A pullback functor for (un)-reduced L^2-cohomology
aa r X i v : . [ m a t h . DG ] F e b A pullback functor for (un)-reduced L -cohomology Stefano SpessatoFebruary 18, 2021
Abstract
In this paper we study the reduced and unreduced L -cohomology groupsof manifolds of bounded geometry and their behavior under uniformly properlipschitz maps. A uniformly proper map is a map such that the diameter ofthe preimage of a subset only depends on the diameter of the subset. Ingeneral, the pullback map along a uniformly proper lipschitz map doesn’tinduce a morphism in, reduced or not, L -cohomology. Then our goal isto introduce two contravariant functors between the category of manifoldsof bounded geometry and uniformly proper lipschitz maps and the category Vec of complex vector spaces and linear maps. As consequence we obtainthat the, reduced or not, L -cohomology is a lipschitz-homotopy invariant.Moreover these functors coincide with the pullback, when the pullback doesinduce a map between the (un)-reduced L -cohomologies. Introduction
This is the first of two papers about lipschitz-homotopy invariants. In the first onewe study L -cohomology, in the second one we concentrate on the Roe Index ofthe signature operator. As reported in the abstract, our goal in this paper is to definea pullback functor , since pullback doesn’t induce an operator between L -spacesnor a morphism in L -cohomology or in reduced L -cohomology. To do this endwe define, for each uniformly proper lipschitz map f : M −→ N between man-ifolds of bounded geometry, an L -bounded operator T f : L ( N ) −→ L ( M ) which induce morphisms in (un)-reduced L -cohomology.If there is an action of a group of isometries Γ on N and on M such that the quo-tients are manifolds of bounded geometry and if f is Γ -equivariant, then also T f is Γ -equivariant respect to the action of Γ induced on the L -spaces. This property of T f will be useful in the second paper.The L -cohomology of a complete Riemannian manifold ( M, g ) is defined as H l ( M, g ) := ker ( d l ) im ( d l − ) (1)1here d l is the closure in L ( M ) of the exterior derivative defined on compactlysupported smooth l -forms. Moreover one can also define the reduced L -cohomologyas H l ( M, g ) := ker ( d l ) im ( d l − ) . (2)If M is a compact manifold we have that both, reduced or not, L -cohomologiescoincide with the de Rham Cohomology and so, in particular, they don’t depend onthe choice of the metric g in M . It’s a well-known fact, indeed, that if two metrics g and h on M are quasi-isometric, i.e. there are two constants C and C such that C h ≤ g ≤ C h, (3)then the (un)-reduced L -cohomologies coincides. It’s easy to check that if M isa compact manifold, then every couple of Riemannian metrics g and h are quasi-isometric.Differently from the case of de Rahm cohomology, the pullback along a differen-tial map f between Riemannian manifolds, in general, doesn’t induce a morphismbetween the L -cohomology groups.Then our goal is to define an operator T f between the L -spaces related to a uni-formly proper, lipschitz map between manifolds of bounded geometry such thatit induces a morphism in L -cohomology. In particular, we will show that thefunctorial properties hold and that if f ∗ induces a map in L -cohomology, then f ∗ [ α ] = [ T f α ] (4)for each close square-integrable differential form.Let us suppose, moreover, that there is a group Γ of isometries that acts on themanifolds such that the quotients are manifolds of bounded geometry. Then, if f is Γ -equivariant, also T f will be Γ -equivariant and the same will hold for the mor-phism in L -cohomology induced by T f .The idea of T f comes from the operator T f defined by Hilsum and Skandalis in[7] to build a perturbation for the signature operator which makes it invertible. Ouroperator, indeed, it’s very similar to their one and basically it is an adaptation tothe bounded geometry case.As a consequence of this fact, it follows that the L -cohomology is a lipschitz-homotopy invariant for manifolds of bounded geometry, i.e. if a map f is a homo-topy equivalence such that• f is lipschitz,• its inverse g is lipschitz and• the homotopies between g ◦ f and f ◦ g with their respective identities arelipschitz, 2hen T f induces an isomorphism between the L -cohomologies. This result isalready claimed by J.Lott in [8], but there is currently no a detailed proof.Let us describe briefly the content of this article: we proceed section by section. Maps between manifolds of bounded geometry
The objects of our study are manifolds of bounded geometry, which are manifoldswith strictly positive injectivity radius and some uniform bounds on the covariantderivatives of the curvature. Some examples of manifolds of bounded geometrycan be compact manifolds, coverings of manifolds of bounded geometry (so, forexample, the euclidean space and the hyperbolic space) and products of manifoldsof bounded geometry. Moreover Greene, in [5], proved that every differential man-ifold admits a metric of bounded geometry. The main results that we will use aboutmanifolds of bounded geometry can be found in [2] and in [10].On the other hand, the maps that we consider are the uniformly proper lipschitz maps between metric space, which are lipschitz maps such that the diameter ofthe preimage of a subset A is controlled by the diameter of A itself. We show, inparticular, that every lipschitz-homotopy equivalence is an uniformly proper map.We conclude the section by introducing a particular family of actions of a group Γ on a metric space. We call these uniformly proper, discontinuous and free actions(or u.p.d.f. actions). We will see that if the metric space is a manifold of boundedgeometry, then we have an u.p.d.f. action if and only if the quotient is a manifoldof bounded geometry. L -cohomology and pull-back In this section we introduce the notions of L -cohomology and reduced L -cohomology.The pullback along a map f , in general, doesn’t induce a map in (un)-reduced L -cohomology, even if it is a lipschitz-homotopy equivalence. An example is φ : R −→ R defined as φ ( x, y ) = ( x, if y ∈ [ − , and φ ( x, y ) =( x, y − sign ( y )1) otherwise.Then, given a map f : ( M, g ) −→ ( N, h ) , we define the Fiber Volume of f as theRadon-Nicodym derivative V ol f := f ⋆ µ M µ N , (5)where µ M and µ N are the measures on M and N induced by their metrics. We willcall R.-N.-lipschitz map a lipschitz map f which has bounded Fiber Volume. Thenwe show that the pullback along a R.-N.-lipschitz map induces an L -boundedoperator. In particular, we also show that R.-N.-lipschitz map are exactly the v.b.-maps defined by Thomas Schick in his Ph.D. Thesis.We conclude the section by showing that the Fiber Volume of a submersion can beexpressed as the integration along the fibers of some particular differential forms.In particular we choose the name Fiber Volume because in the case of a Riemannian3ubmersion one can easily check that
V ol f in a point q in N is exactly the volumeof its fiber F q respect to the metric induced by g . Submersion related to lipschitz maps
Consider a uniformly proper lipschitz map f : ( M, g ) −→ ( N, h ) . The main goalof this section is to define a lipschitz submersion p f : ( M × B k , g × g eucl ) −→ ( N, h ) such that p f ( x,
0) = f ( x ) and if f has some uniform bounds on the deriva-tives when it is written in normal coordinates, then also p f has some bounds onthe derivatives. In particular these bounds are important in Chapter 2. Moreoverwe also require that if there is a u.p.d.f. action of a group Γ on M and N and if f is Γ -equivariant, then p f is Γ -equivariant respect to the action of Γ in M × B k induced by the action of Γ on M .To do this we need to introduce the notion of Sasaki metric on a vector bundle, de-fine an isometric embedding of f ∗ ( T N ) in M × R k and compose two submersion:the first one is a totally geodesic Riemannian submersion M × B k −→ f ∗ ( T N ) ,the second one, that we denote by ˜ p f , is just the geodesic flow after f , i.e. ˜ p f : f ∗ ( T N ) −→ N ( p, w f ( p ) ) −→ exp f ( p ) ( w f ( p ) ) . (6) The pull-back functor
We start the last section proving that the submersion p f has bounded Fiber Volumeand so it is a R.-N.-lipschitz map. Then we define the operator T f : L ( N ) −→L ( M ) as T f ( α ) := Z B k p ∗ f α ∧ ω, (7)where ω is a compactly supported k -differential form on B k such that its integralon B k equals one. It follows the proof of the L -boundedness of T f , which easilyfollows since p f is a R.-N.-lipschitz map.In the subsection 4.3 we prove two important lemmas: the first one, if f : ( M, m ) −→ ( N, h ) and g : ( S, l ) −→ ( M, m ) are uniformly proper lipschitz map betweenmanifolds of bounded geometry, gives a formula which relates p f ◦ g with the sub-mersion p f and p g . The second one gives a formula which relates f and p f when f is a R.-N.-lipschitz map.We conclude the section by showing that, for every z in N , the association F de-fined as ( F ( M, g ) = H z ( M ) F (( M, g ) f −→ ( N, h )) = H z ( N ) T f −→ H z ( M ) (8)is a controvariant functor between the category C of the manifolds of bounded ge-ometry with uniformly proper lipschitz maps and the category Vec of the vectorspaces and linear maps. In particular, we will show that if two maps f and f are4niformly proper lipschitz maps such that they are homotopy with a lipschitz ho-motopy H , then T f = T f in L -cohomology. Moreover if f is a R.-N.-lipschitzmap, then f ∗ = T f in L -cohomology. The same holds considering the reduced L -cohomology instead of the L -cohomology. Acknowledgements:
I am grateful to Paolo Piazza, my advisor, and to Vito FeliceZenobi for the several discussions that we had and for the competences they sharedwith me. I also would like to thank Francesco Bei and Thomas Schick for theiradvice.
Contents Γ -Lipschitz-homotopy equivalence . . . . . . . . . . . . . . . . . 61.2 Uniformly proper maps . . . . . . . . . . . . . . . . . . . . . . . 61.3 Manifolds of bounded geometry . . . . . . . . . . . . . . . . . . 71.4 Uniformly proper and discontinuous actions . . . . . . . . . . . . 9 L -cohomology and pull-back 10 L -forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 L -cohomology and reduced L -cohomology . . . . . . . . . . . 102.3 Fiber Volume and Radon-Nicodym-Lipschitz maps . . . . . . . . 122.4 Quotients of differential forms . . . . . . . . . . . . . . . . . . . 152.5 Fiber Volume of a submersion . . . . . . . . . . . . . . . . . . . 16 ˜ p f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 A submersion related to a lipschitz map . . . . . . . . . . . . . . 26 p f . . . . . . . . . . . . . . . . . . . . . . . 284.2 The T f operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Lemmas about homotopy . . . . . . . . . . . . . . . . . . . . . . 334.4 Lipschitz-homotopy invariance of (un)-reduced L -cohomology . 38 Bibliography 47 Maps between manifolds of bounded geometry Γ -Lipschitz-homotopy equivalence Let us consider two metric spaces ( X, d X ) and ( Y, d Y ) and let Γ be a group actingon X and Y . Definition 1.1.
Two maps f and f : ( X, d X ) −→ ( Y, d Y ) are Γ -lipschitz-homotopy if they are Γ -invariant maps and they are homotopic with a lipschitzhomotopy H : ( X × [0 , , d X × d [0 , ) −→ ( Y, d Y ) which is Γ -invariant .We will denote it by f ∼ Γ f . (9) Definition 1.2.
A map f : ( X, d X ) −→ ( Y, d Y ) is a Γ -lipschitz-homotopy equiv-alence if f is Γ -equivariant, lipschitz and there is a Γ -equivariant map g such that• g is a homotopy inverse of f ,• g is lipschitz,• f ◦ g is Γ -lipschitz-homotopy to id N and g ◦ f is Γ -lipschitz-homotopy to id M . Let us introduce the notion of uniformly proper map.
Definition 1.3.
Consider two metric spaces ( X, d X ) and ( Y, d Y ) . Let f : ( X, d X ) −→ ( Y, d Y ) a map. We will say that f is uniformly (metrically) proper , if there is acontinuous function α : [0 , + ∞ ) −→ [0 , + ∞ ]) such that for every subset A ⊆ Y we have diam ( f − ( A )) ≤ α ( diam ( A )) (10) Remark . One can observe that if f is a uniform proper lipschitz map, then α canbe chosen such that for all s, t ≥ , then α ( t ) < α ( t + s ) , indeed if A ⊆ B then f − ( A ) ⊆ f − ( B ) . This means that the composition of two uniformly proper mapis still uniformly proper, indeed diam (( g ◦ f ) − ( A )) = diam ( f − ( g − ( A ))) ≤ α ( diam ( g − ( A ))) ≤ α ( β ( diam ( A ))) = ( α ◦ β )( diam ( A )) . (11) Remark . Consider two lipschitz maps
F, f : (
X, d X ) −→ ( Y, d Y ) such that f ∼ Γ F . Then, if f is a uniformly proper map, also the homotopy h is uniformly Here we are considering d X × d [0 , as the product distance between d X and the euclideandistance [0 , . F is uniformly proper.To prove this we have to observe that d ( f ( x ) , h ( x, t )) ≤ d ( h ( x, , h ( x, t )) ≤ C h · t. (12)Then, if A is a subset of Y and h t : X −→ Y is defined as h t ( p ) := h ( p, t ) , wehave that h − t ( A ) = { p ∈ X | h ( p, t ) ∈ A }⊆ { p ∈ X | f ( p ) ∈ B C h · t ( A ) } = f − ( B C h · t ( A )) (13)where B C h · t ( A ) are the points y of Y such that d ( y, A ) ≤ C h · t . Then we havethat h − ( A ) = G t ∈ [0 , h − t ( A ) ⊆ f − ( B C h ( A )) × [0 , (14)and so diam ( h − ( A )) ≤ diam ( f − ( B ( A ))) + diam ([0 , ≤ α ( diam ( A ) + 2 C h ) + 1 . (15) Lemma 1.1.
Let f : ( M, d M ) −→ ( N, d N ) be a Γ -lipschitz-homotopy equiva-lence between metric spaces. Then f is uniformly proper.Proof. Since H is a lispchitz map we have that if y is a point in N , then d M ( x, g ◦ f ( x )) = d M ( H ( x, , H ( x, ≤ C H d M × [0 , (( x, , ( x, C H . (16)Now, let x and x be two points in f − ( A ) , then d M ( x , x ) ≤ d M ( x , g ◦ f ( x )) + d M ( g ◦ f ( x ) , g ◦ f ( x )) + d M ( x , g ◦ f ( x )) ≤ C H + C g d N ( f ( x ) , f ( x )) ≤ C H + C g diam ( A ) . (17) In this section we will introduce the notion of manifolds of bounded geometry. Allthe definitions and propositions below can be found in [2].Let ( M, g ) be a Riemannian manifold. 7 efinition 1.4. The Riemannian manifold ( M, g ) has k -bounded geometry if:• the sectional curvature K of ( M, g ) and its first k -covariant derivatives arebounded, i.e. ∀ i = 0 , ..., k there is a constant V i such that ∀ x ∈ M |∇ i K ( x ) | ≤ V i . (18)• there is a number C > such that for all p in M the injectivity radius i g ( p ) satisfies i g ( p ) ≥ C. (19)The maximal number wich satisfies this inequality will be denoted by r inj ( M ) .When we talk about a manifold M with bounded geometry, without specifying the k , we mean that M has k -bounded geometry for all k in N . Proposition 1.2.
Let ( M, g ) be a Riemannian manifold of k -bounded geometry.Then there exists a δ > such that the metric up to its k -th order derivatives andthe Christoffel symbols up to its ( k − -th order derivatives are bounded in normalcoordinates of radius δ around each x ∈ M , with bounds that are uniform in x . Proposition 1.3.
Let ( M, g ) be a Riemannian manifold of k ≥ -bounded geome-try. For every C > there exists a δ > such that the normal coordinate charts φ x are defined on B δ ( x ) for each x ∈ M and the Euclidean distance d E on thenormal coordinates is uniformly C -equivalent to the metric distance d induced by M , that is, ∀ x , x ∈ B δ ( x ) C − d ( x , x ) ≤ d E ( φ x ( x ) , φ x ( x )) ≤ Cd ( x , x ) . (20) Proposition 1.4.
Let ( M, g ) be a Riemannian manifold of k -bounded geometrywith k ≥ . There exists a δ with < δ < r inj ( M ) and a constant C > suchthat for all x , x ∈ M with d ( x , x ) < δ we have that the coordinate transitionmap φ , = exp − x ◦ exp − x : U −→ T x M with U = exp x ( B δ ( x ) ∩ B δ ( x )) ⊂ T x M. (21) is C k − -bounded with | φ , | k − ≤ C .Remark . Consider a Riemannian manifold ( M, g ) and let us denote by µ M themeasure induced by g on M . We have µ M ( B r ( p )) = β n r n (1 − k n + 2) r + O ( r )) , (22)where β n is the volume of an euclidean ball with radius . If M has k ≥ -boundedgeometry, then we have that for each r > there is a constant C r such that foreach p ∈ M µ M ( B r ( p )) ≤ C r . (23)Then this means that if A ⊂ M has diam ( A ) = l , then, fixed a p in A we havethat µ M ( A ) ≤ µ M ( B l ( p )) ≤ C l . (24) it means that all the derivatives of degree less or equal to k − are bounded. .4 Uniformly proper and discontinuous actions In this subsection we will introduce the notion of uniformly properly discontinu-ous and free (u.p.d.f.) action of a group Γ over a Riemannian manifold ( M, g ) , but,before of that, let us explain why these actions are important for us.Let us suppose that there is a uniformly proper lipschitz map f : ( M, g ) −→ ( N, h ) and there is a group Γ acting by isometries on M and N . Consider, moreover, f asa Γ -equivariant map. Then we will see that if the action of Γ is uniformly properlydiscontinuous and free, then the operator T f is Γ -equivariant respect to the actionsinduced by Γ on L ( M ) and L ( N ) .This property will be useful in our second paper about lipschitz-homotopy invari-ants.More in general, if the action of Γ is not uniformly properly discontinuous and free T f can be not Γ -equivariant, but it is still true that T f is Γ -equivariant as operatorbetween the (un)-reduced L -cohomology groups. Definition 1.5.
Consider Γ a group which acts by isometries on a Riemannianmanifold ( M, g ) . We say that the action of Γ is uniformly properly discontinuousand free (u.p.d.f.) if• the action of Γ is free and properly discontinuous,• there exists a number δ > such that d M ( p, γp ) ≤ δ = ⇒ p = γp. (25) Remark . Let us suppose that ( M, g ) is a manifold of bounded geometry. Thenthe following are equivalent:1. Γ induce a u.p.d.f. action,2. the quotient M Γ has bounded geometry.The implication ⇒ can be proved observing that if δ is the constant of theu.p.d.f. action of Γ , then for each p in M we have that B δ ( p ) is a trivializing openof M Γ and so inj M Γ ≥ min { inj M , δ } and the curvature of M Γ has the same boundsof the curvature of M .The implication ⇒ is proved as follow. Let us suppose that for each δ > there is a point p and a γ ∈ Γ such that d M ( p, γp ) ≤ δ . Consider, then δ < min { inj M , inj M Γ } . Then there is a vector v in T p M with norm less then δ such that exp p ( v ) = γp . So if we consider the Riemannian covering s : M −→ M Γ ,then we can apply the formula s ◦ exp p = exp s ( p ) ◦ ds (26)to v . We have that s ( p ) = s ◦ exp p ( v ) = exp s ( p ) ◦ ds ( v ) . (27)9hen, since ds is an isometry, we have that the norm of ds ( v ) is less or equal to theinjectivity radius of M Γ . This means that ds ( v ) has to be null, then in particular v is null and so γp = exp p ( v ) = exp p (0) = p. (28) L -cohomology and pull-back L -forms Let us consider a Riemannian manifold ( M, g ) and let us denote by Ω kc ( M ) thespace of complex differential forms with compact support. The Riemannian metric g induces for every k ∈ N a scalar product on Ω kc ( M ) as follows: consider α and β in Ω ∗ c ( M ) then < α, β > L Ω ∗ ( M ) := i | α | ( m −| α | ) i | β | ( m −| β | ) Z M α ∧ ⋆ ¯ β, (29)where ⋆ is the Hodge star operator given by g .This scalar product gives a norm on Ω kc ( M ) : k α k L Ω k ( M ) : = < α, α > M = i | α | ( m −| α | ) i | α | ( m −| α | ) Z M α ∧ ⋆ ¯ α < + ∞ . (30) Definition 2.1.
We will denote by L Ω k ( M ) the Hilbert space given by the closureof Ω ∗ c ( M ) respect the norm | · | L Ω k ( M ) . Moreover we can also define the Hilbertspace L ( M ) given by L ( M ) := M k ∈ N L Ω k ( M ) . (31)The norm of L ( M ) will be denoted by | · | L or | · | L ( M ) . Remark . Since Ω kc ( M ) is dense in L Ω k ( M ) , then Ω ∗ c ( M ) is dense in L ( M ) . L -cohomology and reduced L -cohomology If a Riemannian manifolds ( M, g ) is complete (as in the bounded geometry case)then we have only one closed extension of the exterior derivative on compactlysupported smooth forms [4]. We will denote it by d .It’s easy to check that d ( dom ( d )) ⊆ ker ( d ) ⊆ dom ( d ) and so we have thatthe composition d ◦ d is well defined and equals to zero. Moreover, similarlywith the smooth forms we have that if α ∈ dom ( d ) ∩ L (Ω k ( M )) then dα is in10 (Ω k +1 ( M )) .Let us consider now the sequence −→ L (Ω ( M )) d −→ L (Ω ( M )) d −→ L (Ω ( M )) d −→ ... (32)It is, following the definition given in [1], an Hilbert complex. Definition 2.2.
We will define i -th group of L -cohomology the group H iL ( M ) := ker ( d i ) im ( d i − ) . (33) Definition 2.3.
We will define i -th group of reduced L -cohomology the group H iL ( M ) := ker ( d i ) im ( d i − ) . (34) Remark . In general for every closed extension d we have different groups of(reduced or not) L -cohomology. Even if in our case all the closed extensionscoincides, it will be useful, in the next proposition, the definition of minimal exten-sion. The minimal extension d min of the exterior derivative has domain given bythe L -forms α such that there is a sequence of compactly supported differentialforms { α k } which satisfies α = lim k → + ∞ α k and such that the limit of the sequence { dα k } exists. Then we have that d min α := lim k → + ∞ dα k . (35) Proposition 2.1.
Let ( N, h ) and ( M, g ) be two Riemannian manifolds. Let B : L ( N ) −→ L ( M ) be a continuous operator such that B (Ω ∗ c ( N )) ⊆ Ω ∗ c ( M ) and Bd = dB over Ω ∗ c ( N ) . Then we have that B ( dom ( d min )) ⊆ dom ( d min ) (36) and Bd min = d min B on the minimal domain of d .Proof. Let α be an element in dom ( d min ) . This means that there is a sequence { α n } in Ω ∗ c ( N ) such that α = lim n → + ∞ α n dα = lim n → + ∞ dα n . (37)Using that B is continuous we have Bα = lim n → + ∞ Bα n (38)11here { Bα n } is a sequence in Ω ∗ c ( M ) . Moreover one can see that the limit of dBα n exists: lim n → + ∞ dBα n = lim n → + ∞ Bdα n = B lim n → + ∞ dα n = Bdα. (39)So d min is well defined in Bα and Bd min α = d min Bα. (40)
Remark . In general, to induce a map in L -cohomology we need an operator A : dom ( A ) ⊆ L ( N ) −→ L ( M ) such that• A ( dom ( d N )) ⊆ dom ( d M ) ,• A and d commute.Moreover to have a morphism induced by A in reduced L -cohomology, we alsoneed that A is L -bounded on im ( d ) , indeed, given an α in dom ( d N ) we have that A ( α + lim k → + ∞ dβ k ) = A ( α ) + A ( lim k → + ∞ dβ k )= A ( α ) + lim k → + ∞ A ( dβ k )= A ( α ) + lim k → + ∞ dA ( β k ) (41)and so, in reduced L -cohomology, [ A ( α + lim k → + ∞ dβ k )] = [ A ( α )] . (42) Corollary 2.2.
Given an operator B as in Proposition 2.1, then B induces a mapin (un)-reduced L -cohomology. Let ( M, ν ) and ( N, µ ) be two measured spaces and let f : ( M, ν ) −→ ( N, µ ) bea function such that the pushforward measure f ⋆ ( ν ) is absolutely continuous withrespect to ν . Definition 2.4.
Let ( N, µ ) be σ -finite, then the Fiber Volume is the Radon-Nicodymderivative
V ol f,ν,µ := ∂f ⋆ ν∂µ . (43)Consider ( M, d M , ν M ) and ( N, d N , µ N ) two measured and metric spaces. Definition 2.5.
A map f : ( M, ν ) −→ ( N, µ ) is Radon-Nikodym-lipschitz or R.-N.-lipschitz if 12 f is lipschitz• f has a well-defined, bounded Fiber Volume. Remark . Consider f : ( M, d M , ν M ) −→ ( N, d N , µ N ) an R.-N.-lipschitz mapand let C be the supremum of ∂f ⋆ µ M ∂µ N . Then for all measurable set A ⊆ N , we havethat µ M ( f − ( A )) = Z A ∂f ⋆ µ M ∂µ N dµ N ≤ C Z A dµ N = Cµ N ( A ) . (44)A measurable, lipschitz map which satisfies the above inequality is called a v.b.-map : these maps are defined by Thomas Schick in his Ph.D. thesis [ ? ]. Then thismeans that all the R.-N.-maps are v.b.-maps. Moreover it is also true that a v.b.-map is R.-N.-lipschitz. To prove this we can start observing that the Fiber Volumeis well-defined since µ N << µ M . Consider C the costant such that µ M ( f − A ) ≤ C · µ N ( A ) : then | ∂f ⋆ µ M ∂µ N | < C. (45)Indeed if the above inequality is not satisfied, then there is A ⊆ N such that µ M ( f − A ) µ N ( A ) > C . But this implies µ M ( f − ( A )) ≥ C · Z A dµ N = C · µ N ( A ) (46)and so we have the contradiction.Let us prove some properties of R.-N.-lipschitz maps. Proposition 2.3.
Consider f : ( M, d M , ν M ) −→ ( N, d N , µ N ) and g : ( N, d N , µ N ) −→ ( W, d W , µ W ) two R.-N.-lipschitz maps. Then ( g ◦ f ) : ( M, d M , ν M ) −→ ( W, d W , µ W ) is a R.-N.-lipschitz map.Proof. Since the equivalence of the definitions, we can check that the compositionof two v.b.-maps is a v.b.-map. We can start observing that the composition oflipschitz map is lipschitz. Moreover we also have that µ M (( g ◦ f ) − A ) ≤ C f µ N ( g − ( A )) ≤ C f · C g µ N ( A ) (47)and so it means that g ◦ f is a v.b.-map. Proposition 2.4.
Let us consider two R.-N.-lipschitz maps f : ( M, d M , µ M ) −→ ( X, d X , µ X ) and g : ( N, d N , µ N ) −→ ( Y, d Y , µ Y ) . Then the map ( f, g ) : ( M × N, d M × d N , µ M × µ N ) −→ ( X × Y, d X × d Y , µ X × µ Y ) (48) is a R.-N.-lipschitz map. roof. Again we will show that ( f, g ) is a v.b.-map. We can observe that ( f, g ) islipschitz. Moreover we can also consider a subset A × B of X × Y . Then we havethat µ M × N (( f, g ) − ( A × B )) = µ M ( f − ( A )) · µ N ( g − ( B )) ≤ C f · C g µ X ( A ) · µ Y ( B ) ≤ C f · C g µ X × Y ( A × B ) . (49)Since the sets { A × B } are generators if the σ -algebra of X × Y we can conclude. Remark . We can also prove the Proposition 2.4 checking that the Fiber Volumeof ( f, g ) in a point ( p, q ) is given by V ol f ( p ) · V ol g ( q ) . Lemma 2.5.
Let f : ( M, g ) −→ ( N, h ) be a lipschitz map between Riemannianmanifolds. Then for all α p in T ∗ p ( N ) and for all q in f − ( p ) we have that || f ∗ α p || kq M ≤ C kf || α p || kp N , (50) where || · || kx X is the norm induced by the metric of X on Λ kx X . Proposition 2.6.
Let ( M, g ) and ( N, h ) be Riemannian manifolds. Let f : ( M, g ) −→ ( N, h ) be a R.-N.-lipschitz map. Then f induces an L -bounded pullback. In par-ticular the norm of f ∗ is less or equal to K f · √ C V ol , where C V ol is the maximumof the Fiber Volume.Proof.
Let ω be a smooth form with compact support in L ( N ) and let K f :=max { , C nf } , where n = dim ( N ) . Then || f ∗ ω || L ( M ) = Z M || f ∗ ω || dµ g ≤ Z M K f f ∗ ( || ω || ) dµ g = K f Z N || ω || d ( f ⋆ µ g )= K f Z N || ω || V ol f,g,h dµ h ≤ K f C V ol Z N || ω || dµ h = K f C V ol || ω || L ( N ) . (51)14 .4 Quotients of differential forms We want to show that if we have a submersion p : M −→ N between two ori-entable manifolds, then there exists an orientation for the bundle given by the sub-mersion, i.e. there is a smooth form η on M such that for each q in N , the pullbackof η on the fiber of q is a Volume form. To do this we need the notion of quotientof differential forms Definition 2.6.
Let us consider a Riemannian manifold M and let π be the projec-tion π : Λ ∗ ( M ) −→ M . Given two differential forms α ∈ Ω k ( M ) , β ∈ Ω n ( M ) we define a quotient between α and β as a section of Λ k − n ( M ) such that for all p in M α ( p ) = αβ ( p ) ∧ β ( p ) . (52) Remark . There are no condition about the continuity, or smoothness, of αβ .In general, given two differential forms, there isn’t a quotient between them:for example we can consider dx and dx ∧ dx in R . Moreover if there is aquotient between α and β it may not be unique: for example if we consider α = dx ∧ dx and β = dx + dx in R then ( dx − dx ) and dx are both quotients.However there are some useful formulas concerning quotients. Proposition 2.7.
Let us consider a differential manifold M and let α, β, γ, δ ∈ Ω ∗ ( M ) . Then, if αγ , αδ , βγ , and βδ are well-defined, then the following formulashold α + βγ = αγ + βγ ,2. if γ as a g -form, β a b -form and δ a d -form. Then α ∧ βγ ∧ δ = ( − g ( b − d ) αγ ∧ βδ .
3. If γ is closed, and αγ is a smooth form, then dαγ = d ( αγ ) .
4. if γ as a g -form, β a b -form then α ∧ ββ ∧ γ = ( − gb αγ .5. if αδ γ exists then αγ ∧ δ = αδ γ .
6. if αγ ∧ δ exists, then αδ γ = αγ ∧ δ . Proof.
It is a direct calculus.
Proposition 2.8.
Let ( M, g ) be a Riemannian manifold of dimension m and let α ∈ Ω k ( M ) be a differential form such that α p = 0 for each p . Then there is asmooth quotient V ol M α and it is given by ( − s ( m − s ) || α || ⋆ M α. (53) in these formulas the = have to be read as ”exists and one of the possible quotient is given by” hen we will not specify about the choice of a quotient, we will consider (53) asquotient.Proof. It is a direct calculus.As proved in [ ? ], in particular in Proposition 16.21.7, when we have a submer-sion f : X −→ Y , then if F q is the fiber of f in q and i q : F q −→ X is theimmersion of the fiber, then i ∗ q ( βf ∗ α ) doesn’t depend by the choice of the quotient.So, it means that a submersion can be considered an oriented fiber bundle withorientation given by V ol X f ∗ V ol Y . It means that for all p in N we have the orientation of F p such that Z F p i ∗ p V ol X V ol Y > . (54) Proposition 2.9.
Let p be a submersion p : X −→ Y . Consider α ∈ Ω ∗ CV ( X ) and β ∈ Ω ∗ ( X ) . Let us suppose that there is αp ∗ β . Then we have that R F αβ = Z F αp ∗ β . (55) Remark . Given a submersion p : X −→ Y we denote the operator of integrationalong the fibers as R F , where F is a generic fiber of p , or as p ⋆ . In this section we will study the Fiber Volumes of lipschitz submersions betweenorientable manifolds.
Proposition 2.10.
Let ( M, g ) and ( N, h ) two orientable, Riemannian manifolds.Let p : ( M, g ) −→ ( N, h ) be a proper lipschitz submersion. Then we have that V ol p,g,h ( q ) = Z F V ol M p ∗ V ol N ( q ) (56) if q is in p ( ˚ M ) , otherwise. again = have to be read as ”exists and one of the possible quotient is given by”: this propertyholds for all possible choice of the quotient. , unless α and β are top-degree forms: in that case the = actually means the equality as differential forms. roof. Let A a measurable set of N . Then we have that f ⋆ µ M ( A ) = Z p − ( A ) dµ M = Z p − ( A ) V ol M = Z p − ( A ) p ∗ ( V ol N ) ∧ V ol M V ol N = Z p − ( A ∩ p ( ˚ M )) p ∗ ( V ol N ) ∧ V ol M V ol N + Z A ∩ [ p ( ˚ M )] c dµ N = Z A ∩ p ( ˚ M ) V ol N ( Z F V ol M V ol N ) + Z A ∩ [ p ( ˚ M )] c dµ N = Z A ∩ p ( ˚ M ) ( Z F V ol M V ol N ) V ol N + Z A ∩ [ p ( ˚ M )] c dµ N = Z A ∩ p ( ˚ M ) ( Z F V ol M V ol N ) dµ N + Z A ∩ [ p ( ˚ M )] c dµ N (57) Remark . We can observe that the Fiber Volume doesn’t depend on the choiceof
V ol M p ∗ V ol N . It is coherent with the uniqueness of the Radon-Nicodym derivative. Remark . If the submersion f : X → Y is in particular a diffeomorphismbetween oriented manifold, then we have that the integration along the fibers of f is pullback f ∗ with a + if f preserves the orientation and a − otherwise. Indeedthe orientation of the fibers is given posing Z f − ( p ) V ol X f ∗ V ol X > (58)and we know that the integration over a -chain is just the evaluation on the orientedpoints. So f ⋆ ( α ) p ( v ,p , ..., v k,p ) = ± α f − ( p ) ( df − f − ( p ) ( v ,p ) , ..., df − f − ( p ) ( v k,p ))= ± ( f − ) ∗ α ( v ,p , ..., v k,p ) , (59)where we have a + if f preserves the orientation and − otherwise. Then, thismeans that q F = ± ( f − ) ∗ . (60)and so, in particular, the Fiber Volume of f is given by ± ( f − ) ∗ V ol X f ∗ ( V ol Y ) = | ( f − ) ∗ V ol X f ∗ ( V ol Y ) | . (61)17onsider ( M, g ) a Riemannian, orientable manifold. Let us denote by ⋆ M theHodge operator on M . We define the chiral operator of M the operator τ M : L ( M ) −→ L ( M ) for each smooth form as τ M := i dim ( M ) ⋆ M if dim ( M ) is even and τ M := i dim ( M )+1 ⋆ M . It’s easy to check that the chiral operator isunitary, bounded operator with norm equal to 1. Proposition 2.11.
Given a fiber bundle p : ( X, g ) −→ ( Y, h ) where X and Y areorientable, we have that if p ∗ induce a map between the L -space, then the same istrue for p ⋆ , the operator of integration along the fibers of p . Moreover, if we denoteby ( p ⋆ ) ∗ the adjoint of p ⋆ , we have that, if τ X and τ Y are the chiral operators of X and Y ( p ⋆ ) ∗ = ( i ) j τ X ◦ p ∗ ◦ τ Y (62) for some integer j .Proof. Let α be in Ω ∗ CV ( X ) and let β be in Ω ∗ ( Y ) , if n = dim ( Y ) we have that h p ⋆ α, β i Y = Z Y [ Z F α ] ∧ ⋆ Y β (63)Now, applying the Projection Formula, we have that h p ⋆ α, β i Y = Z X α ∧ p ∗ ( ⋆ Y β )= i | β | ( n −| β | ) ( − | β | ( n −| β | ) Z X α ∧ ( p ∗ τ Y β )= i | β | ( n −| β | ) ( − | β | ( n −| β | ) i | β | ( n +4 k −| β | ) Z X α ∧ ⋆ X ( τ M p ∗ τ Y β )= h α, i j τ X ◦ p ∗ ◦ τ Y ( β ) i X . (64) Remark . We have that the norm as operator between L -spaces of p ⋆ is the sameof the norm of p ∗ . Corollary 2.12.
Consider p : ( M, g ) −→ ( N, h ) a R.-N.-lipschitz submersion.Then the operator p ⋆ is a L -bounded operator. We conclude this section giving a formula which allows to compute the FiberVolume of the composition of two submersions.
Proposition 2.13.
Let f : ( M, g ) −→ ( N, h ) and g : ( N, h ) −→ ( W, l ) be twosubmersions between oriented Riemannian manifolds. Then we have that V ol g ◦ f,µ M ,µ W ( q ) = Z g − ( q ) ( Z f − g − ( q ) V ol M f ∗ ( V ol N ) ) V ol N g ∗ V ol N (65)18 roof. We can observe that, as quotients, we have that
V ol M ( g ◦ f ) ∗ V ol W = V ol M f ∗ V ol N ∧ f ∗ V ol N ( g ◦ f ) ∗ V ol W (66)and, in particular one can observe that we can choose as quotient f ∗ V ol N ( g ◦ f ) ∗ V ol W = f ∗ ( V ol N g ∗ V ol W ) . (67)Then we can conclude applying the Projection Formula, indeed V ol g ◦ f,µ M ,µ W ( q ) = Z ( g ◦ f ) − ( q ) V ol M ( g ◦ f ) ∗ V ol W = Z ( g ◦ f ) − ( q ) V ol M f ∗ V ol N ∧ f ∗ ( V ol N g ∗ V ol W )= Z g − ( q ) ( Z f − g − ( q ) V ol M f ∗ ( V ol N ) ) V ol N g ∗ V ol N . (68) Let us consider be a Riemannian manifold ( N, h ) of dimension n , π E : E −→ N a vector bundle of rank m endowed with a bundle metric H E ∈ Γ( E ∗ ⊗ E ) anda linear connection ∇ E which preserves H E . Fix { W i } a local frame of E : wehave that if { x i } is a system of local coordinate over U ⊆ N , then we can definethe system of coordinates { x i , µ j } on π − E ( U ) , where the µ j are the componentsrespect to { W j } .Then we can denote by K the map K : T E −→ E defined as K ( b i ∂∂x i + z j ∂∂µ j ) := ( z l + b i z j Γ lij ) s l , (69)where the Γ lij are the Christoffel symbols of ∇ E . Definition 3.1.
The
Sasaki metric on E is the Riemannian metric h E defined forall A, B in T E as h E ( A, B ) := h ( dπ E ( A ) , dπ E ( B )) + H E ( K ( A ) , K ( B )) . (70) Remark . Let us consider the system of coordinate { x i } on N and { x i , µ j } on E . We have that the components of h E are given by h Eij = h ij + H αγ Γ αβi Γ γηj µ β µ η h Eiσ = H σα Γ αβi µ β h Eστ = H σ,τ , (71)19here i, j = 1 , ..., n and σ, τ = n + 1 , ..., n + m . We can observe that if in a point x = ( x , ..., x n ) of M we have that all the Christoffel symbols are zero, then, inlocal coordinate the matrix of h E in a point ( x , v ) is given by (cid:20) h i,j H σ,τ (cid:21) (72) Example 3.1.
Let us consider a Riemannian manifold ( N, h ) . We can consider as E the tangent bundle T M and as h E the metric g itself. In this case the connection ∇ E is the Levi-Civita connection. Let us consider a differential map f : ( M, g ) −→ ( N, h ) . Definition 3.2.
The pullback bundle f ∗ E is a bundle over ( M, g ) given by f ∗ ( E ) := { ( p, v ) ∈ M × E | π E ( t ) = f ( p ) } . (73) Remark . It’s easy to check that given a smooth map f : ( M, g ) −→ ( N, h ) ,then there is a bundle map F induced by f between f ∗ ( E ) and E : this map is justdefined as F ( p, v ) = ( f ( p ) , v ) . (74)Using this map it’s possible to define an euclidean scalar product on f ∗ ( E ) : it issufficient to define F ∗ ( H )( A p , B p ) := H ( F ( A p ) , F ( B p )) (75)An important property about the pullback bundles is that, given a section σ of E , one can define the section f ∗ σ := f ∗ σ ( p ) = σ ◦ f ( p ) . (76)Then we can observe that the pullback bundle of the trivial bundle is again thetrivial bundle.A consequence of the existence of the pullback of a section of E is that it is possibleto pullback also a connection ∇ E . Definition 3.3.
The pullback connection f ∗ ∇ E on f ∗ ( E ) is uniquely definedimposing that ( f ∗ ∇ E ) f ∗ σ = f ∗ ( ∇ E σ ) . (77) Remark . The condition (77) is sufficient to uniquely define a connection on f ∗ ( E ) . An equivalent condition is to impose that the local Christoffel symbols ˜Γ αβ,i are given by ˜Γ αβ,i := ∂f l ∂x i f ∗ (Γ αβ,l ) . (78)20 emark . Let us consider a map f : ( M, g ) −→ ( N, h ) and consider, in par-ticular, the bundle π : f ∗ ( T N ) −→ M . We can consider, on each point p in M ,the scalar product on fiber of f ∗ ( T N ) p given by h f ( p ) , where h is the Riemannianmetric on N . Moreover we can also consider the pullback connection f ∗ ( ∇ LCN ) .Then we can consider the Sasaki metric on f ∗ ( T N ) . Fix then some normal coor-dinates x i around a point p and fix a frame E j of f ∗ ( T N ) . Then we can considerthe fibered coordinates { x i , y j } of f ∗ ( T N ) . Let us observe that in ( p, w f ( p ) ) wehave the orthogonal decomposition T ( p,w f ( p ) ) f ∗ ( T N ) =
Span { ∂∂x i | ( p,w f ( p ) ) } ⊕ ker ( dπ ( p,w f ( p ) ) )= Span { ∂∂x i | ( p,w f ( p ) ) } ⊕ Span { ∂∂y j | ( p,w f ( p )) } (79)and, moreover, we have that Span { ∂∂x i | ( p,w f ( p ) ) } is isometric to T p M , while Span { ∂∂y j | ( p,w f ( p )) } is isometric to T f ( p ) N . This fact follows by (71) and (78)and observing that the Christoffel symbols of ∇ LCN respect to normal coordinatesaround f ( p ) are null in f ( p ) .This fact, in particular, implies that the projection π : f ∗ T N −→ M is a Rieman-nian submersion. Consider M and N two manifolds of bounded geometry and Γ a group which actsu.p.d.f. on them. Suppose that there is a map f : M −→ N which is Γ -equivariant.Let us denote by Y := N Γ and by S N : N −→ Y the Riemannian covering.Let us fix an isometric embedding I : Y −→ R k . We know since the Nash embed-ding theorem that such an immersion exists.Let us denote by A : M −→ R k the composition A := I ◦ S N ◦ f, (80)by B := I ◦ S N . (81)Then, consider the following map I : f ∗ T N −→ M × R k ( p, w f ( p ) ) −→ ( p, exp A ( p ) [ dB f ( p ) w f ( p ) ] − A ( p )) . (82)This map is an embedding and, moreover, it is also an isometric embedding. Letus prove it.Since Γ acts u.p.d.f., we have that Y is a manifold of bounded geometry and so wecan choose a number δ ≤ min { inj N , inj Y } . Then, fixed an orthonormal frame { E i } on a δ -ball around q in N , we know that { dS N ( E i ) } is a local orthonormal21rame on B δ ( S N ( q )) .Then fix p in M and consider { x i } some normal coordinates in B σ ( p ) where σ issuch that f ( B σ ( p )) ⊆ B δ ( f ( p )) .We can choose as coordinates around ( p, f ( p ) ) in f ∗ T N the coordinates { x i , y j } ,where y j are the coordinate referred to E i .On R k let us consider, for each q in B δ ( p ) , the vectors W i ( q ) := exp A ( q ) [ dB f ( q ) E i ( q )] − A ( q ) . (83)Consider, moreover, for each q in B σ ( p ) ⊂ M some vectors Z l ( q ) such that { W i ( q ) , Z l ( q ) } forms an orthonormal basis of { q } × R k . For example we canconsider an orthonormal frame ˜ Z l of T ⊥ I ( Y ) and so we can impose Z l ( q ) := exp A ( q ) ( ˜ Z l ◦ A ( q )) − A ( q ) . (84)Then we have the coordinates given by { x i , ˜ y j , z l } . Then, respect to these coordi-nates, I ( x, y ) = ( x, y, , (85)indeed for each fixed q in B σ ( p ) , we have that I q : f ∗ T N q = T f ( q ) N −→ { q }× R k is a linear map, and so, since I q ( E i ( q )) = W i ( q ) , we have that the vector ofcoordinate ( y , ..., y n ) goes on the vector ( y , ..., y n , , ..., .To conclude that it is a Riemannian embedding we can observe that the tangent of M × R k in the point ( p, t ) is orthogonal decomposed in T ( p,t ) M × R k = Span { ∂∂x i | p } ⊕ Span { ∂∂ ˜ y j | ( p,t ) } ⊕ Span { ∂∂z l | ( p,t ) } (86)where Span { ∂∂x i | p } is isometric to T p M and Span { ∂∂ ˜ y j | ( p,t ) } is isometric to T f ( p ) N .Observing that ( d I ( p,w f ( p ) ) ( ∂∂x i | p,w f ( p ) ) = ∂∂x i | p d I ( p,w f ( p ) ) ( ∂∂y j | ( p,w f ( p ) ) ) = ∂∂ ˜ y j | ( p,t ) , (87)and recalling (79) we can conclude that this embedding is isometric. Starting bynow we will consider f ∗ T N as Riemannian submanifold on M × R k and in par-ticular we will see f ∗ ( T N ) p = T f ( p ) N as a linear subspace of { p } × R k . Remark . It’s very important to observe that if we define an action of Γ on M × R k posing γ ( p, t ) = ( γp, t ) , then the immersion I is Γ -equivariant. Indeed dS γf ( p ) ( γw f ( p ) ) = dS f ( p ) ( w f ( p ) ) , (88)and so exp A ( γp ) [ dB γf ( p ) γw f ( p ) ] − A ( p ) = exp A ( p ) [ dB f ( p ) w f ( p ) ] − A ( p ) (89)and so this means that I ( γ ( p, w f ( p ) )) = ( γp, exp A ( p ) [ dB f ( p ) w f ( p ) ] − A ( p )) = γ I ( p, w f ( p ) ) . (90)22 .4 Totally geodesic Riemannian submersions Let us define the map P : M × R k −→ f ∗ ( T N )( p, t ) −→ ( p, π p ( t )) , (91)where π p : R k −→ T f ( p ) N is the orthogonal projection on T f ( p ) N . We can ob-serve that P is a C ∞ -map, indeed, fixed a local orthonormal frame E i of f ∗ ( T N ) around p we have that locally the map is given by P ( q, t ) = ( q, h t ; E i ( q ) i E i ( q )) . (92)In particular we have that P is a submersion, indeed, using again the coordinate of M × B k , then we have that P ( x, ˜ y, z )) = ( x, ˜ y, (93)and using the decomposition (79) and (86) we can observe that it is, in particular,a Riemannian submersion.Our next step is to show that locally exp P ( p,t ) ,f ∗ T N ◦ P ◦ exp − p,t ) ,M × R k = dP ( p,t ) . (94)To do this we need some definitions and a result of Vilms [12]. Definition 3.4.
Consider E a subbundle of the tangent bundle T M of a smoothmanifolds M . Then E is said to be integrable at p if there exists a rank ( E ) dimensional immersed, connected submanifold Q such that for all q ∈ Q , T q Q = E q . N will be called an integral manifold of E passing through p . E is said to be integrable if it has integral manifolds through every point in M . Definition 3.5. A C ∞ -map f : X −→ Y of finite-dimensional connected C ∞ Riemannian manifolds is defined to be totally geodesic if for each geodesic x t in X the image f ( x t ) is a geodesic in Y . Definition 3.6.
A submanifold F of M is said to be a geodesic submanifold ifevery geodesic in F is also a geodesic in M . Proposition 3.1 (Vilms) . A nontrivial Riemannian submersion is totally geodesicif and only if the fibers are totally geodesic submanifolds and the horizontal sub-bundle is integrable. We can observe that the fiber of a point ( p, w f ( p ) ) is a ( k − dim ( N )) -plane in { p } × R k and so it is a totally geodesic submanifold.Moreover, we have that the horizontal subbundle H is given by H ( p,t ) := Span { ∂∂x i | p } ⊕ Span { ∂∂ ˜ y j | ( p,t ) } . (95) with nontrivial Vilms in [12] means a submersion which is not an immersion ( p, t ) in M × R k the vector −→ v in R k given by −→ v = t − π p ( t ) . (96)Consider the local coordinate ( x, ˜ y, z ) of M × B k we have that −→ v has coordinates (0 , , a ) . Then, if we define the diffeomorphism φ ( x, ˜ y, z ) = ( x, ˜ y, z + a ) . (97)we obtain that the set Q described in coordinates by z = a is locally diffeomorphicto f ∗ T N (which is described by z = 0 ) and so it is a manifold.In particular we have that ( p, t ) ∈ Q ( p,t ) and, moreover, one can easily check that ( dφ ( ∂∂x i ) = ∂∂x i dφ ( ∂∂ ˜ y j ) = ∂∂ ˜ y j , (98)and so we have that T ( q,s ) Q = H ( p,t ) . This means that the horizontal bundle isintegrable and, in particular, we have that P is a totally geodesic submersion.Let us now introduce the following lemma: Lemma 3.2.
Let us consider a totally geodesic Riemannian submersion s : E −→ M . Then we have that exp − s ( p ) ◦ s ◦ exp p = ds p . (99) This, in particular, implies that all the derivatives of s in normal coordinates areuniformly bounded.Proof. Consider v p in T p E . Then we have that γ ( t ) = s ◦ exp p ( t · v p ) is the geodesisdefined by γ (0) = s ( p ) and γ ′ (0) = ds p v p . We also have that σ ( t ) = exp s ( p ) ◦ ds p is the same geodesis γ . Then we have that s ◦ exp p = exp s ( p ) ◦ ds p (100)and so we have (99). Let us choose an orthonormal basis { v i } in ker ( ds p ) ⊥ and { w j } an orthonormal basis on ker ( ds p ) . We have that { ds p ( v i ) } forms an or-thonormal basis on T s ( p ) M . This means that if we consider the coordinates { a i , b j } and { z i } the coordinates relative to these basis, then we have that s ( a i , b j ) = a i , (101)and so the derivatives can only be or . Remark . The map P : M × B k −→ f ∗ ( T N ) is, in particular, a lipschitz map. Remark . We can observe that the map P is Γ -equivariant. Indeed since theRemark 19 we have that the embedding of f ∗ ( T N ) in M × R k is Γ -equivariant.In particular this means that if f ∗ ( T N ) p ⊂ { p } × R k is defined as the span of d ( I ◦ S N ) E i , then f ∗ ( T N ) γp ⊆ d ( I ◦ S N ) γE i = d ( I ◦ S N ) E i . So this means thatthe projection π p = π γp and so P is Γ -equivariant.24 .5 The map ˜ p f In this subsection we will define a submersion ˜ p f : f ∗ ( T δ N ) −→ N where f ∗ ( T δ N ) := { ( p, w f ( p ) ) ∈ f ∗ ( T N ) || w f ( p ) | ≤ inj N } . (102) Lemma 3.3.
Let us consider f : ( M, g ) −→ ( N, h ) a lipschitz map betweenmanifolds of bounded geometry. Let us denote by F the bundle morphism inducedby f between f ∗ ( T N ) and T N and by g s the Sasaki metric on f ∗ ( T N ) . Thenthere is a map ˜ p f : ( f ∗ ( T δ N ) , g s ) −→ ( N, h ) such that:1. ˜ p f is a submersion,2. ˜ p f ( x,
0) = f ( x ) ,3. ˜ p f is Γ -equivariant,4. ˜ p f = ˜ p id N ◦ F ,5. if f , respect to local normal coordinate { x i } on M and { y j } on N , satisfies sup s =0 ,...,k | ∂ s y j ◦ f∂x i ...∂x i s ( x ) | ≤ L (103) for some k and L , then there is a constant C such that, respect to normalcoordinates { z i } on f ∗ ( T BN ) , we have sup s =0 ,...,k | ∂ s y j ◦ p f ∂z i ...∂z i s ( x, v ) | ≤ C (104) where C doesn’t depend by x or j .Proof. We can define ˜ p f : ( f ∗ ( T δ N ) , g s ) −→ ( N, h )( p, w f ( p ) ) −→ exp f ( p ) ( w f ( p ) ) . (105)Then we have that1. ˜ p f is a submersion, indeed for each fixed p in M we have that ˜ p f ( p, · ) : f ∗ ( T δ N ) p = T δf ( p ) N −→ N is the exponential map in f ( p ) . We know thatthe exponential map is a local diffeomorphism and so ˜ p f is a submersion,2. ˜ p f ( p,
0) = f ( p ) : this follows immediately by the definition of exponentialmap, in particular this fact implies that ˜ p f is a lipschitz map ˜ p f is Γ -equivariant, indeed, since Γ acts by isometries, ˜ p f ( γp, dγw f ( p ) ) = exp f ( γp ) dγw f ( p ) = exp γf ( p ) dγw f ( p ) = γexp f ( p ) w f ( p ) = γ ˜ p f ( p, w f ( p ) ) . (106)4. It’s obvious: we have that F ( p, w f ( p ) ) = ( f ( p ) , w f ( p ) ) ,5. Since the last point we can show the assertion just for p id . To study p id innormal coordinates means fix v p in T N , a point q in N and study exp − q ◦ p id ◦ exp v p , (107)where exp v p : V ⊆ T v p T δ N −→ T δ N is the exponential map of T N .Let us consider V small enough such that exp v p ( V ) ⊆ U where U can beidentified with an open of R m + n in which the fibered coordinates { x i , y j } are well defined. Let us suppose, moreover, that { x i } are normal coordinateson N .We know, since | v p | ≤ δ for each v p in T δ N , that the components of thematrix related the Sasaki metric and its derivatives are uniformly bounded.Then, applying the Lemma 3.8 of [10], we obtain that the derivatives of exp v p are uniformly bounded.Let us study now exp − q ◦ p id restricted to V . We have that it can be see as π ◦ φ ( x, y ) where π ( x, y ) = x and φ is the flow of the system of differentialequations given by ( ¨ x k = c k c k = − Γ kij x i x j (108)Then, applying the Lemma 3.4 of [10] we have that the partial derivativesof φ are uniformly bounded. Then we can conclude that the derivatives of exp − q ◦ p id ◦ exp v p are uniformly bounded. Theorem 3.4.
Consider ( M, g ) and ( N, h ) two manifolds of bounded geometryand let f : M −→ N be a lipschitz map. Consider, moreover, a group Γ which actsu.p.d.f. and suppose f as Γ -equivariant. Then there is a map p f : M × B k −→ N such that1. p f is a submersion. In particular one can observe that for each p in M wehave that also p f ( p, · ) : B k −→ N is a submersion,2. p f ( x,
0) = f ( x ) , . p f is Γ -equivariant,4. p f = p id N ◦ ( f, id B k ) ,5. if f , respect to local normal coordinate { x i } on M and { y j } on N , satisfies sup s =0 ,...,k | ∂ s y j ◦ f∂x i ...∂x i s ( x ) | ≤ L (109) for some k and L , then there is a constant C such that, respect to normalcoordinates { z i } on M × B k , we have sup s =0 ,...,k | ∂ s y j ◦ p f ∂z i ...∂z i s ( x, v ) | ≤ C (110) where C doesn’t depend by x or j .6. f ∼ Γ p f where f : M × B k −→ N is defined as f ( p, t ) := f ( p ) .Proof. Let us denote by X := M Γ and Y := N Γ . Since Γ acts u.p.d.f. we know that X and Y are manifolds of bounded geometry. Consider δ := min { inj N , inj M , inj Y , inj X } .We can define p f as p f = ˜ p f ◦ P ◦ ( id M , δ · Id B k ) , (111)where P is the Riemannian submersion defined in section 3.4 and ( id M , δ · Id B k ) : M × B k −→ M × R k is the map which sends ( p, t ) in ( p, δ · t ) .Then we have that1. holds since p f is composition of submersions. Moreover we can observe thatalso p f ( p, · ) : B k −→ N is a composition of submersion.2. We have that p f ( x,
0) = ˜ p f ( x, f ( p ) ) = f ( p ) (112)3. We have that p f is Γ -equivariant since it is composition of Γ -equivariantmaps.4. Since ( id M , δ · Id B k ) has uniformly bounded derivatives, it is enough tostudy ˜ p f ◦ P restricted to M × B δ (0) ⊆ M × R k . We know that exp − p f ( p,t ) ◦ ˜ p f ◦ P ◦ exp ( p,t ) = exp − p f ( p,t ) ◦ ˜ p f ◦ exp P ( p,t ) ◦ exp − P ( p,t ) ◦ P ◦ exp ( p,t ) . (113)And so we know since Lemma 3.2 that exp − P ( p,t ) ◦ P ◦ exp ( p,t ) = dP ( p,t ) hasuniformly bounded derivatives and by Lemma 3.3 we know that if f has thefirst k -derivatives in normal coordinates uniformly bounded, then the samehappens to ˜ p f . Then p f satisfies (110). in particular this fact implies that p f is a lipschitz map
27. This fact follows by the lipschitzianity of p f , indeed we can define H : M × B k × [0 , −→ N as H ( p, t, s ) = p f ( p, s · t ) , (114)and we obtain a Γ -equivariant lipschitz homotopy. Remark . It is important to observe that f ∼ Γ p f implies that if f is a uniformlyproper map, then also p f is uniformly proper. p f Lemma 4.1.
Let us consider a lipschitz map f : ( M, g ) −→ ( N, h ) betweenmanifolds of bounded geometry, a gropu Γ acting by isometries on M and on N and consider δ = min { inj M , inj N , inj M Γ , inj N Γ } . Then we have that the map P ◦ ( id M , δ · id B k ) : M × B k −→ f ∗ ( T δ N ) is a R.-N.-lipschitz map.Proof. Let us observe that ( id M , δ · id B k ) : M × B k −→ M × R k is a Lipschitzmap (its constant is δ ) and its Fiber Volume is V ol ( id M ,δ · id Bk ) ( q ) = 1 δ k . (115)Then if also P is a R.-N.-lipschitz map, then we can conclude.We already know, since Section 3, subsection 3.4, that P | M × Bδ (0) is a Riemanniansubmersion. Then we just have to show that the Fiber Volume is bounded. Fix v p in f ∗ T δ N . Then we have that its fiber is given by a ( k − dim ( N )) -disk of radius δ . Using the Lemma 3.2 we can see that respect to the coordinate { x i , ˜ y j , z l } on M × B k and respect to the coordinates { x i , y j } on f ∗ ( T δ N ) , we have that P ( x, ˜ y, z ) = ( x, ˜ y ) . (116)Then we have that V ol M × B k P ∗ ( V ol f ∗ ( T δ N ) ) ( p, t ) = s det ( G ij ) det ( H rk ) ( p, t ) dz ∧ ... ∧ dz k − n , (117)where G ij and H rk are matrix related to the metrics on M × B k and on f ∗ ( T δ N ) .We can observe that G ij ( p, t ) and H rk ( p, v p ) are both the identity because { x i } are normal coordinates centered in p . and so this means that Z F ( p,wf ( p )) V ol M × R k P ∗ ( V ol f ∗ ( T N ) ) = Z B δ (0) ⊂ R k − n dz ∧ ... ∧ dz k − n = C · δ k − n . (118)28 emma 4.2. Consider f : ( M, g ) −→ ( N, h ) a lipschitz map between Riemannianmanifolds of bounded geometry. Then the map ˜ t f : f ∗ ( T δ N ) −→ M × N definedas ˜ t f ( p, w f ( p ) ) = ( p, ˜ p f ( p, w f ( p ) )) (119) is an R.-N.-lipschitz diffeomorphism with its image.Proof. We can start observing that ˜ t f is a diffeomorphism: first we have to observethat dim ( f ∗ ( T δ N )) = m + n = dim ( M ) + dim ( N ) = dim ( M × N ) . Then fixsome fibered coordinates { x i , a j } on f ∗ ( T δ N ) and let { x i , y j } be some normalcoordinate on M × N . Then the Jacobian of ˜ t is given by J t f ( x , µ ) = (cid:20) ⋆ J exp x ( a ) (cid:21) (120)Then, since the exponential map is a diffeomorphism for each x , we have that J ˜ t f is invertible. Moreover ˜ t f is also injective, indeed if ( p, w f ( p ) ) and ( q, v f ( q ) ) havethe same image, then p = q and exp f ( p ) w f ( p ) = exp f ( p ) v p = ⇒ w p = v p , (121)since their norm is less than δ and δ ≤ inj N . We proved that ˜ t f is a diffeomor-phism.Moreover we also know that ˜ t f is a lipschitz map because p f is a lipschitz map.Finally, we have to prove that ˜ t f is a R.-N.-lipschitz map: consider a point ( p, q ) in M × N . Then its fiber is just its preimage ( p, w f ( p ) ) . This means, following theRemark 13 that the Fiber Volume of ˜ t f is just given by | ˜ t − ∗ V ol T δ N ˜ t ∗ V ol M × N | . (122)Then if V ol
TδN ˜ t ∗ V ol M × N is a bounded function, we can conclude that ˜ t f is a R.-N.-lipschitz map. Consider an orthonormal frame w j ( q ) in N around f ( p ) . Then,considering on M a system of normal coordinates { x i } around p , then choosing anorthonormal frame E j around f ( p ) in N , we can define some fibered coordinates { x i , a j } on f ∗ ( T N ) . Fix now f ( p ) in N and consider the basis { E i ( f ( p )) } in T f ( p ) N . Then we can consider the normal coordinates { x i , ˜ a j } around ( p, f ( p )) .One can easily check, using the definition of exponential map, that the image of ˜ t f is contained in a δ -neighborhood of the Graph ( f ) ∈ M × N . This means thatusing the coordinates { x i , ˜ a j } we can cover all the image of ˜ t f . Out of the imageof ˜ t f we already know since Proposition 2.10 that the Fiber Volume is null.One can observe that, using this coordinates, we have ˜ t f (0 , a j ) = (0 , a j ) . (123)Consider V ol M × N ( x, a ) = q det ( G ij )( x, a ) dx ∧ ... ∧ da n (124)29nd V ol T δ N ( x, ˜ a ) = q det ( H ij )( x, ˜ a ) dx ∧ ... ∧ d ˜ a n , (125)where G ij is the matrix of the metric on f ∗ ( T δ N ) respect to this coordinate and H ij is the matrix of the metric on M × N . Then the Fiber Volume of ˜ t f in (0 , ˜ a ) is given by q G ij (0 ,a ) H ij (0 ,a ) .Then we can observe that on (0 , a ) ∈ f ∗ ( T δ N ) G ij (0 , a ) = (cid:20) (cid:21) (126)and so p det ( G ij ) = 1 . Moreover, we also have that on (0 , a ) in M × N , we havethat H ij (0 , ˜ a ) = (cid:20) h ij (0 , ˜ a ) (cid:21) (127)where h ij is the matrix related to the Riemannian metric h in normal coordinates.Then we have that det ( H ij ) − (0 , a ) = det ( h ij ) − (0 , ˜ a ) ≤ C (128)because N is a manifold of bounded geometry [10]. This means that s G ij (0 , a ) H ij (0 , a ) ≤ C (129)and so the Fiber Volume of ˜ t f is bounded. Corollary 4.3.
Let f : ( M, g ) −→ ( N, h ) be a uniformly proper lipschitz mapbetween riemannian manifolds of bounded geometry. Then p f is a R.-N.-lipschitzmap and p ∗ f is L -bounded.Proof. Let us consider S := ˜ t f ◦ P ◦ ( id M , δ · id B k ) . We know since the previ-ous two lemmas that S is a R.-N.-lipschitz map since it is composition of R.-N.-lipschitz maps.We can observe that p f = pr N ◦ S . We can observe that since p f ∼ Γ f , usingRemark 2, in particular (13), we can see that there is a C > such that p − f ( q ) ⊂ A q := f − ( B C ( q )) = f − ( B C ( q )) × B k . (130)This means that fixed a q in N , then the Fiber Volume of S in a point ( p, q ) can bedifferent from zero only if p ∈ f − ( B C ( q )) .Then, using the Proposition 2.13, we have that the Fiber Volume of p f in a point q os given by V ol p f ( q ) = Z M V ol S ( p, q ) dµ M = Z f − ( B C ( q )) V ol S ( p, q ) dµ M ≤ K · µ M ( f − ( B C ( q ))) . (131)30here K is the maximum of the Fiber Volume of S . Let us observe that since f is uniformly proper, then the diameter of f − ( B C ( q )) is uniformly bounded andso, since the curvature of M is bounded from below, we also have that there is aconstant V such that µ M ( f − ( B C ( q ))) ≤ V (132)and so V ol p f ( q ) ≤ K · V (133)and p f is a R.-N.-lipschitz map. T f operator To define T f we need of a particular n -differential form ω on B k such that• R B k ω = 1 ,• ω is in Ω c ( B k ) .Now we can define the operator T f as follow T f ( α ) = Z B k p ∗ f ( α ) ∧ ω. (134) Proposition 4.4.
Let us consider a uniformly proper lipschitz map f between Rie-mannian manifolds of bounded geometry. Then the operator T f is a bounded op-erator.Proof. We have that T f = pr M,⋆ ◦ e ω ◦ p ∗ f , (135)where e ω ( α ) = α ∧ ω and pr M,⋆ is the integration along the fibers of pr M : M × B k −→ M .We know, by the previous Proposition, that p ∗ f is L -bounded.Moreover applying Corollary 2.12, we can also prove that pr M,⋆ is L -bounded,indeed pr M is a lipschitz map and its Fiber Volume is for every point p in M .Then || q B k || = || pr ∗ M || = 1 . (136)Finally the operator e ω is L -bounded. Fix on M × B k a product atlas { ( U γ , x i , t j ) } on M × B k and consider for all γ an orthonormal frame on pr M ( U γ ) . A differentialform α can be locally written as α = α I,J ( x, t ) ǫ I ∧ dt J (137)where I and J are multi-index and α ∧ ω = ψ ( t ) α I, ( x, t ) ǫ I ∧ V ol B k (138)31here V ol B k = dt ∧ ...dt k and ω = ψ ( t ) V ol B k .Given a partition of unity of M { ρ γ } related to { U γ } , we have that || α || = || X γ Z M × B k ρ γ ( α I,J , α
I,J ) V ol M × B k || . (139)On the otherhand we have that, if C ω is the max of ψ over B k , || e ω ( α ) || = || X γ Z M × B k ρ γ ψ ( t ) ( α I, , α I, ) V ol M × B k ||≤ || X γ Z M × B k ρ γ C ω ( α I, , α I, ) V ol M × B k ||≤ C ω || X γ Z M × B k ρ γ ( α I, , α I, ) V ol M × B k ||≤ C ω || α || . (140) Remark . In particular the previous Proposition holds if f is a lipschitz-homotopyequivalence. Corollary 4.5.
Given a uniformly proper lipschitz map f : ( M, g ) −→ ( N, h ) between two Riemannian manifolds of bounded geometry, then we have T f ( dom ( d min )) ⊆ dom ( d min ) (141) and T f d = dT f . This, in particular, means that T f induces a morphism in L -cohomology. Moreover, since T f is L -bounded, it also induces a morphism be-tween the reduced L -cohomology groups.Proof. We can observe that the operator e ω ◦ p ∗ f satisfies the hypothesis of Propo-sitions 2.1 and 2.2. In particular we have that e ω ◦ p ∗ f (Ω ∗ c ( N )) ⊆ Ω ∗ CV ( M × B k ) . So we can apply the Proposition X, pg. 304 of [6], which allow us to say pr M,⋆ (Ω ∗ CV ( M × B k ) ⊆ Ω c ( M ) and that Z B k dη = d Z B k η, (142)if η is in Ω ∗ CV ( M × B k ) . Then, using that T f is a L -bounded operator we canconclude applying the Proposition 2.1 and the Corollary 2.2. Remark . Since p f is a Γ -equivariant map, then p ∗ f is Γ -equivariant and, in par-ticular T f is Γ -equivariant. 32 .3 Lemmas about homotopy In this section we will study the L -boundedness of the pullback of some homo-topies. Lemma 4.6.
Let f : ( M, g ) −→ ( N, h ) be a lipschitz map. Then consider thehomotopy H : M × B k × [0 , −→ N defined as H ( p, w, t ) := p f ( p, t · w ) . (143) Then, if f is a R.-N.-lipschitz map, then also H and ( H, id [0 , ) are R.-N.-lipschitzmaps.Remark . The map H above is important for us because it is a Γ -equivariantlipschitz-homotopy between p f and f : M × B k −→ N . In particular, provingthis Lemma, we have that the maps p id : N × B k −→ N and the projection pr N : N × B k −→ N are Γ -lipschitz-homotopy with an homotopy H which is a R.-N.-lipschitz map, and so H induces a morphism in (un)-reduced L -cohomology. Proof.
We know that M × B k × { , } is a subset of null-measure, then we willconsider H restricted to M × B k × (0 , .We can also observe that since f is a lipschitz map, then also H and ( H, id [0 , ) arelipschitz maps. Then we just have to prove that they have bounded Fiber Volume.We can observe that if pr N : N × [0 , −→ N is the projection on the firstcomponent, then H = pr N ◦ ( H, id (0 , ) . It follows by the definition of p f that ( H, id (0 , ) is a submersion and so, applying the Proposition 2.13, we have that V ol H ( q ) = Z V ol ( H,id [0 , ) ( q, t ) dt = Z Z F q,t ×{ t } V ol M × B k × [0 , ( H, id [0 , ) ∗ V ol N × [0 , ( q, t ) dt. (144)This fact implies that, if we are able to show that the Fiber Volume of ( H, id (0 , ) is bounded, then also the Fiber Volume of H is bounded.Let us consider the projection pr N × (0 , : M × N × (0 , −→ N × (0 , : we canobserve that ( H, id (0 , ) = pr N × (0 , ◦ ( id M , H, id (0 , ) . (145)One can easily check that ( id M , H, id (0 , ) is a lipschitz submersion .We recall that p f = ˜ p f ◦ P ◦ ( id M , δ · id B k ) . Then if we define ˜ h : f ∗ ( T δ N ) × [0 , −→ N (146)as ˜ h ( p, w f ( p ) , s ) = ˜ p f ( p, s · w f ( p ) ) , (147) it follows observing that the map p f ( p, s · ) : B k −→ N is a lipschitz submersion for each ( p, s ) in M × (0 , .
33e obtain that H = ˜ h ◦ ( P ◦ ( id M , δ · id B k ) , id [0 , ) (148)and, in particular, ( id M , H, id (0 , ) = ( pr M , ˜ h, id (0 , ) ◦ ( P ◦ ( id M , δ · id B k ) , id (0 , ) (149)because P is a bundle morphism between bundles over M .We already know that P ◦ ( id M , δ · id B k ) are R.-N.-lipschitz maps (Lemma 4.1),Moreover one can also observe that id (0 , is a R.-N.-lipschitz. Then we can ap-ply the Proposition 2.4 to conclude that ( P ◦ ( id M , δ · id B k ) , id (0 , ) is an R.-N.-lipschitz map.This means that if we are able to show that pr N × (0 , ◦ ( pr M , ˜ h, id (0 , ) has boundedFiber Volume, then also ( H, id (0 , ) and H will have bounded Fiber Volume.Our next step will be to calculate the Fiber Volume of ( pr M , ˜ h, id (0 , ) .Similarly to the Lemma 4.2, we can consider for each p in M some normal co-ordinates { x i } around p , an, taking an orthonormal frame { w i } of f ∗ ( T N ) , wehave the coordinates { x i , a j , t } on f ∗ ( T N ) × [0 , around ( p, , .Moreover we can also consider the normal coordinates { ˜ a j } around f ( p ) in N related to the basis { w i ( p ) } of T f ( p ) N and so we have the normal coordinates { x i , ˜ a j , t } on M × N × [0 , .We can observe that, the coordinate { x i , ˜ a j , t } are enough to cover the image of ( id M , H, id [0 , ) which is contained in a δ -neighborhood of Graph ( f ) × [0 , .Finally, respect to these coordinates, we have that ( pr M , ˜ h, id [0 , )(0 , a j , t ) = (0 , t · a j , t ) . (150)Similarly as we did in Lemma 4.2 one can show that V ol f ∗ ( T N ) × [0 , (0 , a, t ) = dx ∧ ... ∧ dx m ∧ da ∧ ... ∧ da n ∧ dt (151)and that V ol M × N × [0 , (0 , ˜ a, t ) = q det ( H ij (0 , y )) dx ∧ ... ∧ dx m ∧ d ˜ a ∧ ... ∧ d ˜ a n ∧ dt, (152)where H ij = R ikjl ˜ a j ˜ a l + O (˜ a ) 00 0 1 (153)Then we have that ( pr M , ˜ h, id [0 , ) is a diffeomorphism with its image and so ap-plying Remark 13, its Fiber Volume is given by | ( pr M , ˜ h, id [0 , ) − V ol f ∗ T N × [0 , ( pr M , ˜ h, id [0 , ) ∗ V ol M × N × [0 , | . (154)34hen we can observe that in a point (0 , ˜ a j , t ) we have that the Fiber Volume of ( pr M , ˜ h, id [0 , ) is V ol f ∗ T N × [0 , ( pr M , ˜ h, id [0 , ) ∗ V ol M × N × [0 , = 1 t n (1 + C ( t, ˜ a )) , (155)where C is a bounded function.Let us consider now the projection pr N × [0 , : M × N × (0 , −→ N × (0 , .Following the Proposition 2.13 we have that V ol ( H,id (0 , ) ( q, t ) = Z M V ol ( id M ,H,id [0 , ) ( p, q, t ) dµ M . (156)We know that outside the image of ( id M , H, id [0 , ) the Fiber Volume V ol ( id M ,H,id [0 , ) is null. So if we denote by H t := H ( · , · , t ) , the Fiber Volume it is null also outside pr M ( H − t ( q )) × { q } × { t } . Since H is lipschitz, applying Remark 2 and we have H − t ( q ) ⊆ f − ( B C H t ( q )) × B k × { t } (157)and so pr M ( H − t ( q )) × { q } × { t } ⊂ f − ( B C H t ( q )) × { q } × { t } . (158)Then using that f is a R.-N.-lipschitz map, we have that µ ( f − ( B C H · t ( q ))) ≤ K µ ( B C H × t ( q )) ≤ K C nH t n (1 + L ( t )) , (159)where L is a bounded function which depends by t . Then we have that the FiberVolume of ( H, id (0 , ) is given by V ol ( H,id (0 , ) ( q, t ) = Z f − ( B CH · t ( q )) V ol ( id M ,H,id [0 , ) ( p, q, t ) dµ M ≤ t n (1 + C ( t, ˜ y )) · ( K C nH t n (1 + L ( t ))) ≤ K · t n · t n ≤ K. (160)and so ( H, id (0 , ) is a R.-N.-lipschitz map and so also H = pr N ◦ ( H, id (0 , ) (161)is an R.-N.-lipschitz map. Proposition 4.7.
Consider two uniformly proper lipschitz maps g : ( S, l ) −→ ( M, g ) and f : ( M, g ) −→ ( N, h ) between manifolds of bounded geometry. Thenthe map H : S × B k × B j × [0 , −→ N defined as H ( p, w , w , t ) −→ p f ( p g ( p, t · w ) , w ) (162) and the map ( H, id [0 , ) are R.-N.-lipschitz maps. emark . The map H above is important for us because it is a Γ -equivariantlipschitz-homotopy between p f ◦ ( p g , id B k ) and p f ◦ g : S × B k × B j −→ N defined as p f ◦ g ( p, t , t ) = p f ◦ g ( p, t ) . (163)In particular, proving this Lemma, we prove that the homotopy H is a R.-N.-lipschitz map, and so H induces a morphism in (un)-reduced L -cohomology. Proof.
Exactly as in the previous Proposition, we can just consider the restrictionof H to S × B j × B k × (0 , . Moreover, again as in the previous Proposition,if we are able to show that ( H, id [0 , ) is a R.-N.-lipschitz map, then also H is aR.-N.-lipschitz map.If we denote by π the projection S × B k × N × [0 , −→ N × [0 , on the lasttwo components, then we can observe that ( H, id [0 , ) = π ◦ ( id S × B k , H, id [0 , ) and, applying the Proposition 2.13 we have that V ol
H,id (0 , ( q, s ) := Z S × B k V ol id S × Bk ,H,id (0 , ( p, t , q, s ) dµ S × B k . (164)We will start studying V ol id S × Bk ,H,id [0 , . To do this we have to define the map ˜ g : g ∗ ( T δ M M ) × [0 , −→ M ( p, w g ( p ) , s ) −→ ˜ p g ( p, s · w g ( p ) ) (165)Then we will denote by ˜ h := p f ◦ (˜ g, id B k ) , (166)and so we can observe that H := ˜ h ◦ ( P S ◦ ( id S , δ M · id B j ) , id B k × [0 , ) , (167)where P S and δ M · id B j are the map such that p g = ˜ p g ◦ P S ◦ ( id S , δ M · id B j ) . (168)In particular we have that, if we denote by A := ( P S ◦ ( id S , δ M · id B j ) , id B k × [0 , ) (169)then we have that ( id S × B k , H, id (0 , ) = ( id S × B k , ˜ h, id [0 , ) ◦ A . (170)Our next goal is to compute the Fiber Volume of ( id S × B k , ˜ h, id [0 , ) to study theFiber Volume of ( id S × B k , H, id [0 , ) .In a similar way we did in the previous Proposition and in Lemma 4.2, we canconsider some coordinates { x i , a r , y l , t } around a point ( p, , , in g ∗ ( T δ M ) × k × [0 , and some normal coordinates { x i , ˜ a l , y l , t } on S × N × B k × [0 , .Similarly to the previous Proposition, we have that, in this coordinates, ( id S × B k , ˜ h, id [0 , )(0 , a, y , t ) = (0 , ta, y , t ) . (171)Then exactly as we did in the previous Proposition, one can check that the FiberVolume in a point (0 , ˜ y , y , t ) is given by V ol ( id S × Bk , ˜ h,id [0 , ) (0 , ˜ y , y , t ) ≤ t n C (172)for some constant C .Moreover, we can observe that, using the Proposition 2.4 and the Lemma 4.1, wecan prove that the Fiber Volume of A is bounded by a constant J . Then, using thisresult, we can apply the Proposition 2.13 and so we have that V ol ( id S × Bk ,H,id [0 , ) ≤ t n C · J ≤ t n C . (173)and so, following the proof of the previous Proposition, one can easily show that V ol ( id S × Bk ,H,id [0 , ) ≤ t n C . (174)We will conclude this proof in a very similar way we concluded the previous one:we have that the Fiber Volume of ( H, id [0 , ) is given by V ol ( H,id [0 , ) ( q, t ) = Z S × B k V ol ( id S × Bk ,H,id [0 , ) ( p, w , q, t ) dµ S × B k , (175)but we know that outside the image of ( id S × B k , H, id [0 , ) , its Fiber Volume isnull. In particular this means that if we denote by H t the map S × B j × B k −→ N defined as H t := H ( · , · , · , t ) , (176)then the Fiber Volume of ( id S × B k , H, id [0 , ) restricted to S × B k × { q } × { t } isnull outside pr S × B k ( H − t ( q )) × { q } × { t } . Then, we can observe that since H islipschitz, we can apply the Remark 13 and we have H − t ( q ) ⊆ H − ( B C H t ( q ))= p − f ◦ g ( B C H t ( q ))= p − f ◦ g ( B C H t ( q )) × B j (177)and so pr S × B k ( H − t ( q )) × { q } × { t } ⊆ p − f ◦ g ( B C H t ( q )) . (178)Then we can observe that, since f ◦ g is a uniformly proper lipschitz map, p f ◦ g isa R.-N.-lipschitz map, and so µ ( p − f ◦ g ( B C H t ( q ))) ≤ K µ ( B C H t ( q )) ≤ C t n (1 + L ( t )) , (179)37here L ( t ) is a bounded function. Then we can conclude observing that V ol ( H,id [0 , ) ( q, t ) = Z S × B k V ol ( id S × Bk ,H,id [0 , ) ( p, w , q, t ) dµ S × B k ≤ ( 1 t n C )( C t n (1 + L ( t ))) = K t n t n = K. (180)Then ( H, id [0 , ) is R.-N.-lipschitz and the same holds for H . L -cohomology Consider the category C which has manifolds of bounded geometry as objects anduniformly proper lipschitz maps as arrows. Consider, moreover, the category Vec which has complex vector spaces as objects and linear maps as arrows. In thissection we will show that, for every z in N , the association F : C −→
Vec definedas ( F ( M, g ) = H z ( M ) F (( M, g ) f −→ ( N, h )) = H z ( N ) T f −→ H z ( M ) (181)is a controvariant functor. Moreover, we will show that if two maps f and f are uniformly proper lipschitz-homotopy then T f = T f in (un)-reduced L -cohomology and that if f is a R.-N.-lipschitz map, then f ∗ = T f in (un)-reduced L -cohomology.This fact will imply that if f is a lipschitz-homotopy equivalence between man-ifolds of bounded geometry, then the L -cohomology groups are isomorphic. Inother words, the L -cohomology is a lipschitz-homotopy invariant.Let us introduce, now the operator R L : it will be the main tool that we will use tostudy homotopies. Lemma 4.8.
Let ( M, g ) be a Riemannian manifold and consider ([0 , , g [0 , ) .Then there is a L -bounded operator R L : Ω ∗ ( M × [0 , ∩ L ( M × [0 , −→ Ω ∗ ( M ) ∩ L ( M ) such that for all smooth α ∈ L ( M × [0 , we have i ∗ α − i ∗ α = Z L dα + d Z L α (182) Moreover R L sends compactly-supported differential forms on Ω ∗ ( M × [0 , into Ω ∗ c ( M ) .Proof. Let α be in Ω ∗ c ( M × [0 , with compact support and let p : M × [0 , −→ M be the projection on the first component. There are two possibilities α = f ( x, t ) p ∗ ω (183)or α = f ( x, t ) dt ∧ p ∗ ω, (184)38or some ω in Ω c ( M ) and for some C ∞ -class function f : M × [0 , −→ C . Thenwe can define the operator R , L as follow: if α is a -form then Z , L α := 0 (185)otherwise Z , L α := ( Z f ( x, t ) dt ) ω. (186)This operator is very similar to the operator q [0 , (the integration along the fiber of p : M × [0 , −→ M ), but they are different for the signs. Indeed if we consider α = f ( x, t ) dt ∧ p ∗ ω we have that q [0 , α = Z [0 , f ( x, t ) dt ∧ p ∗ ω = Z [0 , ( − deg ( p ∗ ω ) p ∗ ω ∧ f ( x, t ) dt. (187)Then applying the Projection Formula we obtain q [0 , α = ( − deg ( p ∗ ω ) ω · ( Z [0 , f ( x, t ) dt )= ( − deg ( p ∗ ω ) ω · ( Z f ( x, t ) dt )= ( − deg ( p ∗ ω ) Z , L α. (188)Moreover the operator R , L doesn’t commute with the exterior derivative d on Ω CV ( M × [0 , , indeed if we consider for example M = U an open set of R n d Z , L α = d ( Z f ( x, t ) dt )= ( Z ∂f∂x i dt ) ∧ dx i (189)but Z , L dα = Z L ( ∂f∂x i dx i ∧ dt )= Z L ( − ∂f∂x i dt ∧ dx i )= − ( Z ∂f∂x i dt ) dx i . (190)39ince the operator R L , up to signs, is the operator of integration along the fibers,then we have that the norm is the same of q [0 , = 1 . We know from Lemma 11.4of [ ? ] that R L : Ω ∗ ( M × [0 , −→ Ω ∗− ( M ) and that for all differential formswe have that Z , L dα + d Z , L α = i ∗ α − i ∗ α. (191) Proposition 4.9.
Let id : ( N, h ) −→ ( N, h ) be the identity map on a manifold ofbounded geometry and consider p id the submersion related to the identity. Then,if pr N : N × B k −→ N is the projection on the first component, then exists a L -bounded operator K : L ( N ) −→ L ( N × B k ) such that for every smoothform α p ∗ id α − pr ∗ N α = d ◦ K α + K ◦ dα. (192) Moreover, if α is in Ω ∗ c ( N ) , then K α ∈ Ω ∗ ( N × B k ) has compact support.Proof. First one can observe that pr N is a lipschitz submersion with bounded FiberVolume and so pr ∗ N is a bounded operator.Now we can observe that p id and the projection pr N : ( N × B k , h + g eucl ) −→ ( N, h ) are lipschitz-homotopy. Indeed, if we take H ( p, t, s ) = p id ( p, st ) (193)this is a lipschitz homotopy. Moreover, since Lemma 4.6 we know that H is R.-N.-lipschitz and so H ∗ is an L -bounded operator.This means that for all α in Ω ∗ c ( N ) , using Lemma 11.4. of [ ? ] we have that p ∗ id − pr ∗ N ( α ) = i ∗ H ∗ α − i ∗ H ∗ α = Z L H ∗ dα + d Z L H ∗ α. (194)Then K := Z , L ◦ H ∗ , (195)satisfies (192) Proposition 4.10.
Consider g : ( M, m ) −→ ( N, h ) and f : ( N, h ) −→ ( S, r ) twouniformly proper lipschitz maps between manifolds of bounded geometry. Let usdenote by p g : M × B s −→ N and p f : N × B j −→ S the submersion related to g and f . Then there is a L -bounded operator K : L ( S ) −→ L ( M × B s × B j ) such that for every smooth form α ( p g , id B j ) ∗ ◦ p ∗ f α − p ∗ f ◦ g α = d ◦ K α + K ◦ dα, (196) where p f ◦ g : M × B s × B j −→ S is the map defined as p f ◦ g ( p, s, t ) = p f ◦ g ( p, t ) . (197) Moreover, if α is in Ω ∗ c ( S ) , then K α ∈ Ω ∗ ( M × B s × B j ) has compact support. roof. Let us define the map H : M × B s × B j × [0 , −→ S ( p, t , t , τ ) −→ p f ( p g , id B j ( p, A ( t , τ ) , t )) . (198)Observe that H is the homotopy between p f ◦ g and p f ◦ ( p g , id B j ) : indeed, follow-ing the Theorem 3.4, we have that p f ◦ g = p f ◦ ( g, id B j ) (199)and so H ( p, t , t ,
0) = p f ( p g , id B j ( p, , t )) = p f ◦ ( g, id B j )( p, t ) = p f ◦ g ( p, t , t ) . (200)and H ( p, t , t ,
1) = p f ( p g , id B j ( p, t , t )) . (201)Moreover, since the Lemma 4.7, H is a R.-N.-lipschitz map. Let us define theoperator K := Z L ◦ H ∗ . (202)It is a L -bounded operator because it is composition of L -bounded operators.Then we can observe that for every smooth form α we have ( p g , id B j ) ∗ ◦ p ∗ f α − p ∗ f ◦ g α = ( i ∗ − i ∗ ) H ∗ α = ( d ◦ Z L ) H ∗ α + ( Z L ◦ d ) H ∗ α = d ◦ K α + K ◦ dα. (203)Finally, since H is a proper map (it is composition of proper maps), if α ∈ Ω ∗ c ( S ) the support of K α ∈ Ω ∗ ( M × B s × B j ) is compact. Proposition 4.11.
Let f and f : ( M, m ) −→ ( N, h ) be two uniformly properlipschitz maps between manifolds of bounded geometry. Let us suppose that f and f are uniformly proper lipschitz-homotopy. Then there is a L -bounded operator K : L ( S ) −→ L ( M × B s × B j ) such that for all smooth form αp ∗ f α − p ∗ f α = d ◦ K α + K ◦ dα. (204) Moreover if α ∈ Ω ∗ c ( N ) then the support of K α ∈ Ω ∗ ( M × B k ) is compact.Proof. To prove the assertion it is sufficient to observe that h , the homotopy suchthat h ( p,
0) = f ( p ) and h ( p,
1) = f ( p ) , is a uniformly proper lipschitz map. Thismeans, since p h : M × [0 , × B k −→ N is a R.-N.-lipschitz map (Corollary(4.3))and so p ∗ h is a L -bounded operator.Moreover, up to switch [0 , and B k , p h is a lipschitz-homotopy between p f and41 f . This fact follows directly by the definition of submersion related to a lipschitzmap in Theorem 3.4.So we can conclude exactly as we did in the first and second points taking K := Z L ◦ p ∗ h (205)and using that p h is a proper map. Proposition 4.12.
Consider ( M, g ) , ( N, h ) and ( S, l ) three manifolds of boundedgeometry and consider f : ( M, g ) −→ ( N, h ) , F : ( M, g ) −→ ( N, h ) and g :( S, l ) −→ ( M, g ) three uniformly proper lipschitz maps. Then, in L -cohomology,we have that1. T id M = Id H ∗ ( M ) ,2. T f ◦ g = T g ◦ T f ,3. if f ∼ Γ F then T f = T F ,4. if f is an R.-N.-lipschitz map then f ∗ = T f .Moreover, since the operator T f is bounded, then the identities above also holds inreduced L -cohomology.Proof. Point 1).
Let us consider the standard projection pr M : M × B k −→ M .Then, since the previous proposition we have that for all smooth forms α in Ω ∗ c ( M ) , pr ∗ M − p ∗ id ( α ) = d ◦ K + K ◦ d ( α ) . (206)Now, we can write the identity map in L ( M ) as α ) := Z B k pr ∗ M α ∧ ω = pr M⋆ ◦ e ω ◦ pr ∗ M ( α ) (207)where pr M⋆ is the operator of integration along le fibers of pr M and ω ∈ Ω kc ( B k ) is a differential form such that its integral equals to 1.We have that for every α ∈ Ω ∗ c ( M )1 − T id M ( α ) = pr M⋆ ◦ e ω ◦ pr ∗ M ( α ) − pr M⋆ ◦ e ω ◦ p ∗ id ( α )= pr M⋆ ◦ e ω ◦ ( pr ∗ M − p ∗ id ) α = pr M⋆ ◦ e ω ◦ ( d ◦ K + K ◦ d ) α. (208)Now we can observe that since ω is closed, we have that d ( α ∧ ω ) = ( dα ) ∧ ω .Moreover α ∧ ω is in Ω ∗ CV ( M × B k ) . This means that the exterior derivative canbe switched with pr M⋆ and then we have − T id M ( α ) = d ◦ pr M⋆ ◦ e ω ◦ K + pr M⋆ ◦ e ω ◦ K ◦ d ( α )= d ◦ Y + Y ◦ d ( α ) , (209)42here Y := pr M⋆ ◦ e ω ◦ K . (210)It’s possible to observe that Y is an L -bounded operator: if we are able to showthat the equation (209) holds for all β in dom ( d min ) , then we conclude the firstpoint. To do this it is necessary to show that Y ( dom ( d min ) ⊂ dom ( d min ) .Consider, then β in dom ( d min : we have that there is a sequence { α j } in Ω ∗ c ( M ) such that β = lim j → + ∞ α j dβ = lim j → + ∞ dα j . (211)We can observe that the support of K α j is compact for all j . Then it means that Y α j is a smooth differential form in M . Since M has not boundary, it means that Y α ∈ Ω ∗ c ( M ) . Then we can observe that Y β = Y ( lim j → + ∞ α j ) = lim j → + ∞ Y α j (212)and, moreover lim j → + ∞ dY α j = lim j → + ∞ Y dα j + α j + T id M α j = Y dβ + β + T id M β ∈ L ( M ) . (213)Now we can see that the equation (213) implies that Y ( dom ( d min ) ⊂ dom ( d min ) and, moreover, we also have that β + T id M β = dY β + Y dβ (214)for every β in dom ( d min ) . Then in L -cohomology we have that Id H ∗ ( M ) = T id M . (215) Point 2.
Consider the subermersions p f : M × B j −→ N , p g : S × B k −→ M , p f ◦ g : S × B j −→ N related to f , g , and f ◦ g . Then, since the previous Proposi-tion, we have that for every α in Ω ∗ c ( N )( p g , id B j ) ∗ ◦ p ∗ f − p ∗ f ◦ g ( α ) = d ◦ K + K ◦ d ( α ) . (216)Then if ω and ω ′ are in Ω ∗ c ( B j ) and in Ω c ( B k ) such that their integral equal to 1,then T f ◦ g = pr S⋆ ◦ e ω ◦ p ∗ f ◦ g = pr S⋆ ◦ e ω ◦ pr S × B j ⋆ ◦ e ω ′ ◦ pr ∗ S × B j ◦ p ∗ f ◦ g , (217)43here pr S : S × B k × B j −→ S and pr S × B j : S × B k × B j : −→ S × B j are thestandard projections. We can observe that p f ◦ g = p f ◦ g ◦ pr S × B j , (218)and so we have that T f ◦ g = pr S⋆ ◦ e ω ◦ pr S × B j ⋆ ◦ e ω ′ ◦ p ∗ f ◦ g . (219)Now we will focus on T g ◦ T f . We have that T g ◦ T f = pr S⋆ ◦ e ω ′ ◦ p ∗ g ◦ pr M⋆ ◦ e ω ◦ p ∗ f . (220)It’s possible to apply the Proposition VIII of Chapter 5 in [6] to the fiber bundles ( S × B k × B j , pr S × B k , S × B k , B j ) and ( M × B j , pr M , M, B j ) and the bundlemorphism ( ψ, Ψ) = ( p g , ( p g , id B j )) . Then we obtain that p ∗ g ◦ pr M⋆ = pr S × B j ,⋆ ◦ ( p g , id B j ) ∗ . (221)Finally, since ω ′ = id ∗ B j ω ′ , we have that also e ω ′ and ( p g , id B j ) ∗ commute. Thismeans that T g ◦ T f = pr S⋆ ◦ e ω ′ ◦ p ∗ g ◦ pr M⋆ ◦ e ω ◦ p ∗ f = pr S⋆ ◦ e ω ′ ◦ pr S × B j ⋆ ◦ ( p g , id B j ) ∗ ◦ e ω ◦ p ∗ f = pr S⋆ ◦ e ω ′ ◦ pr S × B j ⋆ ◦ e ω ◦ ( p g , id B j ) ∗ ◦ p ∗ f = pr S⋆ ◦ e ω ′ ◦ pr S × B j ⋆ ◦ e ω ◦ p f ( p g , id B j ) ∗ . (222)This means that, on dom ( d min ) , since Proposition 9.2, we have T f ◦ g − T g ◦ T f = pr S⋆ ◦ e ω ′ ◦ pr S × B j ,⋆ ◦ e ω ◦ ( p ∗ f ◦ g − p f ( p g , id B j ) ∗ )= pr S⋆ ◦ e ω ′ ◦ pr S × B j ,⋆ ◦ e ω ◦ ( K ◦ d − d ◦ K ) . (223)Now, with the same arguments we did in the previous point, one can easily checkthat pr S⋆ ◦ e ω ′ ◦ pr S × B j ⋆ ◦ e ω ◦ d = d ◦ pr S⋆ ◦ e ω ′ ◦ pr S × B j ⋆ ◦ e ω . (224)Let us define the L -bounded operator Y := pr S⋆ ◦ e ω ′ ◦ pr S × B j ⋆ ◦ e ω ◦ K . (225)Then we have that for all α in Ω ∗ c ( N ) the equality T f ◦ g − T g ◦ T f ( α ) = d ◦ Y + Y ◦ d ( α ) (226)holds. To conclude the proof we have to show that the equation above also holdsfor every β in dom ( d min ) . However the proof of this fact is exactly the same proofwe did to show that β + T id M β = dY β + Y dβ (227)44or every β in dom ( d min ) . Then, in L -cohomology, T f ◦ g = T g ◦ T f . (228) Point 3.
Let us consider the homotopy h between f and F . We know, sinceProposition 9.3 that there is a L -bounded operator K , such that for every α in Ω ∗ c ( N ) p ∗ f − p ∗ F ( α ) = d ◦ K + K ◦ d ( α ) . (229)This means that if we consider the L -bounded operator Y := pr M⋆ ◦ e ω ◦ K , (230)then for all α in Ω ∗ c ( N ) we have that T f − T F ( α ) = pr M⋆ ◦ e ω ◦ ( p ∗ f − p ∗ F )= pr M⋆ ◦ e ω ◦ ( d ◦ K + K ◦ d )( α )= d ◦ Y + Y ◦ d ( α ) . (231)To show that the identity above holds for all β in dom ( d min ) it is sufficient to doexactly the same thing we did to prove that β + T id M β = dY β + Y dβ (232)for every β in dom ( d min ) . Then, in L -cohomology, T f = T F . (233) Point 4.
To prove this statement we have to observe that, since Theorem 3.4,we have that p f = p id ◦ ( f, id B k ) . (234)Let us consider, now, a form α ∈ Ω ∗ c ( N ) : we have that T f α = pr M⋆ ◦ e ω ◦ p ∗ f α = pr M⋆ ◦ e ω ◦ ( f, id B k ) ∗ ◦ p ∗ id α = pr M⋆ ◦ ( f, id B k ) ∗ ◦ e ω ◦ p ∗ id α. (235)45he last equality is true since ω = id ∗ B k ω . Now, using the Proposition VIII ofChapter 5 of [6] we have that T f α = pr M⋆ ◦ ( f, id B k ) ∗ ◦ e ω ◦ p ∗ id α = f ∗ ◦ pr N⋆ ◦ e ω ◦ p ∗ id α = f ∗ ◦ T id N α (236)Since f is R.-N.-lipschitz, we have that f ∗ is L -bounded. This means that (236)implies that T f = f ∗ ◦ T id N . (237)Then we have that on dom ( d min ) we have that f ∗ − T f = f ∗ ◦ (1 − T id N )= f ∗ ◦ ( d ◦ Y + Y ◦ d ) . (238)But now we can observe that f ∗ (Ω ∗ c ( N )) ⊆ Ω ∗ c ( M ) since f is proper and for allsmooth form α we have that f ∗ dα = df ∗ α . Then we have that on dom ( d min ) f ∗ − T f = f ∗ ◦ ( d ◦ Y + Y ◦ d )= d ◦ W + W ◦ d, (239)where W = f ∗ ◦ Y . (240)And so in L -cohomology we have that f ∗ = T f . (241)To conclude the proof we have to show that all the identities above also holdsin reduced L -cohomology. To do this is sufficient to show that if Z is an operatorsuch that dZ + Zd (242)is a bounded operator on dom ( d ) , then, in reduced L cohomology, it is the nulloperator. After that, using Corollary 9.6 and the previous points, we can concludethe proof.Consider a differential form α + lim k → + ∞ dβ k on ker ( d ) . We have that dZ + Zd ( α + lim k → + ∞ dβ k ) = dZ + Zd ( lim k → + ∞ α + dβ k )= lim k → + ∞ dZ + Zd ( α + dβ k )= lim k → + ∞ dZ ( α + dβ k ) ∈ im ( d ) . (243)This means that on reduced L -cohomology dZ + Zd is the null operator.46 emark . Let us consider the category B which has manifolds of bounded ge-ometry as objects and uniformly proper R.-N.-lipschitz maps as arrows.We can define the functor I : B −→ C defined as ( I ( M, g ) = (
M, g ) I (( M, g ) f −→ ( N, h )) = (
M, g ) f −→ ( N, h ) . (244)Consider moreover, for all z ∈ N , the functor G : B −→
Vec defined as ( G ( M, g ) = H z ( M ) G (( M, g ) f −→ ( N, h )) = H z ( N ) f ∗ −→ H z ( M ) . (245)As consequence of the last point we have that G = F ◦ I . (246)All this also holds if we replace H z with H z . Corollary 4.13.
Let ( M, g ) and ( N, h ) be two manifolds of bounded geometry. Let f : ( M, g ) −→ ( N, h ) be a lipschitz-homotopy equivalence. Then T f induces anisomorphism in (reduce or not) L -cohomology.Proof. We will prove this fact for the unreduced L -cohomology. The reducedcase can be proved exactly in the same way.Observe that if f is a lipschitz-homotopy equivalence: then it is a uniformly properlipschitz map. For the same reason also its homotopy inverse g is a uniformlyproper lipschitz map. Moreover since g ◦ f is lipschitz-homotopy to id M then,using Lemma 7.2, we also have that g ◦ f is uniformly proper lipschitz homotopyto id M . Then we have that in L -cohomology. H ∗ ( M ) = T id M = T g ◦ f = T f ◦ T g . (247)Using the same arguments one can see that H ∗ ( N ) = T g ◦ T f . (248) References [1] J. Bruning, M. Lesh,
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