y)dy^{1}\wedge...dy^{m}=\int_{B_{\delta}(0_{q})}f(y)d\mu_{E_{q}}(y)$
(102)

where $f:B_{\delta}(0_{q})\longrightarrow\Lambda^{*}_{q}(M)\otimes\mathbb{C}$
and $|f(y)|=|\alpha_{0}|_{y}$.
In order to prove (100), we want to apply the Multivariate Jensen’s Inequality
(Lemma 3.3). Let us define for each $q$ in $M$ the measure on the fiber of $q$
defined for each measurable set $A$ of $E_{q}$ as

$\mu_{E_{q},J}(A):=\frac{\mu_{E_{q}}(A)}{Vol_{\pi,\mu_{E},\mu_{M}}(q)}.$ (103)

Then $\mu_{E_{q},J}$ makes $B_{\delta}(0_{q})\subset E_{q}$ a probability
space. Let us define the map
$F:B_{\delta}(0_{q})\longrightarrow\Lambda^{*}_{q}M\otimes\mathbb{C}\cong\mathbb{R}^{2n}$

$F(y):=Vol_{\pi,\mu_{E},\mu_{M}}(q)\cdot f(y),$ (104)

Then, by the Jensen’s Inequality, considering $\phi(x)=|x|^{p}$, we obtain
that

$|\int_{B_{\delta}(0_{q})}F(y)d\mu_{E_{q},J}|^{p}\leq\int_{B_{\delta}(0_{q})}|F(y)|^{p}d\mu_{E_{q},J}.$
(105)

Observe that the left part of (105) is

$\begin{split}&|\int_{B_{\delta}(0_{q})}F(y)d\mu_{E_{q},J}(y)|^{p}\\\
&=|\int_{B_{\delta}(0_{q})}Vol_{\pi,\mu_{E},\mu_{M}}(q)\cdot
f(y)\cdot\frac{1}{Vol_{\pi,\mu_{E},\mu_{M}}(q)}d\mu_{E_{q}}(y)|^{p}\\\
&=|\int_{B_{\delta}(0_{q})}f(y)d\mu_{E_{q}}(y)|^{p}=|\int_{E_{q}}\alpha|^{p}=|\pi_{\star}\alpha|_{q}^{p}.\end{split}$
(106)

The right part of (105) is

$\begin{split}\int_{B_{\delta}(0_{q})}|F(y)|^{p}d\mu_{E_{q},J}&=\int_{B_{\delta}(0_{q})}|Vol_{\pi,\mu_{E},\mu_{M}}(q)|^{p}\cdot|f(y)|^{p}\frac{1}{Vol_{\pi,\mu_{E},\mu_{M}}(q)}d\mu_{E_{q}}(y)\\\
&=\int_{B_{\delta}(0_{q})}|Vol_{\pi,\mu_{E},\mu_{M}}(q)|^{p-1}\cdot|f(y)|^{p}d\mu_{E_{q}}(y)\\\
&\leq C^{p-1}\int_{E_{q}}|\alpha|_{y}^{p}d\mu_{E_{q}}(y).\end{split}$ (107)

Then we conclude by showing that

$\begin{split}&||\pi_{\star}\alpha||_{p}=\int_{M}|\pi_{\star}\alpha|_{q}^{p}d\mu_{M}(q)\\\
&\leq
C^{p-1}\int_{M}[\int_{E_{q}}|\alpha|_{y}^{p}d\mu_{E_{q}}(y)]d\mu_{M}(q)=C^{p-1}\int_{M}Vol_{M}\cdot[\int_{E_{q}}|\alpha|_{y_{q}}^{p}\frac{Vol_{E}}{\pi^{*}Vol_{M}}]\\\
&=C^{p-1}\int_{M}\int_{E_{q}}|\alpha|_{y}^{p}\pi^{*}Vol_{M}\wedge\frac{Vol_{E}}{\pi^{*}Vol_{M}}=C^{p-1}\int_{E}|\alpha|_{v}^{p}Vol_{E}(v)=C^{p-1}||\alpha||_{p}\end{split}$
(108)

######  Remark 20.

Consider $(M,g)$ and $(N,h)$ two manifolds of bounded geometry and let
$f:(M,g)\longrightarrow(N,h)$ be a $C^{k}_{b}$-map for each $k$ in
$\mathbb{N}$. Fix on $f^{*}(TN)$ the Sasaki metric $g_{Sf}$ induced by $g$,
$f^{*}h$ and $f^{*}\nabla^{LC}_{h}$. If we denote by
$\nabla:=\nabla^{LC}_{g_{S}}$ and by $R$ the Riemann tensor on $f^{*}(TN)$, we
obtain that for each $i$ in $\mathbb{N}$ there is a continuous function
$C_{i}:\mathbb{R}\longrightarrow\mathbb{R}$ such that
$|\nabla^{i}R(v_{p})|\leq C_{i}(||v_{p}||)$. This is a consequence of (89), of
(91) and of Theorem 2.5 of the paper of Schick flow .

###  3.2 A submersion related to a uniform map

Let $(N,h)$ and $(M,g)$ be two manifolds of bounded geometry and let
$\delta\leq inj_{N}.$ Fix, moreover, a smooth map
$f:(M,g)\longrightarrow(N,h)$.
In this subsection we define a submersion
$p_{f}:(f^{*}(T^{\delta}N),g_{Sf})\longrightarrow(N,h)$ where

$f^{*}(T^{\delta}N):=\\{(p,w_{f(p)})\in f^{*}(TN)\mbox{ such that
}|w_{f(p)}|\leq\delta\\}.$ (109)

######  Lemma 3.5.

Let us consider $f:(M,g)\longrightarrow(N,h)$ a smooth Lipschitz map between
two oriented Riemannian manifolds. Suppose that $(N,h)$ is a manifold of
bounded geometry.
Let us denote by $F:f^{*}TN\longrightarrow TN$ the bundle morphism induced by
$f$, i.e. $F(p,w_{f(p)}):=w_{f(p)}$. Fix on $f^{*}TN$ the Sasaki metric
$g_{Sf}$ induced by $f^{*}\nabla^{LC}_{h}$, $f^{*}h$ and $g$. Let us denote by
$\pi:f^{*}T^{\delta}N\longrightarrow M$ be the projection of the bundle. Then
there is a map $p_{f}:(f^{*}(T^{\delta}N),g_{Sf})\longrightarrow(N,h)$ such
that:

  1. 1.

$p_{f}$ is a submersion,

  2. 2.

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

  3. 3.

${p}_{f}$ is $\Gamma$-equivariant888We are considering on $f^{*}TN$ the action
of $\Gamma$ given by $\gamma\cdot(p,w_{f(p)}):=(\gamma\cdot
p,d\gamma(w_{f(p)}))$,

  4. 4.

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

  5. 5.

Assume that for each point $p$ in $M$ there are some local coordinates
$\\{U,x^{i}\\}$ around a point $p$ in $M$ and some normal coordinates
$\\{V,y^{j}\\}$ around $f(p)$ on $N$ such that $V$ contains a
$\delta$-neighborhood of $f(U)$. Assume, moreover, that for each $k$ in
$\mathbb{N}$

$\sup\limits_{x\in U}\sup\limits_{s=0,...,k}|\frac{\partial^{s}y^{j}\circ
f}{\partial x^{i}_{1}...\partial x^{i_{s}}}(x)|\leq L_{k}$ (110)

for some $L_{k}$ which does not depend on the choice of $p$. 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

$\sup\limits_{(x,\mu)\in\pi^{-1}(U)}\sup\limits_{s+t=0,...,k}|\frac{\partial^{s}y^{j}\circ
p_{f}}{\partial x^{i}_{1}...\partial
x^{i_{s}}\partial\mu^{j_{1}}...\partial\mu^{j_{t}}}(x,\mu)|\leq C_{k}$ (111)

where $C_{k}$ only depends on the bounds $L_{1},...,L_{k}$. In particular if
$f$ is a Lipschitz map, then also $p_{f}$ is a Lipschitz map.

Proof: Let us define

$\begin{split}p_{f}:(f^{*}(T^{\delta}N),g_{s})&\longrightarrow(N,h)\\\
(p,w_{f(p)})&\longrightarrow exp_{f(p)}(w_{f(p)}).\end{split}$ (112)

Then

  1. 1.

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

  2. 2.

$p_{f}(p,0_{f(p)})=f(p)$. This follows by the definition of exponential map,

  3. 3.

$p_{f}$ is $\Gamma$-equivariant. Recall that $\Gamma$ acts by isometries. Then

$\begin{split}p_{f}(\gamma p,d\gamma w_{f(p)})&=exp_{f(\gamma p)}d\gamma
w_{f(p)}=exp_{\gamma f(p)}d\gamma w_{f(p)}\\\ &=\gamma
exp_{f(p)}w_{f(p)}=\gamma{p}_{f}(p,w_{f(p)}).\end{split}$ (113)

  4. 4.

${p}_{f}={p}_{id_{N}}\circ F$. It is obvious, indeed $F(p,w_{f(p)})=w_{f(p)}$
and $p_{id}:T^{\delta}N\longrightarrow N$ is $p_{id}(v_{p}):=exp_{p}(v_{p})$,

  5. 5.

Because of the previous point and because of (110) it is sufficient to prove
the assertion only for $p_{id}$. Moreover, in the case of the identity, we can
also suppose that $\\{x^{i}\\}$ and $\\{y^{j}\\}$ are the same normal
coordinates.
Consider $p_{id}$ restricted to $\pi^{-1}(U)$. It can be seen as
$\pi\circ\phi(x,\mu)$ where $\phi$ is the flow of the system of differential
equations given by

$\begin{cases}\dot{x}^{k}=\mu^{k}\\\
\dot{\mu}^{k}=-\Gamma^{k}_{ij}(x)\mu^{i}\mu^{j}\end{cases}$ (114)

Because of Lemma 3.4 of the paper of Schick flow , the partial derivatives of
$\phi$ are uniformly bounded. Then we conclude that the derivatives of
$p_{id}$ are uniformly bounded.

##  4 The pull-back functor

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

######  Lemma 4.1.

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

${t}_{f}(p,w_{f(p)})=(p,p_{f}(p,w_{f(p)}))$ (115)

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

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

$dim(f^{*}(T^{\delta}N))=m+n=dim(M)+dim(N)=dim(M\times N).$ (116)

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 $f^{*}(T^{\delta}N)$. Consider on $M\times N$ the normal
coordinates $\\{x^{i},y^{j}\\}$. Then the Jacobian of $t_{f}$ is given by

$Jt_{f}(x,\mu)=\begin{bmatrix}1&&\star\\\ 0&&Jexp_{x}(\mu)\end{bmatrix}$ (117)

Then, since the exponential map is a diffeomorphism for each $x_{0}$, $Jt_{f}$
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

$exp_{f(p)}w_{f(p)}=exp_{f(p)}v_{p}\implies w_{p}=v_{p},$ (118)

since their norm is less than $\delta$ and $\delta\leq inj_{N}$. We proved
that ${t}_{f}$ is a diffeomorphism with its image.
Since $p_{f}$ is a Lipschitz map, also ${t}_{f}$ 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 $\\{(p,w_{f(p)})\\}$. Because of this and
because of Remark 15, we obtain that the Fiber Volume of ${t}_{f}$ is given by
$|{t_{f}}^{-1^{*}}\frac{Vol_{T^{\delta}N}}{{t_{f}}^{*}Vol_{M\times N}}|$ on
the image of $t_{f}$ and it is null otherwise.
In particular, 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 $M\times N$. Because of the definition of
exponential map, the image of ${t}_{f}$ is contained in a
$\delta$-neighborhood of the $Graph(f)\in M\times N$. Then we can cover all
the image of ${t}_{f}$ using the normal coordinates $\\{x^{i},y^{j}\\}$ around
$(p,f(p))$.
Observe that, with respect to these coordinates, we have
${t}_{f}(0,\mu^{j})=(0,\mu^{j})$.
Consider
$Vol_{f^{*}T^{\delta}N}(x,\mu)=\sqrt{det(G_{ij})}(x,\mu)dx^{1}\wedge...\wedge
d\mu^{n}$ and $Vol_{M\times
N}(x,y)=\sqrt{det(H_{ij})}(x,y)dx^{1}\wedge...\wedge dy^{n}$, 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

$\frac{Vol_{T^{\delta}N}}{t^{*}Vol_{M\times
N}}(x,\mu)=\frac{\sqrt{det(G_{ij})}}{t_{f}^{*}(\sqrt{det(H_{ij})})}(x,\mu)\cdot\frac{1}{det(Jexp_{f(x)}(\mu))}.$
(119)

Observe that in $(0,\mu)$ the matrix $Jexp_{0}(\mu)$ is the identity. Moreover
we also have that $G_{ij}(0,\mu)$ is the identity matrix and so
$\sqrt{det(G_{ij})}(0,y)=1$. Finally we obtain

$H_{ij}(0,\mu)=\begin{bmatrix}1&&0\\\ 0&&h_{ij}(\mu)\end{bmatrix}$ (120)

where $h_{ij}$ is the matrix related to the Riemannian metric $h$ in normal
coordinates. Then $det(H_{ij})^{-1}(0,y)\leq C$ because $N$ is a manifold of
bounded geometry (Theorem 2.5 of flow ). This means that

$\frac{Vol_{T^{\delta}N}}{t^{*}Vol_{M\times
N}}(0,\mu)=\sqrt{\frac{det(G_{ij})}{t_{f}^{*}det(H_{ij})}}(0,\mu)\leq C$ (121)

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

######  Corollary 4.2.

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 $\mathcal{L}^{*}$-bounded.

Proof: Because of Lemma 4.1, we know that $t_{f}$ is a R.-N.-Lipschitz map.
Notice that $p_{f}=pr_{N}\circ t_{f}$, where $pr_{N}:M\times N\longrightarrow
N$ is the projection on the second component.
Consider $\overline{f}:f^{*}T^{\delta}N\longrightarrow N$ defined as
$\overline{f}(w_{f(p)}):=f(p)$, Observe that $p_{f}\sim_{\Gamma}\overline{f}$.
Because of Remark 2, in particular (7), there is a $C>0$ such that

$p_{f}^{-1}(q)\subset
A_{q}:=\overline{f}^{-1}(B_{C}(q))=\pi^{-1}f^{-1}(B_{C}(q)),$ (122)

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, as a consequence of Proposition 2.6, the Fiber Volume of $p_{f}$ in a
point $q$ is given by

$Vol_{p_{f}}(q)=\int_{M}Vol_{t_{f}}(p,q)d\mu_{M}=\int_{f^{-1}(B_{C}(q))}Vol_{t_{f}}(p,q)d\mu_{M}\leq
K\cdot\mu_{M}(f^{-1}(B_{C}(q))).$ (123)

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

$f^{-1}(B_{C}(q))\subseteq B_{R}(x_{0}).$ (124)

Moreover, by Remark (3), there is a constant $V$ such that

$\mu_{M}(f^{-1}(B_{C}(q)))\leq\mu_{M}(B_{R}(x_{0}))\leq V$ (125)

and so

$Vol_{p_{f}}(q)\leq K\cdot V$ (126)

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

######  Remark 21.

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. Then 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.

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

Let us introduce the notion of Thom form.

######  Definition 4.1.

Let $\pi:E\longrightarrow M$ be a vector bundle. A smooth form $\omega$ 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 $1$.

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

$e_{\omega}(\alpha):=\alpha\wedge\omega.$ (127)

Our goal, in this subsection, is to find a Thom form $\omega$ such that the
operator $e_{\omega}$ is $\mathcal{L}^{*}$-bounded. To this end we use the
Thom form introduced by Mathai and Quillen in mathai . In their work, indeed,
they compute a Thom form for a vector bundle endowed with a connection and a
metric bundle. In particular, we follow the construction of this form given by
Getzler in Proposition 1.3 of Gez .

First we construct the Thom form of $TN$.
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 fix the connection $\pi^{*}\nabla^{LC}_{h}$. Denote by
$\Omega^{i,j}$ the algebra

$\Omega^{i,j}:=\Omega^{i}(TN,\Lambda^{j}\pi^{*}TN)=\Gamma(TN,\Lambda^{i}T^{*}(TN)\otimes\Lambda^{j}\pi^{*}TN).$
(128)

We define $\Omega^{*,*}:=\bigoplus\limits_{i,j\in\mathbb{N}}\Omega^{i,j}$.
This is a bigraded algebra where $\Omega^{i,j}$ are the graded subspaces. Let
us define the section $X:TN\longrightarrow\pi^{*}TN$ as

$X(v_{p}):=(v_{p},v_{p}).$ (129)

Fix some normal coordinates $\\{U,x^{i}\\}$ on $N$ and let
$\\{\pi^{-1}(U),x^{i},\mu^{j}\\}$ be the coordinates on $TN$ induced by
$\\{x^{i}\\}$ and by $\\{\frac{\partial}{\partial x^{i}}\\}$. Then

$X(x,\mu):=\mu^{i}\frac{\partial}{\partial x^{i}}.$ (130)

Consider the map $\pi^{*}g(X,X)=|X|^{2}:TN\longrightarrow\mathbb{R}$

$|X|^{2}(v_{p}):=h_{p}(v_{p},v_{p})$ (131)

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

$|X|^{2}(x,\mu)=h_{ij}(x)\mu^{i}\mu^{j}.$ (132)

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

$\begin{split}\pi^{*}\nabla^{LC}_{h}(X)&=d\mu^{i}\otimes\frac{\partial}{\partial
x^{i}}+\mu^{i}\nabla\frac{\partial}{\partial x^{i}}\\\
&=d\mu^{i}\otimes\frac{\partial}{\partial
x^{i}}+\mu^{i}\Gamma^{k}_{ij}(x)dx^{j}\otimes\frac{\partial}{\partial
x^{k}}\\\
&=(\delta^{k}_{j}+\mu^{i}\Gamma^{k}_{ij}(x))dx^{j}\otimes\frac{\partial}{\partial
x^{k}}\end{split}$ (133)

Finally let us consider $\Omega$ the curvature form of $N$ induced by the
Levi-Civita connection $\nabla^{LC}_{h}$. This is a $2$-form on $N$ with
values in $SO(TN)$, which is the bundle of the skew-symmetric endomorphisms of
$TM$. Locally it is given by

$\Omega(x):=R^{j}_{kls}(x)dx^{k}\wedge
dx^{l}\otimes(dx^{s}\otimes\frac{\partial}{\partial x^{j}}),$ (134)

where $R^{j}_{kls}$ are the components of the Riemann tensor of $N$.
Let us identify $\Omega^{i}(N)\otimes SO(TN)$ with
$\Omega^{i}(N)\otimes\Gamma(\Lambda^{2}(TN))$ in the following way: for each
$A\in\Omega^{i}(N)\otimes SO(TN)$ we define $\tilde{A}$ as the form locally
defined as

$\tilde{A}(y):=h(e_{i}(y),A(y)e_{j}(y))e_{i}(y)\wedge
e_{j}(y)\in\Lambda^{2}(T_{N})$ (135)

where $\\{e_{i}\\}$ is an orthonormal frame of $TN$. This means that the
curvature $\Omega$, seen as an element of
$\Omega^{i}(N)\otimes\Gamma(\Lambda^{2}(TN))$, is locally given by

$\Omega(x):=R^{ij}_{kl}(x)dx^{k}\wedge dx^{l}\otimes(\frac{\partial}{\partial
x^{i}}\otimes\frac{\partial}{\partial x^{j}})$ (136)

where $R^{ij}_{kl}=h^{is}R^{j}_{kls}$. 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 whose
support is contained in $[\delta_{0},\delta_{0}]$. Assume that

$(-1)^{\frac{n(n+1)}{2}}\int_{\mathbb{R}^{n}}\phi^{(m)}(\frac{|x|^{2}}{2})dx=1.$
(137)

Then we define

$\overline{\omega}:=\sum_{k=0}^{n}\frac{\phi^{(k)}(\frac{|X|^{2}}{2})}{k!}(\pi^{*}\nabla
X+\pi^{*}\Omega)^{k},$ (138)

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
$\delta_{0}$-neighborhood of the zero section. This is a consequence of
$supp(\phi)\subseteq[-\delta_{0},\delta_{0}]$.
In local fibered coordinate $\\{(U,x^{i},\mu^{j})\\}$, the form
$\overline{\omega}$ is given by

$\overline{\omega}(x,\mu):=\alpha_{IJ}^{K}(x,\mu)dx^{I}\wedge
d\mu^{J}\otimes\frac{\partial}{\partial x^{K}}$ (139)

where, because of $N$ is a manifold of bounded geometry, there is a constant
$C$ which does not depend on the choice of $U$, such that
$|\alpha_{IJ}^{K}(x,\mu)|\leq C$.
Let us introduce the Berezin integral B. This is the isometry
$\textbf{B}:\Lambda^{n}(TN)\longrightarrow N\times\mathbb{R}$ defined as

$\textbf{B}(\alpha_{p}):=(p,Vol_{p}(\alpha_{p}))$ (140)

where $n=dim(N)$ and $Vol_{p}$ is the volume form of $N$ in a point $p$
999Actually the definition of Berezin integral is much more general: this is
the definition of Berezin integral for the fiber bundle $TN$.. It can be
extended to $\mathcal{B}:\Omega^{i,j}\longrightarrow\Omega^{i}(TN)$ by setting
$\mathcal{B}(\alpha\otimes\beta):=\textbf{B}(\beta)\alpha$ if $j=n$,
$\textbf{B}(\alpha\otimes\beta):=0$ otherwise.
As showed in Proposition 1.3 of Gez ,

$\omega:=\mathcal{B}(\overline{\omega})$ (141)

is a Thom form. Observe that, in fibered coordinates $\\{x,\mu\\}$ on $TN$,

$\omega(x,\mu)=\alpha_{IJ}(x,\mu)dx^{I}\wedge d\mu^{J},$ (142)

where $\alpha_{IJ}(x,\mu)=det(h_{ij}(x))^{\frac{1}{2}}\alpha_{IJ}^{0}(x,\mu)$
and $\alpha_{IJ}^{0}(x,\mu)$ is the coefficient of $dx^{I}\wedge
d\mu^{J}\otimes(\frac{\partial}{\partial
x^{1}}\wedge...\wedge\frac{\partial}{\partial x^{n}})$ in $\overline{\omega}$.
So we obtain that $|\alpha_{IJ}(x,\mu)|$ are uniformly bounded. Moreover, the
support of $\omega$ is contained in a $\delta_{0}$-neighborhood of the null
section.

######  Proposition 4.3.

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 $\mu^{j}$ refer to $\\{\frac{\partial}{\partial
x^{i}}\\}$, the coefficients of $\alpha$ are uniformly bounded, then the
pointwise norm $|\alpha_{p}|$ is uniformly bounded.

Proof: It is a direct computation. Let $f:(M,g)\longrightarrow(N,h)$ be a
smooth Lipschitz map and consider $(f^{*}(TN),g_{Sf})$ where $g_{Sf}$ is the
Sasaki metric induced by $g$, $f^{*}h$ and $f^{*}\nabla^{LC}_{h}$. Observe
that if $\omega$ is the Thom form on $TN$ defined in (141) and, if we consider
the map $F:f^{*}(TN)\longrightarrow TN$ given by
$F(p,w_{f(p)})=(f(p),w_{f(p)})$, then $F^{*}\omega$ is a Thom form for
$f^{*}(TN)$ and we also have a uniform bound on the norm of
$|F^{*}\omega|_{p}$.

######  Proposition 4.4.

Let us consider a form $\alpha$ on a Riemannian manifold $(M,g)$. Suppose that
there is a number $C$ such that $|\alpha_{p}|\leq C$ for each $p$ in $(M,g)$.
Then the operator $e_{\alpha}(\beta):=\beta\wedge\alpha$ defines an
$\mathcal{L}^{*}$-bounded operator.

Proof: It is a direct consequence of Hadamard-Schwartz inequality Sbordone
which states that, given some linear forms $\alpha_{1}$, …, $\alpha_{k}$ in
$\mathbb{R}^{n}$ with degree $l_{1},...,l_{k}$, then there is a constant
$C_{n}$ (which only depends on $n$) such that

$|\alpha_{1}\wedge...\wedge\alpha_{k}|\leq
C_{n}|\alpha_{1}|\cdot...\cdot|\alpha_{k}|.$ (143)

######  Remark 22.

Consider the bundle $\pi:TN\longrightarrow N$, let $h_{N}$ be a Riemannian
metric of bounded geometry on $N$ and let $\nabla^{LC}_{h}$ be the Levi-Civita
connection. Fix on $TN$ the Sasaki metric $h_{S}$ induced by $h$ and
$\nabla^{LC}_{h}$. Assume that a group $\Gamma$ acts FUPD on $(N,h)$ by
isometries. Observe that on $TN$ there is an action of $\Gamma$ induced by the
differential i.e.

$\gamma\cdot v_{p}:=d\gamma(v_{p}).$ (144)

Then there is a $\Gamma$-equivariant Thom form $\omega$ for $TN$ which
satisfies the assumptions of Proposition 4.4 and its support is contained in a
$\delta$-neighborhood of $0_{TN}$. In order to prove this fact, consider the
Riemannian covering $s:(N,h)\longrightarrow(N/\Gamma,\tilde{h})$ where
$\tilde{h}$ is the Riemannian metric induced by $h$. Then we obtain the map
$ds:(TN,h_{S})\longrightarrow(T(N/\Gamma),\tilde{h}_{S})$. Observe that
$TN/\Gamma$ and $T(N/\Gamma)$ are diffeomorphic and $ds$ can be seen as the
quotient map. Observe that $h_{S}$, which is the Sasaki metric on $TN$ induced
by $h$, is exactly the pullback metric of $\tilde{h}_{S}$. Then $ds$ is a
local isometry.
So $\omega:=(ds)^{*}\alpha$ is a $\Gamma$-equivariant Thom form for $TN$ which
satisfies all the assumptions of Proposition 4.4.

Let us consider a smooth Lipschitz map $f:(M,g)\longrightarrow(N,h)$ between
manifolds of bounded geometry. Suppose that there is a FUPD action of a group
$\Gamma$ by isometries on $M$ and $N$. Assume that $f$ is
$\Gamma$-equivariant. There is an action of $\Gamma$ on $f^{*}TN$ given by

$\gamma\cdot(p,w_{f(p)}):=(\gamma\cdot p,d\gamma(w_{f(p)})).$ (145)

If $\omega$ is a $\Gamma$-equivariant Thom form with support contained in a
$\delta$-neighborhood of $0_{TN}$ which satisfies the assumptions of
Proposition 4.4, we already know that $F^{*}\omega$ is a Thom form with
support contained in a $\delta$-neighborhood of $0_{TN}$ which satisfies the
assumptions of Proposition 4.4. Moreover $F^{*}\omega$ is also
$\Gamma$-equivariant. In order to prove this it is sufficient to prove that
$F:f^{*}TN\longrightarrow TN$ is $\Gamma$-equivariant. This is true, indeed

$F(\gamma p,d\gamma(w_{f(p)}))=d\gamma(w_{f(p)})=\gamma\cdot w_{f(p)}.$ (146)

by definition of $F$.

###  4.3 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
$pr_{M}:(f^{*}(T^{\delta}N),g_{Sf})\longrightarrow(M,g)$, let $\omega$ be a
Thom form of $TN$ defined as in the previous subsection and consider
$p_{f}:(f^{*}(T^{\delta}N),g_{Sf})\longrightarrow(M,g)$ the submersion related
to $f$ which we introduced in the previous section. If $f$ is differentiable,
then we can define the operator $T_{f}$ for each smooth $\mathcal{L}^{p}$-form
as

$T_{f}(\alpha):=pr_{M,\star}\circ e_{F^{*}\omega}\circ
p_{f}^{*}(\alpha)=\int_{B^{\delta}}p_{f}^{*}(\alpha)\wedge F^{*}(\omega)$
(147)

where $B^{\delta}$ denotes the fibers of $pr_{M}$. If $f$ is not a smooth
Lipschitz map, we consider a smooth Lipschitz map $f^{\prime}$ which is
uniformly homotopic to $f$ 101010We know that such a $f^{\prime}$ exists as a
consequence of Proposition 1.3 and we set111111Actually, in this case, the
definition of $T_{f}$ does depend on the choice of $f^{\prime}$. We will not
denote the choice of $f^{\prime}$ because, as we will see later, $T_{f}$
induces some operators in (un)-reduced $L^{q,p}$-cohomology and in
$L^{q,p}$-quotient cohomology which do not depend on the choice of
$f^{\prime}$.  $T_{f}:=T_{f^{\prime}}.$

######  Proposition 4.5.

Let $f$ be a uniform map between Riemannian manifolds of bounded geometry.
Then the operator $T_{f}$ is an $\mathcal{L}^{*}$-bounded operator.

Proof: Let us suppose $f$ is smooth. As a consequence of Corollary 4.2,
$p_{f}^{*}$ is $\mathcal{L}^{*}$-bounded. Because of Corollary 4.4, we also
know that $e_{F^{*}\omega}$ is an $\mathcal{L}^{*}$-bounded operator. Finally
$pr_{M,\star}$ is $\mathcal{L}^{*}$-bounded thanks to Proposition 3.4. Then
$T_{f}$ is a composition of $\mathcal{L}^{*}$-bounded operators and so it is
$\mathcal{L}^{*}$-bounded.

######  Corollary 4.6.

Given a uniform map $f:(M,g)\longrightarrow(N,h)$ between two Riemannian
manifolds of bounded geometry, then $T_{f}(dom(d_{min}))\subseteq
dom(d_{min})$ and $T_{f}d=dT_{f}$. In particular, this means that $T_{f}$
induces a morphism in $L^{q,p}$-cohomology. Moreover $T_{f}$ also induces a
morphism between the reduced $L^{q,p}$-cohomology groups and the
$L^{q,p}$-quotient cohomology groups.

Proof: Let us suppose, again, that $f$ is smooth and Lipschitz. In order to
prove the Corollary we will prove that the operator $T_{f}$ satisfies the
assumptions of Corollary 2.2. We already know that $T_{f}$ is
$\mathcal{L}^{*}$-bounded. Then 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, for each smooth form $\alpha$ in
$\mathcal{L}^{p}(N)$

$diam(supp(e_{F^{*}\omega}\circ p_{f}^{*}(\alpha))\leq C_{\alpha}.$ (148)

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

Observe that $e_{F^{*}\omega}\circ
p_{f}^{*}(\Omega^{*}_{c}(N))\subseteq\Omega^{*}_{vc}(f^{*}(T^{\delta}N))$,
where $\Omega^{*}_{vc}(f^{*}(T^{\delta}N))$ is the space of vertically
compactly supported smooth forms with respect to the projection
$\pi:f^{*}(T^{\delta}(M))\longrightarrow M$.
By Proposition 6.14.1. of the book of Bott and Tu bottu ,
$\pi_{\star}(\Omega^{*}_{vc}(f^{*}T^{\delta}N)\subseteq\Omega_{c}(M)$, we
obtain

$\int_{B^{\delta}}d\eta=d\int_{B^{\delta}}\eta,$ (149)

if $\eta$ is in $\Omega_{vc}^{*}(f^{*}T^{\delta}N)$. Then we conclude by
applying Corollary 2.2.

######  Remark 23.

Let $f:(M,g)\longrightarrow(N,h)$ be a smooth uniform map between manifolds of
bounded geometry. Let $\Gamma$ be a group acting FUPD on $(M,g)$ and $(N,h)$
by isometries and assume $f$ is $\Gamma$-equivariant.
Consider the action of $\Gamma$ on $f^{*}TN$ defined as
$\gamma\cdot(p,w_{f(p)}):=(\gamma\cdot p,d\gamma w_{f(p)})$. Because of Lemma
3.5, we know that $p_{f}$ is a $\Gamma$-equivariant map. This means that, if
$\alpha$ is a $\Gamma$-equivariant differential form on $N$, then also
$p_{f}^{*}\alpha$ is $\Gamma$-invariant. Moreover we also know that
$F^{*}\omega$ is $\Gamma$-equivariant (Remark 22).
Finally $pr_{M,\star}:\Omega^{*}_{cv}(f^{*}TN)\longrightarrow\Omega^{*}(M)$
preserves the $\Gamma$-equivariance of differential forms. In order to prove
this statement consider $X:=M/\Gamma$ and $Y:=N/\Gamma$ and let
$\tilde{f}:X\longrightarrow Y$ be the map induced by $f$. Denote by
$r:M\longrightarrow X$ and $s:N\longrightarrow Y$ the covering maps. Observe
that $f^{*}TN\cong r^{*}(\tilde{f}^{*}TY)$: this follows because $TN$ can be
identified with $s^{*}TY$ and so $f^{*}TN\cong f^{*}s^{*}TY\cong
r^{*}\tilde{f}^{*}TY$. Let us denote by $pr_{X,\star}$ the integration along
the fibers of $\tilde{f}^{*}TY$ and let
$R:r^{*}\tilde{f}^{*}TY\longrightarrow\tilde{f}^{*}TY$ be the bundle map
induced by $r$. Then we can conclude by applying Proposition VII of Chapter 5
in Conn  which allow us to say that

$pr_{M,\star}\circ R^{*}=r^{*}\circ pr_{X,\star}.$ (150)

Then $T_{f}=pr_{M}\circ e_{F^{*}\omega}\circ p_{f}^{*}$ is a
$\Gamma$-equivariant operator.

###  4.4 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 is a known fact that if $f_{0}\sim_{\Gamma}f_{1}$ with a
Lipschitz homotopy $H$, then

$f_{0}^{*}(E)\times[0,1]\cong f_{1}^{*}(E)\times[0,1]\cong H^{*}(E)\mbox{ and
}f_{0}^{*}(E)\cong f_{1}^{*}(E)$ (151)

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

######  Definition 4.2.

Consider two vector bundles $E$, $F$ over a manifold $N$. 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 $\delta$-neighborhood of the zero
sections $0_{E}$ and $0_{F}$. The bundles $(E,e)$ and $(F,f)$ are
R.-N.-Lipschitz equivalent if there is a bundle isomorphism
$\phi:E\longrightarrow F$ such that

  * •

$\phi(E^{\delta})=F^{\delta}$ for each $\delta>0$,

  * •

$\phi_{|E^{\delta}}$ and $\phi^{-1}_{|F^{\delta}}$ are R.N.-Lipschitz maps.

######  Remark 24.

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
$\phi^{*}A=A\circ\phi:(E^{\delta},e)\longrightarrow(M,g)$ is again an
R.-N.-Lipschitz map. This means that, up to isomorphisms, we can see $A$ as an
R.N.-Lipschitz map from $(E^{\delta},e)$ to $(M,g)$.

######  Proposition 4.7.

Let us consider two Riemannian manifolds $(M,g)$ 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 let
$H:(M\times[0,1],g+dt^{2})\longrightarrow(N,l)$ be a smooth Lipschitz homotopy
between them. Suppose that for each $p$ in $M$ and for each $t_{1},t_{2}$ in
$[0,1]$

$d(H(p,t_{1})\mathchar 24635\relax\;H(p,t_{2}))\leq\frac{inj_{N}}{4}.$ (152)

Let us denote by $g_{S,f_{i}}$ the Sasaki metric on $f_{i}^{*}(TN)$ defined by
using $g$, $f^{*}_{i}(l)$ and $f^{*}_{i}(\nabla^{LC}_{l})$ where $i=0,1$.
Then $(f^{*}_{0}(TN),g_{Sf_{0}})$ and $(f^{*}_{1}(TN),g_{S,f_{1}})$ are
R.-N.-Lipschitz equivalent.

Proof: We already know that $f^{*}_{0}TN$ and $f^{*}_{1}TN$ are homeomorphic.
In particular, we consider the homeomorphism $h:f^{*}_{1}TN\longrightarrow
f^{*}_{0}TN$ introduced in Proposition 1.7 of the book of Hatcher AT . Before
to introduce this homeomorphism we need a specific cover $\\{U_{i}\\}$ for
$M\times[0,1]$ and some specific orthonormal frames of $H^{*}TN$ on $U_{i}$.
In his proof, indeed, Hatcher uses a generic cover $\\{U_{i}\\}$ and some
generic frames of $H^{*}TN$, but, in our setting, we need $U_{i}$ and some
frames which satisfy some further conditions.
Let us start by the cover.
Fix two $N$-small numbers $\delta_{1}$ and $\delta_{2}$ such that
$\delta_{2}\leq\frac{inj_{N}}{2}$ and
$\delta_{1}\leq\delta_{2}-\frac{inj_{N}}{4}$. By Lemma 2.16 of bound  we know
that there is a cover $\\{B_{\delta_{1}}(q_{i})\\}$ of $N$ such that a ball
$B_{\delta_{2}}(x)$ in $N$ intersects at most $K$ balls
$\\{B_{\delta_{2}}(q_{i})\\}$. Consider the cover of $M\times[0,1]$ given by
$\\{H^{-1}(B_{\delta_{1}}(q_{i}))\\}$. Observe that

$H^{-1}(B_{\delta_{1}}(q_{i}))\subseteq\pi_{0}^{-1}(\pi_{0}(H^{-1}(B_{\delta_{1}}(q_{i}))))\subseteq
H^{-1}(B_{\delta_{2}}(q_{i}))$ (153)

where $\pi_{0}:M\times[0,1]\longrightarrow M$ is the projection on the first
component. The first inclusion of (153) is obvious. The second one is a
consequence of (152), indeed if $p$ is in $\pi(H^{-1}(B_{\delta_{1}}(q_{i})))$
then there is a $t_{1}$ such that $H(p,t_{1})$ is in $B_{\delta_{1}}(q_{i})$.
Then for all $t_{2}$ in $[0,1]$ the inequality (152) holds. This implies that

$d(H(p,t_{2})\mathchar 24635\relax\;q)\leq d(H(p,t_{1})\mathchar
24635\relax\;H(p,t_{2}))+d(H(p,t_{1}),q)\leq\frac{inj_{N}}{4}+\delta_{1}\leq\delta_{2}$
(154)

and so $(p,t_{2})$ is in $H^{-1}(B_{\delta_{2}}(q_{i}))$ for each $t_{2}$ in
$[0,1]$. Let us define

$U_{i}:=\pi_{0}(H^{-1}(B_{\delta_{1}}(q_{i}))).$ (155)

Let us consider the cover $\\{U_{i}\times[0,1]\\}$. Observe that
$H(U_{i}\times[0,1])\subseteq B_{\frac{inj_{N}}{2}}(q_{i})$ for each $i$.
Moreover there is a number $K$ such that for each $i$ the intersection
$(U_{i}\times[0,1])\cap(U_{j}\times[0,1])$ is not empty.
Let us introduce our orthonormal frame of $TN$ on
$B_{\frac{inj_{N}}{2}}(q_{i})$.
Consider for each $i$ some normal coordinates $\\{y^{j}\\}$ around $q_{i}$.
Fix on $T_{q_{i}}N$ the orthonormal basis $\\{s_{j}(q_{i})\\}$ where
$s_{j}(q_{i}):=\frac{\partial}{\partial y^{j}}(q_{i})$. Using parallel
transport along geodesics emanating from $q_{i}$, we obtain a frame
$\\{s_{r}\\}$. Let us denote by $\\{\theta^{r}\\}$ the dual frame of
$\\{s_{r}\\}$ and let
$a^{r}_{j},b^{r}_{j}:B_{\frac{inj_{N}}{2}}(q_{i})\longrightarrow\mathbb{R}$ be
the function defined by

$\theta^{s}(y):=a^{s}_{j}(y)dy^{j}\mbox{ and }dy^{s}:=b^{s}_{j}(y)\theta^{j}.$
(156)

As shown on pages 6-9 of flow , for each $k$ in $\mathbb{N}$, the partial
derivatives of order $k$ of $|a^{i}_{j}(y)|$ and of $|b^{i}_{j}(y)|$ are
uniformly bounded by a constant $C_{k}$ which does not depends on
$B_{\frac{inj_{N}}{2}}(q_{i})$ or on $\\{y^{j}\\}$. Consider the Riemannian
metric $l$ on $N$. Let us define $A^{s}_{j}:=l^{sr}a_{r}^{m}l_{mj}$ and
$B^{s}_{j}:=l_{sr}b^{r}_{m}l^{mj}$. Then

$s_{r}(y)=A^{w}_{r}(y)\frac{\partial}{\partial y^{w}}(y)\mbox{ and
}\frac{\partial}{\partial y^{w}}(y):=B^{r}_{w}(y)s_{r}(y).$ (157)

Observe that also the partial derivatives of order $k$ of $|A^{i}_{j}(y)|$ and
of $|B^{i}_{j}(y)|$ are uniformly bounded.
The frame $\\{s_{j}\\}$ can be used to identify $\pi:H^{*}TN\longrightarrow
M\times[0,1]$ on $U_{i}\times[0,1]$ and
$U_{i}\times[0,1]\times\mathbb{R}^{n}$. Indeed we can identify
$(p,t,\mu^{j}s_{j}(H(p,t)))$ in $\pi^{-1}(U\times[0,1])$ with
$(p,t,\mu^{1},...,\mu^{n})$ in $U\times[0,1]\times\mathbb{R}^{n}$.

Let us introduce the isomorphism $h$. By Lemma 2.17 of bound , there is a
partition of unity $\\{\xi\\}$ of $N$ referred to the cover
$\\{B_{\delta_{1}}(q_{i})\\}$ where each $\xi$ is a smooth Lipschitz function.
Consider $\\{\phi_{i}\\}$ where $\phi_{i}:=f^{*}_{0}(\xi)$. We know that
$f_{0}^{-1}(B_{\delta_{1}}(q_{i}))\subseteq\pi_{0}(H^{-1}(B_{\delta_{1}}(q_{i}))=U_{i}$.
This means that $\\{\phi_{i}\\}$ is a smooth Lipschitz partition of unity
subordinate to $U_{i}$. Let $\psi_{i}:M\longrightarrow[0,1]$ be

$\psi_{i}:=\phi_{1}+\phi_{2}+...+\phi_{i}.$ (158)

Let $M_{i}\subset M\times[0,1]$ be the graph of $\psi_{i}$ and let
$\pi_{i}:E_{i}\longrightarrow M_{i}$ be the restriction of $H^{*}TN$ to
$M_{i}$. Since $H^{*}TN$ is trivial on $U_{i}\times[0,1]$, then homeomorphism
$e_{i}:M_{i}\longrightarrow M_{i+1}$ defined as
$e_{i}(p,\psi_{i}(p)):=(p,\psi_{i-1}(p))$ lifts to a homeomorphism
$h_{i}:E_{i}\longrightarrow E_{i-1}$. This homeomorphism is the identity
outside $\pi^{-1}(U_{i}\times[0,1])$ and on $p^{-1}(U_{i}\times[0,1])$ it is
defined by

$h_{i}(p,\psi_{i}(p),v):=(p,\psi_{i-1}(p),v).$ (159)

Then homeomorphism $h$ is given by the composition

$h:=h_{1}\circ h_{2}\circ h_{3}\circ....$ (160)

We already know from Proposition 1.7 of AT  that $h$ is an isomorphism of
vector bundles and so, in particular, a diffeomorphism.
Fix on $M\times[0,1]$ the metric $g+dt^{2}$. Let us consider the maps
$\tilde{f}_{i}:M\longrightarrow N$ defined as
$\tilde{f}_{i}(p):=H(p,\psi_{i}(p))$ and consider the Sasaki metric
$g_{S,\tilde{f}_{i}}$ on $E_{i}$ induced by the metric $g$ on $M$, the
pullback metric $\tilde{f}_{i}^{*}l$ and the pullback connection
$\tilde{f}_{i}^{*}\nabla^{Lc}_{l}$. Observe that outside
$\pi^{-1}(U_{i}\times[0,1])$ the map
$h_{i}:(\tilde{f}^{*}_{i}TN,g_{S,f_{i}})\longrightarrow(\tilde{f}^{*}_{i-1}TN,g_{S,f_{i-1}})$
is an isometry. This follows because $\tilde{f}_{i-1}=\tilde{f}_{i}$ outside
$\pi^{-1}(U_{i}\times[0,1])$. Then the norm of $dh_{i}$ and the Fiber Volume
of $h_{i}$ are both equal to $1$ outside $\pi^{-1}(U_{i}\times[0,1])$. Let us
consider $h_{i}$ on $\pi^{-1}(U_{i}\times[0,1])$.

Consider $p$ in $U_{i}$. Recall that we have on
$B_{\frac{inj_{N}}{2}}(q_{i+1})$ some normal coordinates $\\{y^{z}\\}$. Fix a
constant $K$ such that $K>\sup\\{1,||dH_{(}q,t)||\\}$ for each $(q,t)$ in
$M\times[0,1]$. Choose some normal121212With respect to $g$ coordinates
$\\{W,x^{j}\\}$ around $p$ such that in $W\times[0,1]$ the norms of the
derivatives of $H:W\times[0,1]\longrightarrow B_{\frac{inj_{N}}{2}}(q_{i+1})$
and the norm of the Gram matrix of $g$ with respect to $\\{x^{i}\\}$ are
uniformly bounded by $K$. Let $\\{x^{j},\mu^{z}\\}$ be the fibered coordinates
on $\tilde{f}_{i+1}^{*}(TN)_{|U_{i+1}}$ related to the frame
$\\{\frac{\partial}{\partial y^{z}}\\}$. On the other hand we can also define
the fibered coordinates $\\{x^{j},\nu^{z}\\}$ on
$\tilde{f}_{i+1}^{*}(TN)_{|U_{i+1}}$ related to the frame $\\{s_{j}\\}$.
In the same way we obtain $\\{x^{j},\sigma^{z}\\}$ which are the fibered
coordinates on $\tilde{f}_{i}^{*}(TN)$ related to the frame
$\\{\frac{\partial}{\partial y^{j}}\\}$ and the coordinates
$\\{x^{j},\tau^{z}\\}$ which are the fibered coordinates related to the frame
$\\{s_{j}\\}$.
With respect to the coordinates related to the frame $\\{s_{j}\\}$ we have
$h_{i}(x,\nu)=(x,\nu)$, while, with respect to the coordinates related to
$\\{\frac{\partial}{\partial y^{j}}\\}$, we obtain

$h_{i+1}(x,\mu)=(x^{j},A_{r}^{j}(\tilde{f}_{i}(x))B_{w}^{r}(\tilde{f}_{i+1}(x))\mu^{w}).$
(161)

The norms of $\frac{\partial}{\partial x^{i}}$ and
$\frac{\partial}{\partial\mu_{j}^{m}}$ with respect to $g_{S,\tilde{f}_{i+1}}$
and the norms of $\frac{\partial}{\partial x^{i}}$ and
$\frac{\partial}{\partial\sigma_{j}^{m}}$ with respect to
$g_{S,\tilde{f}_{i}}$ are uniformly bounded by a constant $L$ which does not
depend on $i$ (this is a consequence of (89)). Then, since the norms of the
derivatives of $A_{r}^{j}$ and $B_{w}^{r}$ are uniformly bounded by a constant
$C$, we obtain that the Lipschitz constant of $h_{i}$ is less or equal to
$C^{2}\cdot L$.
Let us focus on the Fiber Volume of $h_{i}$ on $U_{i}$.
Observe that, since $h_{i}(x,\nu)=(x,\nu)$,

$\begin{split}Vol_{h_{i+1}}(x,\nu)&=h_{i+1}^{*}\sqrt{\frac{det(G_{rj})}{det(L_{jk})}}(x,\nu)\\\
&=\sqrt{det(G_{rj})det(L_{jk})^{-1}}(x,\nu).\end{split}$ (162)

where $G_{rj}$ is the matrix related to the metric $g_{S,f^{*}_{i+1}}$ with
respect to the coordinates $\\{x,\nu\\}$ and $L_{jk}$ is the matrix related to
$g_{S,f^{*}_{i}}$ with respect to the coordinates $\\{x,\tau\\}$. Observe that

$det(G_{rj})(x,\nu)=det(\hat{G}_{rj}))(A(x,\nu))\cdot det(J_{A}(x,\nu))^{2},$
(163)

where $A$ is the change of coordinates from $(x,\nu)$ to $(x,\mu)$, $J_{A}$ is
its Jacobian and $\hat{G}_{rj}$ is the matrix related to the metric
$g_{S,\tilde{f}^{*}_{i+1}}$ with respect to the coordinates $\\{x,\mu\\}$.
Define

$A(x,\nu):=(x^{l},A^{r}_{j}(f_{i}(x))\nu^{j}).$ (164)

Notice that the determinant of $\hat{G}_{kr}$ is uniformly bounded on each
$\delta$-neighborhood of the $0$-section of $f^{*}_{i+1}TN$ (this is a
consequence of Formula (89)). Then

$|det(G_{rj})(x,\nu)|=|det(\hat{G}_{kr}))(A(x,\nu))|\cdot|det(J_{A}(x,\nu))|^{2}\leq
J(d_{\tilde{f}_{i+1}TN}((x,\mu),0_{f^{*}_{i+1}TN})$ (165)

where $J:\mathbb{R}\longrightarrow\mathbb{R}$ is a function which does not
depend on $i$.
Let us focus on $det(L_{jk})^{-1}$. We have

$det(L_{rj})^{-1}(x,\tau)=det(\hat{L}_{kr}))^{-1}(\overline{A}(x,\tau))\cdot
det(J_{\overline{B}}(x,\tau))^{2},$ (166)

where $\overline{A}$ is the change of coordinates from $(x,\tau)$ to
$(x,\sigma)$, $J_{\overline{B}}$ is the Jacobian of its inverse and
$\hat{L}_{kr}$ is the matrix related to the metric $g_{S,f^{*}_{i}}$ with
respect to the coordinates $\\{x,\sigma\\}$. Let us define

$A(x,\nu):=(x^{l},A^{r}_{j}(\tilde{f}_{i}(x))\tau^{j}).$ (167)

and

$A(x,\nu)^{-1}:=(x^{l},B^{r}_{j}(\tilde{f}_{i}(x))\sigma^{j}).$ (168)

Notice that the determinant of $\hat{L}_{kr}^{-1}$ is uniformly bounded on
each $\delta$-neighborhood of the $0$-section of $f^{*}_{i}(TN)$ (this is a
consequence of Formula (91)). Then

$|det(L_{ij})^{-1}(x,\tau)|=|det(\hat{L}_{kr})^{-1}(\overline{A}(x,\tau))|\cdot|det(J_{\overline{B}}(x,\tau))|^{2}\leq
F(d_{\tilde{f}_{i}}((x,\mu),0_{f^{*}_{i+1}}),$ (169)

where $F:\mathbb{R}\longrightarrow\mathbb{R}$ is a function which does not
depend on $i$. This means that for each $\delta$ the Fiber Volume of
$h_{i|_{\pi_{i}^{-1}(U_{i})}}$ is bounded in each $\delta$-neighborhood of the
$0$-section of $f^{*}_{i}TN$ by a constant $C(\delta)$.
Then, in order to conclude the proof, we just have to observe that, since for
each $p$ in $M$ there are at most $K$ of $U_{i}$’s such that $p\in U_{i}$,
then the Lipschitz constant of $h$ is bounded by $C^{2K}\cdot L^{K}$ and the
Fiber Volume of $h$ is uniformly bounded by $C(\delta)^{K}$. The same also
happens for $h^{-1}$ and so $f^{*}_{0}(TN)$ and $f^{*}_{1}(TN)$ are
R.N.-Lipschitz equivalent.

######  Proposition 4.8.

Let us consider $f_{0},f_{1}:(M,g)\longrightarrow(N,h)$ two smooth 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.

Proof: Let us consider a smooth Lipschitz homotopy $H$ between $f_{0}$ and
$f_{1}$. Consider a finite partition $\\{[s_{i},s_{i+1}]\\}$ of $[0,1]$ such
that $s_{i+1}-s_{i}\leq\frac{inj_{N}}{4C_{H}}$, 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

$d(H_{i}(p),H_{i+1}(p))\leq d(H(p,s_{i}),(p,s_{i+1}))\leq
C_{H}\cdot(s_{i+1}-s_{i})=\frac{inj_{N}}{4}.$ (170)

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

######  Corollary 4.9.

Consider a Lipschitz map $H:(M\times[0,1],g+dt^{2})\longrightarrow(N,h)$.
Suppose that $g$ satisfies the same assumption as Proposition 4.7. For each
$i$ in $[0,1]$, denote by $f_{i}:(M,g)\longrightarrow(N,h)$ the map defined as
$f_{i}(p):=H(p,i)$. Assume that there is an $\varepsilon$-neighborhood of $i$
in $[0,1]$ such that $H(x,t)=f_{i}(x)$ on this neighborhood. Then the vector
bundles $(H^{*}(TN),g_{S,H})$ and
$(f^{*}_{i}(TN)\times[0,1],g_{Sf_{i}}+dt^{2})$ are R.-N.-Lipschitz equivalent.

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
$\overline{f}_{i}^{*}(TN)$ defined by using $g$, $\overline{f}^{*}_{i}(h)$ and
$\overline{f}^{*}_{i}(\nabla^{LC}_{h})$ is the product metric between the
Sasaki metric on $f^{*}_{i}(TN)$ and the metric $dt^{2}$ on $[0,1]$.
Then we can conclude by observing that the map

$\mathcal{H}:(M\times[0,1]\times[0,1],g+ds^{2}+dt^{2})\longrightarrow(N,h)$
(171)

defined as $\mathcal{H}(p,s,t):=H(p,i\cdot t+s\cdot(1-t))$ is a Lipschitz-
homotopy between $H$ and $\overline{f}_{i}$.

######  Corollary 4.10.

Consider two smooth Lipschitz maps $f_{0},f_{1}:(M,g)\longrightarrow(N,h)$,
assume that $g$ satisfies the same assumptions of Lemma 4.7. Suppose that
$f_{0}$ and $f_{1}$ are Lipschitz-homotopic with a smooth Lipschitz homotopy
$H$. Assume that there are two $\varepsilon$-neighborhoods $U_{0}$ and $U_{1}$
of $M\times\\{0\\}$ and $M\times\\{1\\}$ such that $H(p,t)=f_{i}(p)$ for each
$(p,t)$ in $U_{i}$. Let us denote by

$\Phi:(f_{0}^{*}(TN)\times[0,1],g_{S,f_{0}}+dt^{2})\longrightarrow(H^{*}TN,g_{SH})$
(172)

the R.N.-Lipschitz equivalence of vector bundles. Then the restriction

$\phi:=\Phi_{|_{f_{0}^{*}(TN)\times\\{0\\}}}:(f_{0}^{*}(TN),g_{S,f_{0}})\longrightarrow(f_{1}^{*}(TN),g_{S,f_{1}}).$
(173)

is a R.N.-Lipschitz equivalence of vector bundles.

Proof: We know that $\phi$ is a vector bundle isomorphism: its inverse is
given by the inverse of $\Phi$ restricted to the bundle
$H^{*}TN_{|M\times\\{1\\}}$. Moreover the injections
$j_{i}:(f_{i}^{*}(TN),g_{Sf_{i}})\longrightarrow(H^{*}TN,g_{H})$ are isometric
embedding since $H$ is constant around $M\times\\{i\\}$. Then the bounds on
the Fiber Volume and on the Lipschitz constant of $\phi$ are the same of
$\Phi$.

######  Remark 25.

Let $i=0,1$ and consider two smooth Lipschitz maps
$f_{i}:(M,g)\longrightarrow(N,h)$. Let us suppose that $h$ is a Lipschitz
homotopy between $f_{0}$ and $f_{1}$. Let $p_{h}$ and $p_{f_{i}}$ be the
submersion defined in Lemma 3.5 related to $h$ and to $f_{i}$. Then

$p_{h}\circ\Phi:(f_{0}^{*}(TN)\times[0,1],g_{S,f_{0}}+dt^{2})\longrightarrow(N,h)$
(174)

is a smooth Lipschitz homotopy between $p_{f_{0}}$ and

$p_{f_{1}}\circ\phi:(f_{0}^{*}(T^{\delta}N),g_{S,f_{0}})\longrightarrow(f_{1}^{*}(T^{\delta}N),g_{S,f_{1}}).$
(175)

Moreover it is also true that

$\begin{split}T_{f_{1}}&=pr_{M1\star}\circ e_{F_{1}^{*}\omega}\circ
p_{f_{1}}^{*}\\\ &=pr_{M1\star}\circ(\phi^{*})^{-1}\circ\phi^{*}\circ
e_{F_{1}^{*}\omega}\circ(\phi^{*})^{-1}\circ\phi^{*}\circ p_{f_{1}}^{*}\\\
&=pr_{M0\star}\circ
e_{\phi^{*}F_{1}^{*}\omega}\circ(p_{f_{1}}\circ\phi)^{*},\end{split}$ (176)

where $pr_{Mi}:f^{*}_{i}(TN)\longrightarrow M$ is the projection of the
bundle. This means that up to R.N.-Lipschitz isomorphisms, we can consider
$p_{f_{1}}$ as a map defined on $f_{0}^{*}TN$ and $p_{h}$ as a homotopy
between $p_{f_{0}}$ and $p_{f_{1}}$. From now on we will not write the
isomorphisms $\Phi$ or $\phi$ every time.

###  4.5 Lemmas about homotopy

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

######  Lemma 4.11.

Let $f:(M,g)\longrightarrow(N,h)$ be a smooth Lipschitz map and let
$\delta\leq inj_{N}$. Consider the homotopy
$H:(f^{*}(T^{\delta}N)\times[0,1],g_{S}+dt^{2})\longrightarrow(N,l)$ defined
as

$H(p,w,t):=p_{f}(t\cdot w_{f(p)}).$ (177)

Then, if $f$ is a smooth R.-N.-Lipschitz map, then also $H$ and
$(H,id_{[0,1]}):(f^{*}(T^{\delta}N)\times[0,1],g_{S}+dt^{2})\longrightarrow(N\times[0,1],l+dt^{2})$
are R.-N.-Lipschitz maps.

Proof: Observe that $H=pr_{N}\circ(H,id_{[0,1]})$, where
$pr_{N}:N\times[0,1]\longrightarrow N$ is the projection of the first
component. In particular $pr_{N}$ is a R.-N.-Lipschitz map. This means that if
we prove that $(H,id_{[0,1]})$ is a R.-N.-Lipschitz map, then also $H$ is a
R.-N.-Lipschitz map.
Observe that $f^{*}(T^{\delta}N)\times\\{0,1\\}$ is a set of null measure and
so we can consider the interval $[0,1]$ as open.
Let us define the map

$(pr_{M},H,id_{(0,1)}):(f^{*}T^{\delta}N\times(0,1),g_{S}+dt^{2})\longrightarrow(M\times
N\times(0,1),g\times h+dt^{2})$ (178)

given by $(pr_{M},H,id_{(0,1)})(v_{f(p)},s):=(p,p_{f}(s\cdot v_{p}),s)$. This
map is a submersion since we are considering $s\neq 0$. In particular
$(pr_{M},H,id_{(0,1)})$ is Lipschitz since it is a composition of Lipschitz
maps.
Moreover $(H,id_{(0,1)})=pr_{N\times(0,1)}\circ(pr_{M},H,id_{(0,1)})$ and so,
by applying Proposition 2.6, we have

$Vol_{(H,id_{(0,1)})}(q,t)=\int_{M}Vol_{(pr_{M},H,id_{(0,1)})}(p,q,t)d\mu_{M}.$
(179)

Let us calculate the Fiber Volume of $(pr_{M},H,id_{(0,1)})$.
Similarly to the Lemma 4.1, consider for each $p$ in $M$ some normal
coordinates $\\{x^{i}\\}$ around $p$, some normal coordinates $\\{y^{j}\\}$
around $f(p)$ in $N$. Fix the frame $\\{\frac{\partial}{\partial x^{i}}\\}$,
we obtain the coordinates $\\{x^{i},\mu^{j},t\\}$ on $f^{*}(TN)\times[0,1]$
around $(p,0,0)$.
Moreover we can also fix 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, with respect to these coordinates, we obtain

$(pr_{M},H,id_{(0,1)})(0,\mu^{j},t)=(0,t\cdot y^{j},t).$ (180)

Similarly as we did in Lemma 4.1, the Volume forms
$Vol_{f^{*}(TN)\times[0,1]}(0,\mu,t)=dx^{1}\wedge...\wedge dx^{m}\wedge
d\mu^{1}\wedge...\wedge d\mu^{n}\wedge dt$ and $Vol_{M\times
N\times[0,1]}(0,y,t)=\sqrt{det(H_{ij}(0,y))}dx^{1}\wedge...\wedge dx^{m}\wedge
dy^{1}\wedge...\wedge dy^{n}\wedge dt$, where

$H_{ij}(0,y)=\begin{bmatrix}1&&0&&0\\\ 0&&h_{l,s}(y)&&0\\\
0&&0&&1\end{bmatrix}$ (181)

and $h_{l,s}$ are the components of the metric $h$ on $N$. Then
$(pr_{M},H,id_{[0,1]})$ is a diffeomorphism with its image and so, by applying
Remark 15, its Fiber Volume on $im(pr_{M},H,id_{[0,1]})$ is given by

$|[(pr_{M},H,id_{[0,1]})^{-1}]^{*}\frac{Vol_{f^{*}TN\times[0,1]}}{(pr_{M},\tilde{h},id_{[0,1]})^{*}Vol_{M\times
N\times[0,1]}}|.$ (182)

Observe that, similarly to Lemma 4.1, in a point $(0,y^{j},t)$ the Fiber
Volume of $(pr_{M},\tilde{h},id_{[0,1]})$ is

$\frac{Vol_{f^{*}TN\times[0,1]}}{(pr_{M},H,id_{[0,1]})^{*}Vol_{M\times
N\times[0,1]}}=\frac{1}{t^{n}}(1+C(t,y)),$ (183)

where $C$ is a bounded function.
Consider the projection $pr_{N\times[0,1]}:M\times N\times(0,1)\longrightarrow
N\times(0,1)$. Let us recall that, thanks to the Proposition 2.6, we have

$Vol_{(H,id_{(0,1)})}(q,t)=\int_{M}Vol_{(id_{M},H,id_{[0,1]})}(p,q,t)d\mu_{M}.$
(184)

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 $H_{t}:M\times
N\longrightarrow N$ the map defined as $H_{t}(p,q):=H(p,q,t)$. Observe that
$Vol_{(id_{M},H,id_{[0,1]})}$ on $M\times\\{q\\}\times\\{t\\}$ is null outside

$pr_{M}(H^{-1}_{t}(q))\times\\{q\\}\times\\{t\\}\supseteq[im(id_{M},H,id_{[0,1]})]\cap[M\times\\{q\\}\times\\{t\\}].$
(185)

Since $H$ is Lipschitz, by applying Remark 2, we have

$H^{-1}_{t}(q)\subseteq\pi^{-1}(f^{-1}(B_{C_{H}\cdot t}(q)))\times\\{t\\}$
(186)

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

$pr_{M}(H^{-1}_{t}(q))\times\\{q\\}\times\\{t\\}\subset f^{-1}(B_{C_{H}\cdot
t}(q))\times\\{q\\}\times\\{t\\}.$ (187)

Observe that, since $N$ is a manifold of bounded geometry,

$\mu_{N}(B_{C_{H}\cdot t}(q))\leq C_{H}^{n}t^{n}(1+L(t))$ (188)

where $L$ is a bounded function which depends by $t$.
Then because of the fact that $f$ is a R.-N.-Lipschitz map, we obtain

$\mu(f^{-1}(B_{C_{H}\cdot t}(q)))\leq K_{0}\mu_{N}(B_{C_{H}\cdot t}(q))\leq
K_{0}C_{H}^{n}t^{n}(1+L(t)),$ (189)

Then the Fiber Volume of $(H,id_{(0,1)})$ is given by

$\begin{split}Vol_{(H,id_{(0,1)})}(q,t)&=\int_{f^{-1}(B_{C_{H}\cdot
t}(q))}Vol_{(id_{M},H,id_{[0,1]})}(p,q,t)d\mu_{M}\\\
&\leq\frac{1}{t^{n}}(1+C(t,\tilde{y}))\cdot(K_{0}C_{H}^{n}t^{n}(1+L(t)))\\\
&\leq K\cdot\frac{1}{t^{n}}\cdot t^{n}\leq K.\end{split}$ (190)

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

$H=pr_{N}\circ(H,id_{(0,1)})$ (191)

is a R.-N.-Lipschitz map. Let us consider two smooth 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 map
$\overline{g}:g^{*}T^{\sigma}M\longrightarrow M$ defined as
$\overline{g}(v_{g(p)}):=g(p)$ and consider the submersion
$p_{g}:g^{*}T^{\sigma}M\longrightarrow M$ related to $g$.
Consider, moreover, the compositions $f\circ\overline{g}$ and $f\circ
p_{g}:g^{*}T^{\sigma}M\longrightarrow N$. Observe that these maps are
Lipschitz-homotopic. Then, because of Remark 25, the pullback bundles over $S$
given by $(f\circ\overline{g})^{*}TN$ and $(f\circ p_{g})^{*}TN$ are
R.-N.-Lipschitz homotopy equivalent. This means that, up to R.N.-Lipschitz
equivalence of vector bundles, we can suppose

$(f\circ\overline{g})^{*}TN=(f\circ p_{g})^{*}TN.$ (192)

Then we can see the maps $p_{f\circ p_{g}}$ and $p_{f\circ\overline{g}}$ as
defined on the same domain $(f\circ\overline{g})^{*}TN$.
Denote by $h:g^{*}(T^{\sigma}M)\times[0,1]\longrightarrow N$ the homotopy
defined as $h(v_{g(p)},t):=f\circ p_{g}(t\cdot v_{g(p)})$ between $f\circ
p_{g}$ and $f\circ\overline{g}$. Because of Corollary 4.9 the bundles
$(h^{*}(TN),g_{S,h})$ and
$(f\circ\overline{g}^{*}(T^{\delta}N)\times[0,1],g_{S}+dt^{2})$ are
R.-N.-Lipschitz equivalent. In particular, according to Remark 25, we consider

$p_{h}:((f\circ\overline{g})^{*}(T^{\delta}N)\times[0,1],g_{S}+dt^{2})\longrightarrow(N,l)$
(193)

is a R.-N.-Lipschitz map because $h$ is smooth, uniformly proper and Lipschitz
map and because of Remark 21.
Observe that $p_{h}$ is a Lipschitz homotopy between the maps
$p_{f\circ\overline{g}}$ and $p_{f\circ
p_{g}}:f\circ\overline{g}^{*}(T^{\delta}N)\longrightarrow(N,l)$.
Applying the point 4. of Proposition 3.5, we obtain that

$p_{f\circ p_{g}}=p_{id_{M}}\circ F\circ P_{g}=p_{f}\circ P_{g}$ (194)

where $F:f^{*}(T^{\delta}N)\longrightarrow T^{\delta}N$ is the bundle map
induced by $f$ and

$P_{g}:(f\circ\overline{g})^{*}T^{\delta}N=(f\circ
p_{g})^{*}T^{\delta}N\longrightarrow f^{*}T^{\delta}N$ (195)

is the bundle map induced by $p_{g}$. Then we obtain the following
Proposition.

######  Proposition 4.12.

The map
$H:((f\circ\overline{g})^{*}T^{\delta}N\times[0,1],g_{S}+dt^{2})\longrightarrow(N,l)$
defined as

$H(v_{f\circ\overline{g}},t):=p_{f}(P_{g}(t\cdot
v_{f\circ\overline{g}}))=p_{h}$ (196)

is a R.-N.-Lipschitz map and it is the homotopy between $p_{f\circ p_{g}}$ and
$p_{f\circ\overline{g}}$.

###  4.6 Uniform homotopy invariance of (un)-reduced $L^{q,p}$-cohomology

Before the final proof we need some Lemmas.

######  Lemma 4.13.

Let $(M,g)$ be a Riemannian manifold and consider $([0,1],dt^{2})$. Then there
is an $\mathcal{L}^{*}$-bounded operator
$\int_{0\mathcal{L}}^{1}:\mathcal{L}^{p}(M\times[0,1])\longrightarrow\mathcal{L}^{p}(M)$
such that for all smooth $\alpha\in\mathcal{L}^{p}(M\times[0,1])$,

$i_{1}^{*}\alpha-
i_{0}^{*}\alpha=\int_{0\mathcal{L}}^{1}d\alpha+d\int_{0\mathcal{L}}^{1}\alpha$
(197)

Moreover $\int_{0\mathcal{L}}^{1}$ sends compactly-supported differential
forms on $\Omega^{*}(M\times[0,1])$ to $\Omega^{*}_{c}(M)$.

Proof: Let $\alpha$ be in $\Omega^{*}_{c}(M\times[0,1])$ with compact support
and let $p:M\times[0,1]\longrightarrow M$ be the projection on the first
component. Then we can decompose every differential form on $M\times[0,1]$ as
a sum of forms

$\alpha=g(x,t)p^{*}\omega+f(x,t)dt\wedge p^{*}\omega,$ (198)

for some $\omega$ in $\Omega_{c}(M)$ and for some $C^{\infty}$-class functions
$g,f:M\times[0,1]\longrightarrow\mathbb{C}$. Then we can define the linear
operator $\int_{0,\mathcal{L}}^{1}$ as follow: if $\alpha$ is a $0$-form with
respect to $[0,1]$, then

$\int_{0,\mathcal{L}}^{1}\alpha:=\int_{0,\mathcal{L}}^{1}g(x,t)p^{*}\omega=0$
(199)

and, if $\alpha$ is a $1$-form with respect to $[0,1]$,

$\int_{0,\mathcal{L}}^{1}\alpha:=(\int_{0}^{1}f(x,t)dt)\omega.$ (200)

This operator is very similar to the operator $p_{\star}$ (the integration
along the fiber of $p:M\times[0,1]\longrightarrow M$), but they differ by
sign. Consider $\alpha$ in $\Omega^{k}(M\times[0,1])$: we have
$\int_{0\mathcal{L}}^{1}\alpha=\pm p_{\star}\alpha$ where the choice between
$+$ or $-$ depends on the degree of $\alpha$. This implies that the norm of
$\int_{0\mathcal{L}}^{1}$ is equal to the norm of $p_{\star}$ as operators
between the $\mathcal{L}^{q}$-spaces for each $q$ in $[1\mathchar
24635\relax\;+\infty)$. Then, by Proposition 3.4, we can see that $p_{\star}$
and $\int_{0,\mathcal{L}}^{1}$ are $\mathcal{L}^{*}$-bounded.
We know from Lemma 11.4 of the book of Lee lee  that
$\int_{0\mathcal{L}}^{1}:\Omega^{*}(M\times[0,1])\longrightarrow\Omega^{*-1}(M)$
and that for all differential forms the equality (197) is satisfied.

######  Proposition 4.14.

Let $id:(N,h)\longrightarrow(N,h)$ be the identity map on a manifold of
bounded geometry and consider $p_{id}:T^{\delta}N\longrightarrow N$ the
submersion related to the identity defined in Lemma 3.5. Then, if
$pr_{N}:T^{\delta}N\longrightarrow N$ is the projection of the tangent bundle,
then there is an $\mathcal{L}^{*}$-bounded operator
$K_{1}:\mathcal{L}^{*}(N)\longrightarrow\mathcal{L}^{*}(T^{\delta}N)$ such
that for every smooth form $\alpha$

$p_{id}^{*}\alpha-pr_{N}^{*}\alpha=d\circ K_{1}\alpha+K_{1}\circ d\alpha.$
(201)

Moreover, if $\alpha$ is in $\Omega^{*}_{c}(N)$, then
$K_{1}\alpha\in\Omega^{*}(T^{\delta}N)$ has compact support.

Proof: Observe that $pr_{N}$ is a Lipschitz submersion with bounded Fiber
Volume and so $pr_{N}^{*}$ is a bounded operator.
Moreover $p_{id}$ and the projection
$pr_{N}:(T^{\delta}N,g_{S})\longrightarrow(N,h)$ are homotopic. In particular,
the homotopy $H(w_{p},s)=p_{id}(s\cdot w_{p})$ is a Lipschitz map. Moreover,
by Lemma 4.11 we know that $H$ is R.-N.-Lipschitz and so $H^{*}$ is an
$\mathcal{L}^{*}$-bounded operator.
This means that for all $\alpha$ in $\Omega^{*}_{c}(N)$, because of Lemma
11.4. of lee

$p_{id}^{*}-pr_{N}^{*}(\alpha)=i_{1}^{*}H^{*}\alpha-
i_{0}^{*}H^{*}\alpha=\int_{0\mathcal{L}}^{1}H^{*}d\alpha+d\int_{0\mathcal{L}}^{1}H^{*}\alpha.$
(202)

Then $K_{1}:=\int_{0\mathcal{L}}^{1}\circ H^{*}$ satisfies (4.14) and it is
$\mathcal{L}^{*}$-bounded.

######  Proposition 4.15.

Consider $g:(M,m)\longrightarrow(N,h)$ and $f:(N,h)\longrightarrow(S,r)$ two
smooth uniformly proper Lipschitz maps between manifolds of bounded geometry.
Let $\delta\leq inj_{N}$ and $\sigma\leq inj_{S}$. Denote by
$\overline{g}:g^{*}(T\delta N)\longrightarrow S$ the map $g\circ pr_{M}$,
where $pr_{M}:g^{*}(T\delta N)\longrightarrow M$.
Then there is an $\mathcal{L}^{*}$-bounded operator
$K_{2}:\mathcal{L}^{*}(N)\longrightarrow\mathcal{L}^{*}((f\circ\overline{g})^{*}T^{\delta}S)$
such that for every smooth form $\alpha$

$p_{f\circ p_{g}}^{*}\alpha-p_{f\circ\overline{g}}^{*}\alpha=d\circ
K_{2}\alpha+K_{2}\circ d\alpha.$ (203)

Moreover, if $\alpha$ is in $\Omega^{*}_{c}(S)$, then
$K_{2}\alpha\in\Omega^{*}((f\circ\overline{g})^{*}T^{\delta}S)$ has compact
support.

Proof: Consider $h:g^{*}T^{\delta}N\times[0,1]\longrightarrow S$ the map
$h(w_{g(p)},s):=f\circ p_{g}(s\cdot w_{g(p)})$. This is a Lipschitz homotopy
between $f\circ p_{g}$ and $f\circ\overline{g}$. Let us define the submersion
related to $h$ defined in Lemma 3.5

$p_{h}:(f\circ\overline{g})^{*}(T^{\sigma}S)\times[0,1]\longrightarrow S.$
(204)

Actually the domain of $p_{h}$ should be $h^{*}T^{\delta}S$ and the domain of
$f\circ p_{g}$ should be $(f\circ p_{g})^{*}T^{\delta}S$, but we are
considering these maps defined up to R.N.-lipschitz equivalence of vector
bundle (see Remark 25).
Observe that $p_{h}$ is the homotopy between $p_{f}\circ P_{g}=p_{f\circ
p_{g}}$ and $p_{f\circ\overline{g}}$ defined in Lemma 4.12 and recall that
$p_{h}$ is a R.-N.-Lipschitz map. Let us define the operator
$K_{2}:=\int_{0\mathcal{L}}^{1}\circ p_{h}^{*}$. It is an
$\mathcal{L}^{*}$-bounded operator because it is a composition of
$\mathcal{L}^{*}$-bounded operators. Then for every smooth form $\alpha$ we
have

$\begin{split}(p_{f\circ p_{g}})^{*}\alpha-
p_{f\circ\overline{g}}^{*}\alpha&=(i_{0}^{*}-i_{1}^{*})p_{h}^{*}\alpha\\\
&=(d\circ\int_{0\mathcal{L}}^{1})p_{h}^{*}\alpha+(\int_{0\mathcal{L}}^{1}\circ
d)p_{h}^{*}\alpha=d\circ K_{2}\alpha+K_{2}\circ d\alpha.\end{split}$ (205)

Finally, since $p_{h}$ is a proper map (it is a composition of proper maps),
if $\alpha\in\Omega^{*}_{c}(S)$ the support of
$K_{2}\alpha\in\Omega^{*}((f\circ\overline{g})^{*}T^{\delta}S)$ is compact.

######  Proposition 4.16.

Let $f_{0}$ and $f_{1}:(M,m)\longrightarrow(N,l)$ be two smooth, uniformly
proper and Lipschitz maps between manifolds of bounded geometry. Let us
suppose that $f_{1}\sim_{\Gamma}f_{0}$ with a smooth Lipschitz homotopy. Then
there is an $\mathcal{L}^{*}$-bounded operator
$K_{3}:\mathcal{L}^{*}(N)\longrightarrow\mathcal{L}^{*}(p_{f_{1}}^{*}T^{\delta}N)$
such that for all smooth form $\alpha$

$p_{f_{1}}^{*}\alpha-p_{f_{0}}^{*}\alpha=d\circ K_{3}\alpha+K_{3}\circ
d\alpha.$ (206)

Moreover if $\alpha\in\Omega^{*}_{c}(N)$ then the support of
$K_{3}\alpha\in\Omega^{*}(p_{f_{1}}^{*}T^{\delta}N)$ is compact.

Proof: Observe that $h$, the homotopy such that $h(p,0)=f_{1}(p)$ and
$h(p,1)=f_{0}(p)$, is a uniformly proper Lipschitz map. Let us define the
metric $g_{S}$ on $f_{1}^{*}T^{\delta}N$ as the Sasaki metric defined by using
the Riemannian metric $m$, the bundle metric $f_{1}^{*}l$ and the connection
$f^{*}\nabla^{LC,N}$.
By applying Corollary 4.2, we obtain that
$p_{h}:(f_{1}^{*}T^{\delta}N\times[0,1],g_{S}+dt^{2})\longrightarrow(N,h)$ is
a R.-N.-Lipschitz map and so $p_{h}^{*}$ is an $\mathcal{L}^{*}$-bounded
operator.
Moreover $p_{h}$ is a Lipschitz-homotopy between $p_{f_{1}}$ and $p_{f_{0}}$.
This fact follows directly by the definition of submersion related to a
Lipschitz map in Lemma 3.5 and by Remark 25.
So we can conclude as well as we did in the Proposition 4.14 and Proposition
4.15 by considering $K_{3}:=\int_{0\mathcal{L}}^{1}\circ p_{h}^{*}$ and by
using that $p_{h}$ is a proper map.

######  Proposition 4.17.

Consider $(M,g)$,$(N,h)$ and $(S,l)$ three manifolds of bounded geometry and
consider $f,\tilde{f}:(M,g)\longrightarrow(N,h)$,
$F:(M,g)\longrightarrow(N,h)$ and $g:(S,l)\longrightarrow(M,g)$ three uniform
maps, possibly non-smooth. Then, in $L^{q,p}$-cohomology,

  1. 1.

$T_{id_{M}}=Id_{H^{*}_{q,p}(M)}$,

  2. 2.

let us consider a Thom form $\omega$ on $\Omega^{*}(TN)$. Assume that the
pointwise norm of $\omega$ is uniformly bounded and that the support of
$\omega$ is strictly contained in $T^{inj_{N}}N$131313For example as the Thom
form defined in Subsection 4.2.. Let us suppose that $f\sim_{\Gamma}\tilde{f}$
with a smooth Lipschitz homotopy. Then, in $L^{q,p}$-cohomology,

$T_{f}=pr_{M\star}\circ e_{\tilde{F}^{*}\omega}\circ p_{f}^{*},$ (207)

  3. 3.

if $f$ is not differentiable and $f^{\prime}$ is a smooth Lipschitz maps which
is Lipschitz-homotopic to $f$, then $T_{f}$ does not depend on the choice of
$f^{\prime}$,

  4. 4.

if $f\sim_{\Gamma}F$ then $T_{f}=T_{F}$,

  5. 5.

$T_{f\circ g}=T_{g}\circ T_{f}$,

  6. 6.

if $f$ is an $L^{*}$-map141414This means that $f$ is a smooth R.-N.-Lipschitz
map which is also uniformly proper then $f^{*}=T_{f}$,

Moreover the identities above also holds in reduced $L^{q,p}$-cohomology and
in $L^{q,p}$-quotient cohomology.

Proof: Point 1. Let us consider the standard projection
$pr_{M}:TM\longrightarrow M$ and let $\delta\leq inj_{M}$. Because of Lemma
4.14, for all smooth forms $\alpha$ in $\Omega^{*}_{c}(M)$,

$pr_{M}^{*}-p_{id}^{*}(\alpha)=d\circ K_{1}+K_{1}\circ d(\alpha).$ (208)

Then the identity map in $\mathcal{L}^{p}(M)$ can be written as

$1(\alpha):=pr_{M\star}\circ e_{\omega}\circ pr^{*}_{M}(\alpha)$ (209)

where $pr_{M\star}$ is the operator of integration along the fibers of
$pr_{M}$ and $\omega$ is a Thom form with uniformly bounded pointwise norm and
support contained in $T^{\delta}M$ (see subsection 4.2).
Then for every $\alpha\in\Omega_{c}^{*}(M)$

$\begin{split}1-T_{id_{M}}(\alpha)&=pr_{M\star}\circ
e_{\omega}\circ(pr^{*}_{M}-p_{id}^{*})\alpha\\\ &=pr_{M\star}\circ
e_{\omega}\circ(d\circ K_{1}+K_{1}\circ d)\alpha.\end{split}$ (210)

Observe that since $\omega$ is closed,
$d(\alpha\wedge\omega)=(d\alpha)\wedge\omega$. Moreover $\alpha\wedge\omega$
is in $\Omega^{*}_{vc}(TM)$. This means that the exterior derivative can be
switched with $pr_{M\star}$ and so

$\begin{split}1-T_{id_{M}}(\alpha)&=d\circ pr_{M\star}\circ e_{\omega}\circ
K_{1}+pr_{M\star}\circ e_{\omega}\circ K_{1}\circ d(\alpha)\\\ &=d\circ
Y_{1}+Y_{1}\circ d(\alpha),\end{split}$ (211)

where $Y_{1}:=pr_{M\star}\circ e_{\omega}\circ K_{1}$. Observe that $Y_{1}$ is
an $\mathcal{L}^{*}$-bounded operator because it is a composition of
$\mathcal{L}^{*}$-bounded operators. By applying Proposition 2.1, we obtain
that $Y_{1}(dom(d))\subseteq dom(d)$ and so (211) holds for every $\alpha$ in
$dom(d_{min})$. Then in $L^{q,p}$-cohomology we obtain

$Id_{H^{*}_{q,p}(M)}=T_{id_{M}}.$ (212)

Point 2. Consider $\omega$ a Thom form of $TN$ which has support contained in
$T^{\delta}N$ and uniformly bounded pointwise norm (for example the Thom form
of Subsection 4.2) and let $h:M\times[0,1]\longrightarrow N$ be a smooth
Lipschitz homotopy between $f$ and $\tilde{f}$. Consider $g_{S}$ the Sasaki
metric on $T^{\delta}N$ induced by the metric on $N$ and by the Levi-Civita
connection on $TN$. Moreover let us denote by $g_{S,f}$ the Sasaki metric on
$f^{*}TN$ induced by the metrics on $M$ and on $N$ and the pullback of the
Levi-Civita connection on $TN$. Then the fibered map induced by $h$

$\begin{split}H:(h^{*}T^{\delta}N,g_{Sh})&\longrightarrow(T^{\delta}N,g_{S})\\\
(p,t,w_{h(p,t)})&\longrightarrow w_{h(p,t)}\end{split}$ (213)

is Lipschitz. By Remark 25 we can see $H$ as a Lipschitz map defined on
$(f^{*}(T^{\delta}N)\times[0,1],g_{S,f}+dt^{2})$. Observe that
$F^{*}\omega=i_{0}^{*}H^{*}\omega$ and
$\tilde{F}^{*}\omega=i_{1}^{*}H^{*}\omega$ where $i_{j}:M\longrightarrow
M\times[0,1]$ is defined by setting $i_{j}(p)=(p,j)$ for each $j$ in $[0,1]$.
Observe that

$F^{*}\omega-\tilde{F}^{*}\omega=d\int_{0\mathcal{L}}^{1}H^{*}\omega-\int_{0\mathcal{L}}^{1}H^{*}d\omega=d\int_{0\mathcal{L}}^{1}H^{*}\omega.$
(214)

Then, if we consider $\alpha$ an $\mathcal{L}^{p}$ smooth form on
$f^{*}T^{\delta}N$, we obtain

$\begin{split}e_{F^{*}\omega}-e_{\tilde{F}^{*}\omega}(\alpha)&=\alpha\wedge
d\int_{0\mathcal{L}}^{1}H^{*}\omega\\\
&=(-1)^{deg(\alpha)}d(\alpha\wedge\int_{0\mathcal{L}}^{1}H^{*}\omega)-(-1)^{deg(\alpha)}d\alpha\wedge\int_{0\mathcal{L}}^{1}H^{*}\omega\\\
&=(-1)^{deg(\alpha)}d(\alpha\wedge\int_{0\mathcal{L}}^{1}H^{*}\omega)-(-1)^{deg(\alpha)}\int_{0\mathcal{L}}^{1}d\alpha\wedge
H^{*}\omega\\\ &=d\circ K-K\circ d(\alpha)\end{split}$ (215)

where $K(\alpha)=(-1)^{deg(\alpha)}\circ\int_{0\mathcal{L}}^{1}\circ
e_{H^{*}\omega}(\alpha).$ Observe that the pointwise norm of $\omega$ is
uniformly bounded in $TN$: it follows from subsection 4.2. Then, since $H$ is
a Lipschitz map, also the norm of $H^{*}\omega$ is uniformly bounded in
$f^{*}T^{\delta}N\times[0,1]$. This means that $e_{H^{*}\omega}$ is an
$\mathcal{L}^{*}$-bounded operator.
This implies that for each smooth form $\alpha$ in $\mathcal{L}^{p}(N)$ for
all $p\in[1,+\infty)$,

$\begin{split}&T_{f}-pr_{M\star}\circ e_{\tilde{F}^{*}\omega}\circ
p_{f}^{*}(\alpha)=pr_{M\star}\circ(e_{F^{*}\omega}-e_{\tilde{F}^{*}\omega})\circ
p_{f}^{*}\alpha\\\ &=pr_{M\star}\circ(d\circ K-K\circ d)\circ
p_{f}^{*}\alpha=d\circ J-Jd(\alpha),\end{split}$ (216)

where $J:=pr_{M\star}\circ K\circ p_{f}^{*}$. Observe that $J$ is
$\mathcal{L}^{*}$-bounded because it is a composition of
$\mathcal{L}^{*}$-bounded operators. Then, since
$J(\Omega_{c}^{*}(N))\subseteq\Omega_{c}^{*}(M)$, we conclude the proof of
Point 2 by applying Proposition 2.1.
Point 3. Consider two smooth Lipschitz approximations $f_{1}$ and $f_{2}$ of
$f$. This in particular means that $f_{1}\sim_{\Gamma}f_{2}$. Let us consider,
now, a smooth Lipschitz homotopy $H$ between $f_{1}$ and $f_{2}$ as in Lemma
1.5. We know, thanks to Proposition 4.16, that there is an
$\mathcal{L}^{*}$-bounded operator $K_{3}$, such that for every $\alpha$ in
$\Omega_{c}^{*}(N)$

$p_{f_{1}}^{*}-p_{f_{2}}^{*}(\alpha)=d\circ K_{3}+K_{3}\circ d(\alpha).$ (217)

Moreover, by the previous point, if we consider the Thom form
$F_{1}^{*}\omega$ of $f_{1}^{*}TN$ defined as in subsection 4.2, we obtain
that

$T_{f_{2}}=pr_{M\star}\circ e_{F_{1}^{*}\omega}\circ p_{f_{2}}^{*}.$ (218)

Moreover, by the previous point, we can also suppose that

$T_{f_{1}}=pr_{M\star}\circ e_{F_{1}^{*}\omega}\circ p_{f_{1}}^{*}.$ (219)

This means that if we consider the $\mathcal{L}^{*}$-bounded operator151515It
is $\mathcal{L}^{*}$-bounded because it is a composition of
$\mathcal{L}^{*}$-bounded operators $Y_{3}:=pr_{M\star}\circ
e_{F_{1}^{*}\omega}\circ K_{3}$, then for all $\alpha$ in $\Omega_{c}^{*}(N)$
we obtain

$\begin{split}&T_{f_{1}}-T_{f_{2}}(\alpha)=pr_{M\star}\circ
e_{F_{1}^{*}\omega}\circ(p_{f_{1}}^{*}-p_{f_{2}}^{*})\\\ &=pr_{M\star}\circ
e_{F_{1}^{*}\omega}\circ(d\circ K_{3}+K_{3}\circ d)(\alpha)=d\circ