Homological properties of transitive Lie algebroids via Sullivan models

FACULTADE DE MATEM´ ATICAS DEPARTAMENTO DE XEOMETR´ IA E TOPOLOX´ IA Homological properties of transitive Lie algebroids via Sullivan models (Propiedades homol´ogicas de algebroides de Lie transitivos via modelos de Sullivan) Jose Manuel Ribeiro Oliveira 2013 Homological properties of transitive Lie algebroids via Sullivan models (Propiedades homol´ogicas de algebroides de Lie transitivos via modelos de Sullivan) Jose Manuel Ribeiro Oliveira Tesis realizada en el Departamento de Geometria y Topologia de la Facultad de Matem´aticas, bajo la direcci´on del profesor Aleksandr Mishchenko (Moscow State University), siendo tutor Jes´us Antonio ´ Alvarez L´opez (de este Departamento), para obtener el grado de Doctor en Ciencias Matem´aticas por la Universidad de Santiago de Compostela. Autorizaci´on del director: Fdo.: A. Mishchenko Autorizaci´on del tutor: Fdo.: J.A. ´ Alvarez L´opez El autor: Fdo.: J.M. Ribeiro Oliveira Santiago de Compostela a de de 2013. Acknowledgments During the preparation of this work, I always could have the cooperation of some people, whom I now express my gratitude to. First of all, I want to thank Professor Aleksandr Mishchenko who I want to show and express my admiration and appreciation for his level of mathematical knowledge, for his valuable criticism, for his strong intellectual dynamism to suggest ways and theoretical perspectives and to share views, always with great directness and mathematical rigour, which allowed me to grow intellectually. It is also a pleasure to acknowledge Professor Tatiana Mishchenko for all her help and facilities given to me. I want to thank Professor Jesus Alvarez who always was energetically engaged in the success of my training, for his valuable lectures on foliated manifolds, and for scientific ideas on stratified spaces in order to develop my present work in the context of stratified spaces. I want to thank Professor Paula Smith and Professor Lisa Santos for all pedagogical support and their efforts carried out for the success of my professional career. Finally, I tender my warm and deep gratitude to my parents, to my two sisters, to Alina and her parents for the patience and understanding over the time during which this work has been developed. v Homological properties of transitive Lie algebroids via Sullivan models Abstract D. Sullivan considered a new model for the underlying cochain complex of classical cohomologies with rational coefficients for arbitrary simplicial spaces which gives an isomorphism with classical rational cohomologies. This new model is determined by the Rham complex of all rational polynomial forms defined on the simplicial complex triangulating the space. Other cell-like constructions of cochain complexes which induce isomorphisms in cohomology with classical cohomologies had been already presented by H. Whitney. Recent ideas developed by K. Mackenzie and J. Kubarski concerning Lie algebroids are applied to a generalization of a cell-like construction for transitive Lie algebroids over combinatorial manifolds. Namely, given a compact smooth manifold M, smoothly triangulated by a simplicial complex K, and a transitive Lie algebroid Aon M, we define a piecewise smooth form on Ato be a family ω= (ω∆)∆∈Kof differential forms such that, for each simplex ∆∈K,ω∆∈Ω∗(A!! ∆; ∆) is a smooth form defined on the Lie algebroid A!! ∆, restriction of Ato the simplex ∆, satisfying the compatibility condition under restrictions of the form ω∆to all faces of the simplex ∆, that is, if ∆0is a face of ∆, then λ∗ ∆,∆0(ω∆) = ω∆0, in which λ∆,∆0denotes the canonical Lie algebroid morphism induced by the inclusion ∆0,→∆. The set Ω∗(A;K) of all piecewise smooth forms defined on Ais a commutative cochain algebra. We define a map Ω∗(A;M)→Ω∗(A;K) which assigns, to each smooth form ω∈Ω∗(A;M), the piecewise smooth form ξ= (ξ∆)∆∈K∈Ω∗ ps(A;K) defined by the condition ξ∆=λ∗ M,∆(ω) for each simplex ∆ ∈K. This map is a natural morphism of cochain algebras. In this thesis, we prove that, for compact combinatorial manifolds, the cohomology of this construction is isomorphic to the Lie algebroid cohomology of A. We apply this isomorphism in piecewise invariant cohomology of Lie algebroids and piecewise de Rham cohomology of locally trivial Lie groupoids. vii Contents Acknowledgments ................................. v Abstract ...................................... vii Introduction 1 1 Preliminaries on Lie algebroids 5 1.1 Restriction of transitive Lie algebroids . . . . . . . . . . . . . . . . . . . . 5 1.2 Smooth forms and cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 TrivialLiealgebroids.............................. 29 2 Piecewise smooth cohomology 39 2.1 Complex of Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Algebra of piecewise smooth forms . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Mayer-Vietoris sequence for piecewise smooth forms . . . . . . . . . . . . . 49 2.4 Generalization of piecewise smooth context . . . . . . . . . . . . . . . . . . 55 2.5 Sheaves of piecewise smooth forms . . . . . . . . . . . . . . . . . . . . . . 58 3 Main theorems 65 3.1 Mishchenko’stheorem ............................. 65 3.2 Piecewise invariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3 Piecewise de Rham cohomology of Lie groupoids . . . . . . . . . . . . . . . 84 Conclusion 91 ix bundle to M. We recall that a Lie algebroid on base Mis a vector bundle π:A −→ M on Mequipped with a vector bundle morphism γ:A −→ TM, called anchor of A, and a structure of Lie algebra on the vector space Γ(A) of sections of Asuch that the induced map γΓ: Γ(A)−→Υ(M) is a Lie algebra homomorphism and the action of the algebra C∞(M) on Γ(A) satisfies the natural condition: [ξ, fη] = f[ξ, η]+(γΓ(ξ)·f)η for each ξ,η∈Γ(A) and f∈ C∞(M). The Lie algebroid Ais called transitive if the anchor γis surjective. As usually, when there is no ambiguity, we drop the anchor map and the Lie bracket in the notation of the Lie algebroid but, when it is needed to emphasize them, we write (A,[·,·], γ) for denoting this structure. Definition (Morphism of Lie algebroids). Let Mand Ntwo smooth manifolds and (A,[·,·], γ) and (B,[·,·], δ) Lie algebroids over MeNrespectively. A morphism of Lie algebroids from (A,[·,·], γ) to (B,[·,·], δ) consists of a pair of mappings (ψ, ϕ), with ψ:A −→ B and ϕ:M−→ N, such that (ψ, ϕ) is a vector bundle morphism satisfying the equality δ◦ψ=T(ϕ)◦γ, in which T(ϕ) : TM −→ TN means the tangential of ϕ, and preserving the Lie bracket condition for ψ-decompositions, that is, for each ξ, η ∈Γ(A) with decompositions ψ◦ξ= m X i=1 ai⊗ξiψ◦η= m X j=1 bj⊗ηj ξi,ηj∈Γ(B), then ψ◦[ξ, η]A=X i,j aibj⊗[ξi, ηj]B+ m X j=1 (γ◦ξ)(bj)⊗ηj− m X i=1 (γ◦η)(ai)⊗ξj We notice that, when M=N, a simple characterization for a vector bundle morphism between two Lie algebroids on Mto be a Lie algebroid morphism can seen in [7] or [10]. Namely, if (A,[·,·], γ) and (B,[·,·], δ) are Lie algebroids over the same smooth manifold M, then a vector bundle morphism ψfrom Ato Bis a Lie algebroid morphism if, and only if, γ=ψ◦δand the induced map ψΓ: Γ(A)−→ Γ(B) is a Lie algebra morphism. 6 We give some examples of Lie algebroids and algebraic constructions in Lie algebroids. Other examples of Lie algebroids are given in section 3. Example 1 (Lie algebras). Any real finite dimensional Lie algebra gover a one-point space M={∗} (so C∞(M) = R) with anchor equal to zero is a totally intransitive Lie algebroid on M. Any Lie algebra morphism between two Lie algebras is a Lie algebroid morphism for this structure of Lie algebroid. Example 2 (Tangent Lie algebroid). If Mis a smooth manifold then TM is a Lie algebroid on M. The anchor map is the identity map of TM, and the Lie bracket is the usual Lie bracket of vector fields. This is called the tangent Lie algebroid of M. The anchor map γ:A−→ TM is a Lie algebroid morphism from A to the tangent algebroid of M. Let Fbe a regular foliation of M. The tangent Lie algebroid of Fis, by definition, the vector subbundle of TM consisting of the tangent spaces to F, with the usual Lie bracket of vector fields tangent to F, and the inclusion map as the anchor. Conversely, if Ais a Lie algebroid on M, whose its anchor map γis injective, then, setting Ex=γx(Ax) for each x∈M, we obtain a vector subbundle Eof TM defining a foliation of M, and the tangent Lie algebroid of this foliation is isomorphic to the Lie algebroid A(see [7]). Example 3 (Trivial Lie algebroid). Let gbe a real finite dimensional Lie algebra and Ma smooth manifold and consider the trivial vector bundle M×g. On the Whitney sum TM ⊕(M×g) = TM ×M(M×g) we define an anchor map γ:TM ⊕(M×g)−→ TM by taking γto be the projection of TM ⊕(M×g) on TM and a Lie bracket on Γ(TM ⊕(M×g)) by setting [X, u),(Y, v)] = ([X, Y ]T M , X(v)−Y(u)−[u, v] for X, Y ∈TM and u, v :M−→ gsmooth maps. Then, TM ⊕(M×g) is a transitive Lie algebroid on Mand called the trivial Lie algebroid on Mwith structure algebra g. 7 Example 4 (Lie algebroid product). Let Mand Nbe two smooth manifolds and (A,[·,·]A, γ) and (B,[·,·]B,bγ) Lie algebroids over Mand Nrespectively. The product of the Lie algebroids (A,[·,·]A, γ) and (B,[·,·]B,bγ), denoted by A×B, is the vector bundle product A × B over M×N, in which the anchor is γ×bγand the Lie bracket is defined in the following way: for each ξ= (ξ1, ξ2) and η= (η1, η2)∈Γ(A×B) [ξ, η]A×B = ([ξ, η]1,[ξ, η]2)∈Γ(A×B) where [ξ, η]1 (x,y)= [ξ1(−, y), η1(−, y]A(x) + bγ(ξ2(x,y))(η1(x, −)) −bγ(η2(x,y))(ξ1(x.−)) and [ξ, η]2 (x,y)= [ξ2(−, y), η2(−, y]A(x) + γ(ξ1(x,y))(η2(x, −)) −γ(η1(x,y))(ξ2(x.−)) The projections πA:A × B −→ A and πB:A × B −→ B are morphisms of Lie algebroids. Consider now a real finite dimensional Lie algebra gand the tangent Lie algebroid TM. The Lie algebra gis a Lie algebroid over a one-point space N={∗}. We can take the product of Lie algebroids TM ×g, which is defined over M≃M×N. On the other hand, we can consider the trivial Lie algebroid TM ⊕(M×g) over Mand we easily see that the map F:TM ×g−→ TM ⊕(M×g) given by F(x, u, v) = (x, u, x, v) is a (strong) isomorphism of Lie algebroids. Henceforth, we will identify both Lie algebroids. Example 5 (Lie algebroid of covariant differential operators). Let Mbe a smooth manifold and π:E−→ Ma vector bundle on M. For each x∈M, denote by A(E)xthe vector space of all linear maps ψ: Γ(E)−→ Exsuch that there exists a vector u∈TxMsatisfying the equality ψ(fξ) = f(x)ψ(ξ)+(u·f)xξx for all f∈C∞(M) and ξ∈Γ(E). The vector uis unique and so we can define a map γ:G x∈M A(E)x−→ TM 8 We denote by A(E) the disjoint union Fx∈MA(E)x. We define a Lie bracket on the space of the sections of A(E) locally as follows. Fix a local trivialization ϕ:π−1(U)−→ U×F of the vector bundle E, in which Uis an open subset of Mand Fis the fibre type of E. Let gl(F) be the Lie algebra of Fand, for each ξ∈Γ(E), the map ξϕ:U−→ Fdefined by ξϕ(x) = ϕx(ξx), in which ϕx:Ex−→ Fis the linear map induced by ϕ. It can be seen in [7], [5], [8] that the map ϕ:TU ×gl(F)−→ A(E)Udefined by ϕ(u, g)(ξ) = (ϕx)−1(u·ξϕ+ (g◦ξϕ(x)) is bijective. Hence, the Lie bracket and the anchor of the trivial Lie algebroid TU ×gl(F) can be carried to A(E)Uand the space A(E) becomes a transitive Lie algebroid on M, which is denoted by D(E) and called the Lie algebroid of covariant differential operators on the space of sections of the vector bundle E(see [5], [7], [8], [10], [11], [16], [18]). Example 6 (Lie algebra bundles). A Lie algebra bundle over a smooth manifold M is a vector bundle π:K−→ Mequipped with a section [·,·] of the vector bundle V2(K, K) such that, for each x∈M, (Kx,[·,·]x) is a Lie algebra and Kadmits an atlas {ψj:Uj×g−→ π−1(U)}(gis a Lie algebra) in which each ψjxis a Lie algebra isomorphism. A Lie algebra bundle is a totally intransitive Lie algebroid (see [10]). Let π:E−→ Mbe a vector bundle on a smooth manifold M. Then, the vector bundle End(E) = L(E;E), whose fibres, at each point x∈M, are the vector spaces L(Ex;Ex), is a Lie algebra bundle (see [10]). Next example is not used in this work but we include it since it has been a crucial example on the development of integrability theory of Lie algebroids Example 7 (Weinstein’s transformation algebroid). Suppose that we have an action µ:g−→ Υ(M) of a Lie algebra gon a smooth manifold M. Then, we can associate to this action a Lie algebroid which is called the corresponding transformation algebroid 9 and defined as follows. The vector bundle underlying this transformation algebroid is the trivial bundle g×M−→ M. The anchor map γ:g×M−→ TM is defined by γ(X, x) = µ(X)(x) for each X∈gand x∈M, and the Lie bracket on the sections of g×M, considered as maps from Mto g, is defined as [ξ, η](z) = [ξ(z), η(z)] + (µ(ξ(z)))z(η)−(µ(η(z)))z(ξ) This algebroid is usually denoted by gnM. We are going to handle restriction of Lie algebroids. We consider first the case in which the restriction is made to a open subset of the base. We are going to see that the restriction of a Lie algebroid to an open subset of base coincides with the restriction of the underlying vector bundle to that open subset. We note that, for restrictions to open subsets, we do not need the Lie algebroid to be transitive. We begin with the local property of the Lie bracket. Proposition 1.1.1. Let Mbe a smooth manifold and (A,[·,·], γ) a Lie algebroid on M. Let Ube an open subset of Mand X, Y ∈Γ(A) such that Yvanishes on U. Then, the Lie bracket [X, Y ] vanishes on U. Proof. Let x0∈U. Take the closed subset M\Uand the open subset M\F, where F={x0}. Obviously M\U⊆M\Fand then there exists a smooth function g∈C∞(M) such that g(M\U) = {1}and supp g⊆M\F. If x /∈Uthen g(x) = 1 and so g(x)Yx=Yx. The restrictions of both sections to Ualso coincide and then gY =Y. Hence, we have [X, Y ](x0)=[X, gY ](x0) = g(x0)[X, Y ](x0)+(γΓ(X)·g)(x0)Y(x0)=0 because g(x0) = 0 and Yx0= 0. Therefore, the Lie bracket [X, Y ] vanishes for all points of the open subset U. Although next proposition is not used in this section, we will note it here as an application of the local property of Lie bracket. 10 Proposition 1.1.2. Let Mbe a smooth manifold and (A,[·,·], γ) a Lie algebroid on M. For each x∈M, denote by gxthe kernel of the linear map γx:Ax−→ TxM. Then, gxhas a natural structure of Lie algebra defined by [u, v]=[X, Y ](x) with u, v ∈gxand X, Y ∈ΓAsuch that Xx=uand Yx=v. Proof. We are going to check that the definition is coherent. Let b Y∈Γ(A) be a section such that b Yx=v. Fix a local frame of the vector bundle (A, π, M), say us, (s1,· · · , sk) defined on an open subset Uof M. Then, there are smooth functions f1,· · · ,fk∈C∞(U) such that Y/U −b Y/U =Pjfjsj. By using partition of unity, we can extend all maps fj and sections sjto all of Mand we obtain e f1,· · · ,e fk∈C∞(M) and es1,· · · ,esk∈Γ(A) such that the restrictions of e fjand esjto Ucoincide with fjand sjrespectively. Denote by Zthe section of Adefined by Z=Pje fjesj. By the local property of Lie bracket and the third condition of the definition of Lie algebroid we have [X, Y −b Y](x)=[X, Z](x) = X je fj(x)[X, esj](x)+(γΓ(X)·e fj)(x)esj(x) = 0 because e fj(x) = 0 and (γ◦X)(x) = 0.  Let Mbe a smooth manifold and (A,[·,·]A, γ) a transitive Lie algebroid over M. Consider the vector bundle Ker γ. The Lie bracket structure on Γ(A) induces a bracket structure on Γ(Ker γ) and so Ker γis a totally intransitive Lie algebroid on M, called adjoint Lie algebroid of A. We notice that a totally intransitive Lie algebroid may not be a Lie algebra bundle. However, Ker γis a Lie algebra bundle on M(see [7], [10]). Proposition 1.1.3. Let (A,[·,·], γ) be a Lie algebroid on a smooth manifold M. Let Ube an open subset of Mand consider the vector bundle AU, restriction of Ato U. Then, the Lie bracket [·,·] : Γ(A)×Γ(A)−→ Γ(A) 11 restricts to a Lie bracket [·,·]AU: Γ(AU)×Γ(AU)−→ Γ(AU) Proof. We want define a Lie bracket on Γ(AU). For that, fix two sections X, Y ∈Γ(AU) and x∈U. We can take two sections e X,e Y∈Γ(A) and an open subset Vof Msuch that x∈V⊆V⊆Uand X/V =e X/V and Y/V =e Y/V We define [X, Y ]AU(x) = [ e X, e Y](x) This definition is coherent because, if we take other section b X∈Γ(A) and an open subset Wof Msuch that x∈W⊆W⊆Uand XW=b XW, then, taking the difference e X−b X and applying the proposition 1.1.1 to the open subset V∩W, we have that [ e X, Y ](x) = [b X, Y ](x). Therefore, a bracket on Γ(AU) is well defined and we easily can see that [·,·]AU satisfies the conditions of Lie bracket on Γ(AU).  Next, if (A,[·,·], γ) is a Lie algebroid on a smooth manifold Mand Uan open subset of M, we want to define a structure of Lie algebroid in the vector bundle AUdefined on U. From previous proposition, we already have a Lie bracket on Γ(AU). We only need to define an anchor map. Take then b∈Uand u∈ Ab. We have that γ(u)∈(TM)b. Since Uis an open subset of M, (TU)b= (TM)band so γ(u)∈(TU)b. Hence, we may restrict the anchor γ:A −→ TM to a map γU:AU−→ TU. Obviously, γUis a vector bundle morphism. It remains to check that the morphism induced by γUon the sections of AU is a Lie algebra morphism. Take X, Y ∈Γ(AU) and b∈U. Fix e X, e Y∈Γ(A) such that e X/V =X/V and e Y/V =Y/V where Vis an open subset of Msuch that b∈V⊆V⊆U. Firstly, we remark that γ◦e Xand γ◦e Yare extensions of γU◦Xand γU◦Yrespectively. Consequently, we have that [γU◦X, γU◦Y](b) = [γ◦e X, γ ◦e Y](b) 12 and so, (γ/U ◦[X, Y ])(b) = γ/U ([X, Y ](b)) = =γ([X, Y ](b)) = γ([ e X, e Y](b)) = = (γ◦[e X, e Y])(b)) = [γ◦e X, γ ◦e Y](b) = = [γU◦X, γU◦Y](b) Also, for each g∈C∞(U), eg∈C∞(M) such that eg/V =g/V [X, gY ](b) = [ e X, ege Y](b) = =eg(b)[ e X, e Y](b) + ((γ◦e X)·eg)(b)e Y(b) = =g(b)[X, Y ](b) + ((γ/U ◦X)·g)(b)Y(b) We have proved the following proposition. Proposition 1.1.4. The vector bundle AU, with the structures above, becomes a Lie algebroid on U. This Lie algebroid will denoted by (AU,[·,·]AU, γAU) or simply by AU. Moreover, if (e i, i) denote the pair of smooth maps i:U−→ Mand e i:AU−→ Adefined by the inclusions, (e i, i) is a morphism of Lie algebroids, which is fibrewise injective. We consider now the case in which the restriction is made, non necessarily to an open submanifold, but to a general submanifold of the base. In this case, the Lie algebroid restricted to a submanifold may not coincide with the restriction of the underlying vector bundle to that submanifold, but it will be a vector subbundle of the underlying vector bundle restricted to that submanifold, which is given by image inverse of Lie algebroids through the inclusion of submanifolds. The transitivity of Lie algebroids will be needed to show that inverse image always exists for any smooth embedding of manifolds. We begin by noting brief considerations on the construction of image inverse. Let Mand Nbe smooth manifolds and ϕ:N−→ Ma smooth map. Suppose that (A,[·,·], γ) is a transitive Lie algebroid on M. Let π:A −→ Mdenote the vector bundle 13 underlying the Lie algebroid Aand πTM and πTN the canonical projections of the tangent bundles TM and TN respectively. The anchor γ:A −→ TM defines Aas a vector bundle on TM. In order to complete the following diagram A   yγ TN T ϕ −→ TM take the vector bundle ((Tϕ)∗A,bγ, TN), inverse image of the vector bundle (A, γ, TM) by the smooth map Tϕ, where bγdenotes the canonical projection (Tϕ)∗A −→ TN. We notice that the vector bundle (Tϕ)∗Aexists because γis surjective. Obviously, (Tϕ)∗Ais also a vector bundle over N, in which its projection is the composition of the projection bγ with the canonical projection πTN . We obtain the following commutative diagram (Tϕ)∗A// bγ  A γ π ~~ TN T ϕ // πN  TM πM  Nϕ//M Moreover, the vector bundle (Tϕ)∗A, πTN ◦bγ, Nis N-isomorphic to a vector subbundle of the Whitney sum TN ⊕ϕ∗A, whose sections of this vector subbundle are the sections s= (X, ξ) : N−→ TN ⊕ϕ∗A (X∈Γ(TN) and ξ∈Γ(ϕ∗A)) of the vector bundle TN ⊕ϕ∗Acharacterized by the equality T(ϕ)(X) = γ(Φ ◦ξ), where Φ : ϕ∗A −→ A stands for the canonical vector bundle morphism defined by Φ(x, u) = u. The notable fact is that the vector bundle (Tϕ)∗A, πTN ◦bγ, Ninherits a natural structure of transitive Lie algebroid on N. We note this structure of Lie algebroid on next proposition. The details of the proof can be found in ([7],[10]). 14 Proposition 1.1.5. Keeping the same hypothesis and notations as above, if Ais a transitive Lie algebroid on M, then the vector bundle (Tϕ)∗A, πTN ◦bγ, Ncarries a natural structure of transitive Lie algebroid on N, in which the canonical projection bγ: (Tϕ)∗A −→ TN is the anchor map and the Lie bracket is defined in the following way: Fix a local frame (s1,· · · , sk) of the vector bundle (A, π, M) defined on an open subset Uof M. Take now two sections (X, ξ), (Y, η)∈Γ(Tϕ)∗A, where X, Y ∈Γ(TN) and ξ, η ∈Γ(ϕ∗A). Then, on the open subset V=ϕ−1(U), we have the decompositions ξ/V =Pifi(si◦ϕ/V ) and η/V =Pjgj(sj◦ϕ/V ) with fi, gj∈C∞(V). Define a Lie bracket by setting [(X, ξ),(Y, η)]/V = =[X, Y ]/V ,X i,j figj[si,sj]◦ϕ/V +X j (X·gj)(sj◦ϕ/V )−X i (Y·fi)(si◦ϕ/V ) Thus, the pair of mappings (ϕ!!, ϕ), in which ϕ!! : (Tϕ)∗A −→ A is the smooth map defined by ϕ!!(X, a) = a, is a morphism of Lie algebroids. Definition (Inverse image of transitive Lie algebroid). Keeping the same hypothesis and notations as above, the vector bundle (Tϕ)∗A, πT N ◦bγ, Nequipped with this structure of Lie algebroid is called the inverse image Lie algebroid of Aby the map ϕand denoted by (ϕ!!A,bγ, [·,·]ϕ!!A). From now on, we write simply ϕ!!Ainstead of (ϕ!!A,bγ, [·,·]ϕ!!A), dropping the anchor bγand the Lie bracket [·,·]ϕ!!A. The smooth map ϕ!! is called the induced map by ϕand the Lie algebroid morphism (ϕ!!, ϕ) is called the canonical Lie algebroid morphism of an induced Lie algebroid. Example. Let gbe a finite dimensional Lie algebra. The Lie algebra gis a Lie algebroid over a one-point space M={∗}. Let Nbe a smooth manifold and ϕ:N−→ M the constant map. Then, (Tϕ)∗gis equal to TN ⊕(TN ×g) and we easily see that the anchor and the Lie bracket of the Lie algebroid (T(ϕ))∗gcoincides with the ones of the trivial Lie algebroid TN ⊕(TN ×g). We define now restriction of a transitive Lie algebroid to a general submanifold of the 15 Next, we recall the definition of exterior derivative. Let Mbe a smooth manifold and Aa Lie algebroid on M, with anchor γ:A −→ TM and Lie bracket [·,·] on Γ(A). We first consider the algebra Ω0(A;M) = C∞(M). Let f∈C∞(M). We can define the smooth 1-form d(f) : M−→ 1 ^(Γ(TM), C∞(M)) d(f)(X) = Dfx(Xx) = X·f for each X∈Υ(M). Hence, we define d(f)∈V1(Γ(A), C∞(M)) by d(f)(X) = (γ◦X)·f for each X∈Γ(A). Now, for each p≥1 we define dp: Ωp(A;M)−→ Ωp+1(A;M) dpω(X1, X2,···, Xp+1) = p+1 X j=1 (−1)j+1(γ◦Xj)·(ω(X1,···,c Xj,···, Xp+1)) + +X i<k (−1)i+kω([Xi, Xk], X1,···,c Xi,···,c Xk,···, Xp+1) for ω∈Ωp(A;M) and X1, X2,···, Xp+1 ∈Γ(A). The family of differential operators d∗= (dp)p≥0defines, on the commutative graded algebra Ω∗(A;M), a structure of differential graded algebra. Hence, Ω∗(A;M) becomes a commutative cochain algebra, which is defined over R. Definition (Lie algebroid cohomology). Keeping the same hypothesis and notation as above, the Lie algebroid cohomology space of Ais the cohomology space of the algebra Ω∗(A;M) equipped with the structures defined above. This cohomology space is denoted by H∗(A;M). In view of discuss of extensions of smooth forms in Lie algebroids, we recall the definition of inverse image of a smooth form. Let Mand Ntwo smooth manifolds and Aand B Lie algebroids on Mand Nrespectively. Let λ= (F, f) be a morphism of Lie algebroids 22 defined by the smooth maps F:A −→ B and f:M−→ N. If ωis a smooth form on B of degree p, we can consider a smooth form of degree pon A, denoted by λ∗ω, and defined by λ∗ωx(v1, v2,···, vp) = ωf(x)(F(v1), F(v2),···, F(vp)) for each x∈Mand v1,v2, . . . vp∈ Ax. The form λ∗ωis called the pullback or inverse image of ωby the morphism λ. Thus, for each p≥0, there is a map λ∗p: Ωp(B;N)−→ Ωp(A;M) ω−→ λ∗ω The family λ∗= (λ∗p)p≥0is a morphism of cochain algebras. Therefore, we have a contravariant functor from the category of Lie algebroids to the category of cochain algebras. For details, see [4], [6] and [7]. Definition (Restriction of smooth forms). Let Mbe smooth manifold and Aa Lie algebroid on M. Let ϕ:N ,→Mbe an embedded smooth manifold of Mand consider the canonical morphism λ= (ϕ!!, ϕ) in which ϕ!! :A!! N−→ A is defined by ϕ!!(X, a) = a. Consider the cochain algebras Ω∗(A;M) and Ω∗(A!! N;N) and the morphism λ∗: Ω∗(A;M)−→ Ω∗(A!! N;N) ω−→ λ∗ω For each smooth form ω∈Ωp(A;M), the form λ∗(ω)∈Ωp(A!! N;N) is called restriction of ωto Nand denoted by ω!! Nor simply by ωN, when there is no ambiguity with the restriction of ωto the restricted vector bundle AN. In subsequent sections, the smooth embedding ϕ:N ,→Mwill be often denoted by ϕM,N :N ,→Mand the homomorphism λ∗: Ω∗(A;M)−→ Ω∗(A!! N;N) by ϕA M,N . The homomorphism ϕA M,N will be called the homomorphism of cochain algebras generated by the inclusion ϕM,N . We will notice here some considerations on extension of smooth forms. We begin first with some remarks on extension of smooth forms on vector bundles. After those remarks, 23 we then state a proposition concerned to extension of smooth forms on Lie algebroids. In the case of vector bundles, we are going to divide in two cases. The first case is very basic and is the one in which we have a vector bundle (E, π, M) defined over a smooth manifold M,Na submanifold of Mwhich is closed subset of M, and we want to extend smooth forms belonging to the restricted vector bundle EN. In this case, for each x∈N, the fibres of ENare the same as the fibres of E. The second case is the one in which we have a vector subbundle (F, π0, N) of a given vector bundle (E, π, M). In this case, the fibres of Fare vector subspaces of the fibres of Eand, for extension of smooth forms, we need to fix a Riemannian structure. Remark 1. Let (E, π, M) be a vector bundle defined over a smooth manifold Mof dimension nand Na submanifold of Mwhich is closed subset of Min the topological sense. Let ω∈Ωp(EN;N) be a smooth form on the restricted vector bundle ENdefined over Nand suppose there exist an open subset Uof Msuch that N⊂Uand a smooth form eω∈Ωp(EU;U) such that eω/N =ω. Then, the form ωextends to a smooth form ξ∈Ωp(E;M). Proof. We can consider a partition of unity ϕ1,ϕ2: [0,1] −→ Mcorrespondent to the open covering of Mmade by M\Nand Usuch that ϕ1(x) + ϕ2(x) = 1 ∀x∈M, supp ϕ1⊂M\Nand supp ϕ2⊂U. Define then ξ∈Ωp(E;M) by ξx=              ϕ2(x)eωxif x∈U 0∈Vp(Ex;R) if x∈M\supp ϕ2 The form ξis then a smooth extension of ωto the whole M. We can improve last remark and obtain the following. Remark 2. Let (E, π, M) be a vector bundle defined over a smooth manifold Mof dimension nand Na submanifold of Mwhich is closed subset of Min the topological 24 sense. Let ω∈Ωp(EN;N) be a smooth form on the restricted vector bundle ENdefined over Nand suppose there is an open subset Uof M, with N⊂U, such that Uis the domain of a chart of the vector bundle E (π, ψ) : π−1(U)−→ U×Rn Then, the form ωextends to a smooth form ξ∈Ωp(EU;U). Proof. Let (bπ, b ψ) : bπ−1(U)−→ U× ∧p(Rn) be the corresponding chart of the vector bundle (VpE, bπ, M) and consider the smooth map θ=b ψ◦ω:N−→ Vp(Rn). Let e θ:U−→ Vp(Rn) be a smooth extension of θ. Then, the form ξ∈Ωp(EU;M) defined by ξx= (bπ, b ψ)−1(x, e θ(x)) is a smooth extension of ωto the open subset U. We can again improve last remark and obtain the following. Remark 3. Let (E, π, M) be a vector bundle defined over a smooth manifold Mof dimension nand Na submanifold of Mwhich is closed subset of Min the topological sense. Let ω∈Ωp(EN;N) be a smooth form on the restricted vector bundle ENdefined over N. Then, the form ωextends to a smooth form ξ∈Ωp(E;M). Proof. For each x∈N, let Uxan open neighborhood of xin Mwhere it is defined a chart of the vector bundle. We can apply last proposition to the closed submanifold N∩Ux and then there is a smooth form ξx∈Ωp(EUx;Ux) such that ξx/N∩Ux=ω/N∩Ux. We have N⊂Sx∈NUx. Denote by U∗the open subset U∗=M\N. Then (Ux)x∈N∪U∗is an open covering of M. Let (Vj)j∈Jbe a locally finite refinement of the covering (Ux)x∈N∪U∗and consider a partition of unity (ϕj)j∈Jsubordinated to the covering (Vj)j∈J. Define now the index sets J∗={j∈J:Vj⊂M\N}and J0=J\J∗ 25 We are going to check the following: •N⊂Sj∈J0Vj. •For each j∈J0, there exists x∈Nsuch that Vj⊂Ux. For the first statement, if x∈Nthen there is j∈Jwith x∈Vj. If jcould belong to J∗, we would conclude x∈Vj⊂U∗=M\Nand so x /∈N. For the second statement, given j∈J0, we have that Vj⊂Uyfor some y∈N∪U∗. Since j /∈J∗then Vj*M\N and so Vj⊂Uxfor some xwhich doesn’t belong to U∗. Next, we can fix, for each j∈J0, xj∈Nwith Vj⊂Uxj. Denote the smooth form ξxj/Vj∈Ωp(EVj;Vj) by ξj. Finally, define ξ∈Ωp(E;M) by ξ=X j∈J0 ϕjξj where ϕjξjis defined by ϕjξj(x) =              ϕj(x)ξ(x) if x∈Vj 0∈Vp(Ex;R) if x∈M\supp ϕj We have that ξis well defined since the covering is locally finite. Moreover, the form ξis smooth extension of ω. We consider now the case in which we have smooth forms defined on a vector subbundle of a given vector bundle. Let (E, π, M) be a vector bundle defined over a smooth manifold M. We recall that a vector bundle (F, π0, N) is called vector subbundle of (E, π, M) if F is a submanifold of E,Nis a submanifold of M,π(F)⊂Nand π0=π/F :F−→ Nand, for each x∈N,Fxis a vector subspace of Ex. Fix Riemannian structure on the vector bundle (E, π, M). For each x∈N, we have Ex=Fx⊕F⊥ xand denote ψx:Ex−→ Exthe orthogonal projection over the fibre Fx. Then, the map ψ:E/N −→ Fdefined by ψxfor each x∈Nis a N-morphism of vector bundles from (E/N , π, N) to (F, π0, N). Using this discuss, we get immediately the following proposition. 26 Remark 4. Let (E, π, M) be a vector bundle on a smooth manifold M,Na submanifold of Mand (F, π0, N) a vector subbundle of (E, π, M). Let ω∈Ωp(F;N) be a smooth form. Then, the form ωextends to a smooth form ξ∈Ωp(EN;N). Proof. Using the map ψabove, the form ψ∗ωis the required extension.  We consider now the case of extensions smooth forms in Lie algebroids. We begin with two basic remarks. Remark 5. Let Mbe a smooth manifold, Uan open subset of Mand ϕ:U−→ Mthe inclusion map. Let (A,[·,·], γ) be a transitive Lie algebroid on Mand consider the Lie algebroid (AU,[·,·]AU, γU) on Uconstructed in the proposition 1.1.4 as well the Lie algebroid (ϕ!!A,[·,·]ϕ!A,bγ) on Uconstructed in the proposition 1.1.5. Then, for each p≥0, Ωp(AU;U) and Ωp(A!! U;U) are isomorphic. Namely, for the maps ψ:AU−→ A!! U defined by ψ(a) = (γ(a), a) and b ψ:ϕ!!A −→ AUdefined by b ψ((x, u), a) = a, the map ψ∗p: Ωp(A!! U;U)−→ Ωp(AU;U) is the inverse of ( b ψ)∗p: Ωp(AU;U)−→ Ωp(A!! U;U). Remark 6. Let Abe a transitive Lie algebroid on a smooth manifold Mand ϕ:N ,→Man embedded submanifold such that Nis a closed subset in Min the topological sense. Let ϕ!! :ϕ!A −→ A be the canonical morphism defined by ϕ!!(X, a) = a and consider Im ϕ!! equipped with the natural structure of transitive Lie algebroid on N given in the proposition 1.1.9. The map ϕ!! is a N-isomorphism of Lie algebroids between ϕ!!Aand Im ϕ!!. Hence, the map ϕ!! induces an isomorphism between Ωp(A!! N;N) and Ωp(Im ϕ!!;N). Let us notice now a proposition concerning extensions of smooth forms in Lie algebroids. Proposition 1.2.1. Let Mbe a smooth manifold and ϕ:N ,→Man embedded submanifold such that Nis a closed subset in Min the topological sense. Let Abe a transitive Lie algebroid on Mand consider the canonical morphism λ= (ϕ!!, ϕ) in which ϕ!! :A!! N−→ A is defined by ϕ!!(X, a) = a. Consider the cochain algebras Ω∗(A;M) and 27 Ω∗(A!! N;N). Then, the cochain algebra morphism λ∗: Ω∗(A;M)−→ Ω∗(A!! N;N) ω−→ λ∗ω is surjective. Proof. Given a smooth form eω∈Ωp(A!! N;N), we can define a smooth form bω∈Ωp(Im ϕ!!;N) by bω(ξ1, . . . , ξp) = eω((γ◦ξ1, ξ1),...,(γ◦ξ1, ξp)) where γis the anchor of A. We apply the remarks above to the form bωand the result follows since any smooth extension ω∈Ω∗(A;M) of bωsatisfies λ∗(ω) = eω. Previous propositions are used in the statement of the Mayer-Vietoris sequence in Lie algebroids. We recall it briefly. The details can be seen in [6]. We suppose that Uand V are open subsets of Msuch that M=U∪Vand consider the diagrams where all maps are the inclusions U∩V i ||x x x x x x x x xj "" F F F F F F F F FAU∩V b i {{w w w w w w w wb j ## G G G G G G G G U k## F F F F F F F F FV l {{x x x x x x x x xAU b k## G G G G G G G G GAV b l {{w w w w w w w w w MA These inclusions induce, by inverse image, the following cochain maps: i∗: Ω∗(AU;U)−→ Ω∗(AU∩V;U∩V)j∗: Ω∗(AV;V)−→ Ω∗(AU∩V;U∩V) k∗: Ω∗(A;M)−→ Ω∗(AU;U)l∗: Ω∗(A;M)−→ Ω∗(AV;V) which are defined by i∗(ω) = ω/U∩V,j∗(ω) = ω/U∩V,k∗(ω) = ω/U and l∗(ω) = ω/V . We denote by Ω∗(AU;U)×Ω∗(AV;V) the cochain complex product of Ω∗(AU;U) and Ω∗(AV;V) made from the cartesian product Ωp(AU;U)×Ωp(AV;V). Under these conditions, we have an short exact succession of cochain complexes 0−→ Ω∗(A;M)λ∗ −→ Ω∗(AU;U)×Ω∗(AV;V)δ∗ −→ Ω∗(AU∩V;U∩V)−→ 0 28 in which the maps λp: Ωp(A;M)−→ Ωp(AU;U)×Ωp(AV; ) δ: Ωp(AU;U)×Ωp(AV;V)−→ Ωp(AU∩V;U∩V) are defined by λp(ω)=(k∗ω, l∗ω) = (ω/U , ω/V ) and δp(α, β) = j∗β−i∗α= = β/U∩V−α/U∩V. Therefore, for the short exact succession mentioned above, it corresponds the long exact succession in cohomology · · · −→ Hp−1(A/U∩V;U∩V)∂p−1 −→ Hp(A;M) Hp(A;M)Hp(λ∗) −→ Hp(Ω∗(A/U ;U)×Ω∗(A/V ;V)) Hp(Ω∗(A/U ;U)×Ω∗(A/V ;V)) Hp(δ∗) −→ Hp(A/U∩V;U∩V) Hp(A/U∩V;U∩V)∂p −→ Hp+1(A;M)−→ · · · in which ∂∗is the connecting homomorphism. We recall that the cohomology space Hp(Ω∗(A/U ;U)×Ω∗(A/V ;V)) is a product of Hp(A/U ;U) and Hp(A/V ;V), with the projections naturally defined, and hence the space Hp(Ω∗(A/U ;U)×Ω∗(A/V ;V)) is isomorphic to the cartesian product Hp(A/U ;U)×Hp(A/V ;V) by the isomorphism ([ξ],[η]) −→ [(ξ, η)]. 1.3 Trivial Lie algebroids The aim of this section is to state a result concerning the triviality of transitive Lie algebroids over contractible manifolds. This result is a direct consequence of a deep result on actions of certain cohomology space on the set of operators extensions of Lie algebroids by Lie algebra bundles. This deep result is due to Mackenzie (see [10]). In view of the 29 statement of this result, some definitions and examples of Lie algebroids used in the study of this topic are given below. A complete description of these definitions and results can be found in Mackenzie’s book ”General Theory of Lie Groupoids and Lie Algebroids”. The paper ”Comparison of categorical characteristic classes of transitive Lie algebroid with Chern-Weil homomorphism” by Mishchenko and Xiaoyu contains an interesting summary of this study. Example 1 (Lie algebroid of covariant derivatives). Let Mbe a smooth manifold and π:E−→ Ma vector bundle on M. Denote by Φ(E) the Lie groupoid on Mmade by all linear isomorphism ξ:Ex−→ Eyfor each x,y∈M(see [10]). For each n≥1, let πL:Ln(E;E)−→ Mbe the vector bundle whose fibre at z∈Mis the vector space of all p-linear maps from Ez× · · · × Ezto Ez. The canonical action Φ(E)∗Ln(E;E)−→ Ln(E;E) of the Lie groupoid Φ(E) on the vector bundle πL:Ln(E;E)−→ Mis defined by ξ·ϕ=ξ◦ϕ◦(ξ−1× · · · × ξ−1)∈Ln(Ey;Ey) where x,y∈M,ξ:Ex−→ Eyis a linear isomorphism and ϕ∈Ln(Ex;Ex). A section η∈Γ(Ln(E;E)) is stable for this action if, for all x,y∈M, there is a linear isomorphism ξ:Ex−→ Eysuch that ξ·ηx=ηy. For a stable section η∈Γ(Ln(E;E)), the stabilizer subgroupoid of Φ(E) at ηis defined by {ξ∈Φ(E) : ξ·η(α(ξ)) = η(β(ξ))} in which α: Φ(E)−→ Mand β: Φ(E)−→ Mare the source and target projections of the Lie groupoid Φ(E). We notice that a section η∈Γ(Ln(E;E)) need not be stable. Nevertheless, for a Lie algebra bundle π0:K−→ Mon Mwith bracket [·,·]∈V2(K, K), the bracket [·,·] is a stable section for the action above restricted to the vector bundle V2(K;K). Hence, the stabilizer subgroupoid of Φ(K) at [·,·] is well defined and denoted by ΦAut(K) (see [10]). The Lie algebroid of ΦAut(K) is denoted by DDer(K) and its 30 elements are called covariant derivatives of K. The Lie algebroid DDer(K) is transitive (see [10], [18]). Example 2 (Adjoint Lie algebra bundle). Let Mbe a smooth manifold and π:K−→ Ma Lie algebra bundle with fibre type g. Consider the Lie subalgebra Der(g) of gl(g) made by the derivations of g. In [10], it can be seen that the Lie subalgebra Der(g) corresponds to a unique Lie algebra subbundle of the Lie algebra bundle End(K). This Lie algebra subbundle is denoted by Der(K) and its elements are called derivations of K. Thus, the image ad(K) of K, by the Lie algebra bundle morphism ad :K−→ Der(K) defined by adx:Kx−→ Der(Kx) for each x∈M, is a Lie algebra subbundle of Der(K), in which the fibre ad(K)xis an ideal of Der(Kx) (see [10]). Consequently, the Lie algebra bundle quotient Der(K)/ad(K) is well defined. The Lie algebra subbundle ad(K) is called the adjoint Lie algebra bundle of K. The Lie algebra bundle quotient Der(K)/ad(K) usually is denoted by Out (K). Example 3 (Lie algebroid quotient). Let Mbe a smooth manifold and Aa Lie algebroid over M, with anchor γ:A −→ TM and Lie bracket [·,·]Aon Γ(A). Consider the Lie algebra bundle Ker γ. An ideal of Ais a Lie algebra subbundle Kof Ker γsuch that, for all sections ξ∈Γ(A) and η∈ΓK, [ξ, η]Ais a section of K. In these conditions, one defines the Lie algebroid quotient of Aby Kas follows. Let Abe the vector bundle quotient A/K and γ:A −→ TM the map induced by γ. The Lie bracket in the space of the sections of Ais defined by [ξ+ ΓK, η + ΓK]A= [ξ, η]A+ ΓK for each ξ,η∈Γ(A). The Lie algebroid Ais transitive and usually denoted by A/K (see [10] or [11]). A particular case of Lie algebroid quotient is the following. Let π:K−→ Mbe a Lie algebra bundle on a smooth manifold Mand consider the transitive Lie algebroid DDer(K) of all covariant derivatives of a Lie algebra bundle K. The adjoint Lie algebra 31 Chapter 2 Piecewise smooth cohomology In this chapter, we deal with transitive Lie algebroids defined over simplices of a simplicial complex. We briefly discus a class of spaces for which the piecewise smooth cohomology spaces is defined, precisely the class of all complexes of Lie algebroids. Some facts concerning extension of piecewise smooth forms are presented. In what follows, all simplicial complexes considered are geometric and finite. Simplex means always closed simplex and each simplex can be represented as a convex body generated by its vertices. We shall denote the boundary of the simplex ∆ by bd ∆. We shall write s≺∆, if sis a face of the simplex ∆. The notation ϕ:s,→∆, where ϕis the inclusion, will also be used when sis a face of ∆. Many of the definitions and properties stated through the entire chapter can be found, on level of cell spaces, in [1], [9], [22], [24] and [26]. 2.1 Complex of Lie algebroids Consider a simplicial complex Kand two simplices ∆ and ∆0of Ksuch that ∆0is a face of ∆. Let A∆be a transitive Lie algebroid on ∆ and denote respectively by ϕ∆,∆0: ∆0,→∆ 39 and (ϕ∆,∆0)!! : (A∆)!! ∆0−→ A∆the inclusion and the induced maps. We recall that the Lie algebroid (A∆)!! ∆0denotes the restriction of A∆to ∆0, which is, by definition, the Lie algebroid (ϕ∆,∆0)!!A∆, inverse image of A∆by ϕ∆,∆0. Since ∆0is a compact embedded submanifold of ∆, by proposition 1.1.10, the Lie algebroid (A∆)!! ∆0can be identified to the Lie algebroid Im (ϕ∆,∆0)!!. Hence, for each x∈∆0, the fibre (A∆0)xis a vector subspace of the fibre (A∆)x. Definition (Complex of Lie algebroids). Let Kbe a simplicial complex. A complex of Lie algebroids on Kis a family A={A∆}∆∈Ksuch that, for each ∆ ∈K,A∆is a transitive Lie algebroid on ∆ and A∆0= (A∆)!! ∆0for each face ∆0of ∆, that is, the Lie algebroid restriction of A∆to ∆0is the Lie algebroid A∆0. Alternatively, a complex of Lie algebroids on Kmeans a family of transitive Lie algebroids defined on the simplices of Ksuch that the structures of Lie algebroids induced on each intersection of two simplices coincide. We give now some examples of complexes of Lie algebroids. Example 1 (Tangent complex). Let Kbe a simplicial complex. For each simplex ∆∈K, consider the tangent Lie algebroid T∆ defined over ∆. If ∆0is a face of ∆ then, by the proposition 1.1.7, (T∆)!! ∆0=T∆0and consequently we obtain a complex of Lie algebroids {T∆}∆∈K, which is called the corresponding tangent complex on K. Example 2 (Trivial complex). Let Kbe a simplicial complex and ga real finite dimensional Lie algebra. For each simplex ∆ ∈Kconsider the transitive Lie algebroid T∆⊕(∆ ×g). If ∆0is a face of ∆ then (T∆⊕(∆ ×g))!! ∆0=T∆0⊕(∆0×g) by proposition 1.1.8. We conclude that the family {T∆⊕(∆ ×g)}∆∈Kis a complex of Lie algebroids. This complex is called the trivial complex on K. Example 3 (Restriction of complexes of Lie algebroids). Let Kbe a simplicial complex and A={A∆}∆∈Ka complex of Lie algebroids on K. Let Lbe a simplicial 40 subcomplex of K. We can consider a new complex of Lie algebroids, defined over Land denoted by AL, given by restriction of Ato the simplices of L, that is, AL={A∆}∆∈L. Example 4 (Complex corresponding a combinatorial manifold). Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K, and Aa transitive Lie algebroid on M. Each simplex ∆ of Kis a compact embedded submanifold of M. We can consider the Lie algebroid restriction A!! ∆on the compact submanifold ∆. By proposition 1.1.7, if ∆0is a face of ∆, then (A!! ∆)!! ∆0=A!! ∆0and so we obtain a complex of Lie algebroids on Kwhich will be called the corresponding complex of Aover the simplicial complex K and denoted by {A!! ∆}∆∈K. Notation. Let K,Lbe two simplicial complexes and f:K−→ La simplicial map. For each simplex ∆ of Kgenerated by the vertices a0,a1,. . . ,ap, we denote by f(∆) the simplex of Lgenerated by f(a0), f(a1), . . . , f(ap) with redundances removed. Definition (Morphism of complexes of Lie algebroids). Let K,Lbe two simplicial complexes and A={A∆}∆∈Kand B={B∆0}∆0∈Ltwo complexes of Lie algebroids on Kand Lrespectively. Let f:K−→ Lbe a simplicial map and suppose that, for each ∆∈K, a morphism of Lie algebroids F∆:A∆−→ Bf(∆) over f/∆: ∆ −→ f(∆) is given. The family λ=(F∆)∆∈K, fis called a morphism of complex of Lie algebroids from Ato Bif this family is compatible with the restrictions, that is, if ∆ and ∆0are two simplices of K, with ∆0face of ∆, the restriction F∆/∆0:A∆0−→ Bf(∆0)coincides with the Lie algebroid morphism F∆0. This is equivalent to say that the diagram A∆0 F∆0// i∆0,∆  Bf(∆0) if(∆0),f(∆)  A∆ F∆//Bf(∆) is commutative, where i∆0,∆and if(∆0),f(∆) are the inclusions maps. Suppose that Tis other simplicial complex and g:L−→ Ta simplicial map. Let C={C∆00 }∆00∈Tbe a complex of Lie algebroids on T. For each ∆0∈L, let G∆0:B∆0−→ Cg(∆0)be a morphism of Lie 41 algebroids over g/∆0: ∆0−→ g(∆0) such that the family δ=(G∆0)∆0∈L, gis a morphism of complex of Lie algebroids from Bto C. Then, we define the composition δ◦λto be the family δ◦λ=(Gf(∆) ◦F∆)∆∈K, g ◦f. It is routine to verify that the family δ◦λis a morphism of complexes of Lie algebroids from Ato C. Example 1 (Identity morphism). Let Kbe a simplicial complex and A={A∆}∆∈K a complex of Lie algebroids on K. For each ∆ ∈K, let id∆:A∆−→ A∆be the Lie algebroid morphism identity. Then, the family λ=(id∆)∆∈K, idK, where idK:K−→ K is the simplicial morphism identity, is a morphism of complex of Lie algebroids. Example 2 (Inclusion morphism). Let Kbe a simplicial complex and suppose that A={A∆}∆∈Kis a complex of Lie algebroids on K. Let Lbe a simplicial subcomplex of Kand consider the complex of Lie algebroids AL={A∆}∆∈Lgiven by restriction of A to L(example 3, after the definition of complex of Lie algebroids). Consider the family λ=(id∆)∆∈L, iL,K), where iL,K :L−→ Kis the simplicial inclusion and id∆:Aα−→ Aα is the Lie algebroid morphism identity. Then, the family λis a morphism of complexes of Lie algebroids from ALto A. Next proposition is obvious. Proposition 2.1.1. The class of all complexes Lie algebroids and morphisms of complexes of Lie algebroids with the composition indicated as above is a category. This category is called the category of complexes of Lie algebroids. 2.2 Algebra of piecewise smooth forms Let A={A∆}∆∈Kbe a complex of Lie algebroids on a simplicial complex K. For each simplex ∆ of K, we denote, as done on previous sections, by Ω∗(A∆; ∆), d∗ ∆the cochain 42 algebra of all smooth forms on A∆. Let ∆ and ∆0be two simplices of K, with ∆0face of ∆, and ϕ∆,∆0: ∆0,→∆ the inclusion map. By definition of complex of Lie algebroids, we have that (A∆)!! ∆0=A∆0. The homomorphism of cochain algebras generated by the inclusion ϕ∆,∆0is denoted by ϕA∆ ∆,∆0: Ω∗(A∆; ∆) −→ Ω∗(A∆0; ∆0) We give now the definition of piecewise smooth form. The idea of this definition is based in the Whitney book’s [26] or in the Sullivan’s paper [22]. Definition (Piecewise smooth form). Let Kbe a simplicial complex and suppose that A={A∆}∆∈Ka complex of Lie algebroids on K. A piecewise smooth form of degree p(p≥0) on the complex of Lie algebroids Ais a family ω= (ω∆)∆∈Ksuch that the following conditions are satisfied. •For each ∆ ∈K,ω∆∈Ωp(A∆; ∆) is a smooth form of degree pon A∆. •For each ∆, ∆0∈K, with ∆0face of ∆, ϕA∆ ∆,∆0(ω∆) = ω∆0 Let (ϕ∆,∆0)!! :A∆0−→ A∆be the map induced by ϕ∆,∆0. We recall that the spaces Ω∗(A∆0; ∆0) and Ω∗(Im (ϕ∆,∆0)!!; ∆0) are identified and that the fibre (A∆0)xis a vector subspace of the fibre (A∆)x. Hence, the second condition of the definition given above can be stated in the following form: for each x∈∆0and vectors u1, . . . , up∈(A∆0)x ω∆0(x)(u1,···, up) = ω∆(x)(u1,···, up) Thus, a piecewise smooth form is a collection of smooth forms, each one defined on a transitive Lie algebroid over a simplex of K, which are compatible under restriction to faces. The set of all piecewise smooth forms of degree pon the complex of Lie algebroids Awill be denoted by Ωp ps(A;K) or simply Ωp(A;K) . We have then Ωp ps(A;K) = {(ω∆)∆∈K:ω∆∈Ωp(A∆),∆0≺∆ =⇒(ω∆)!! ∆0=ω∆0} 43 Remark. When p= 0, a piecewise smooth form on Aof degree zero is a family (ϕ∆)∆∈K∈Ω0 ps(A;K)⊂Q∆∈KC∞(∆) such that ϕ∆: ∆ −→ Ris smooth and the equality ϕ∆0=ϕ∆/∆0holds for each face ∆0of ∆. The compatibility condition of restrictions to faces gives a map ϕ:|K| −→ Rwhich is continuous. The map ϕmay not be a differentiable map but it is a piecewise smooth function. The set of all maps ϕ∈C(|K|;R) which are compatible with restrictions to the faces of |K|and with smooth restrictions to the faces of |K|is denoted by Cps(|K|;R). Obviously, Ω0 ps(A;K) has a natural structure of algebra over Rand is naturally identified to Cps(|K|;R). Let A={A∆}∆∈Kbe a complex of Lie algebroids on a simplicial complex K. Since restrictions of smooth forms are compatible with sums and products, various operations on Ωp ps(A;K) can be defined by the corresponding operations on each simplex of K. Namely, if ω= (ω∆)∆∈K,η= (η∆)∆∈K∈Ωp ps(A;K) are two piecewise smooth forms of degree p on the complex of Lie algebroids Aand f:|K| −→ Ra continuous map, we may define ω+η,fω and ω∧ηto be ω+η= (ω∆+η∆)∆∈K fω = (f/∆ω∆)∆∈K ω∧η= (ω∆∧η∆)∆∈K The set Ωp ps(A;K), equipped with these operations, becomes a real vector subspace of Q∆∈KΩp(A∆; ∆), for each natural p≥0. Thus, Ωp ps(A;K) is a module over the algebra Cps(|K|;R). When p= 0, Ω0 ps(A;K) = Cps(|K|;R) has a structure of an unitary associative algebra over R. Moreover, the direct sum Ω∗ ps(A;K) = M p≥0 Ωp ps(A;K) equipped with the exterior product defined by the corresponding exterior product on each algebra Ω∗ ps(A∆; ∆) = Lp≥0Ωp(A∆; ∆), is a commutative graded algebra over R. In this section, we shall also include some basic functorial properties. We begin first with the definition of inverse image of a piecewise smooth form. 44 Definition (Inverse image of piecewise smooth forms). Let K,Lbe simplicial complexes and A={A∆}∆∈Kand B={B∆0}∆0∈Lcomplexes of Lie algebroids on Kand Lrespectively. Let f:K−→ Lbe a simplicial map and suppose that, for each ∆ ∈K, a morphism of Lie algebroids F∆:A∆−→ Bf(∆) over f/∆: ∆ −→ f(∆) is given such that the family λ=(F∆)∆∈K, fis a complex of Lie algebroids morphism. For each ∆ ∈K, we have the commutative diagram A∆ F∆// π∆  Bf(∆) πf(∆)  ∆f/∆//f(∆) If ω= (ω∆0)∆0∈L∈Ωp ps(B;L) is a piecewise smooth form, then, for each ∆ ∈K, we can consider the smooth form (F∆, f/∆)∗(ωf(∆))∈Ωp(A∆; ∆). We define λ∗ωto be λ∗ω=(F∆, f/∆)∗(ωf(∆))∆∈K Proposition 2.2.1. On the same conditions above, the form λ∗ωis a piecewise smooth form of degree pdefined on the complex of Lie algebroids A. Proof. It remains to check the compatibility condition of the restriction to the faces. Let sbe a face of ∆. Then f(s) is also a face of the simplex f(∆) and hence (ωf(∆))/f(s)=ωf(s). We also have that the equality (F∆, f/∆)∗ωf(∆)/f(s)=(F∆, f/∆)∗(ωf(∆)/s holds, and therefore (λ∗ω)∆/s=(F∆, f/∆)∗(ωf(∆))/s= (F∆, f/∆)∗ωf(∆)/f(s)= (F∆, f/∆)∗(ωf(s)) Now, by the compatibility of (F∆, f/∆) to the restrictions, we have (F∆, f/∆)∗(ωf(s))=(Fs, f/s)∗(ωf(s)) = (λ∗ω)/s and so (λ∗ω)∆/s= (λ∗ω)/s. 45 Proposition 2.2.2. Let A={A∆}∆∈K,B={B∆0}∆0∈Land C={C∆00 }∆00∈Tthree complexes of Lie algebroids on the simplicial complexes K,Land Trespectively. Let f:K−→ Land g:L−→ Tsimplicial maps and suppose that, for each ∆ ∈Kand ∆0∈L, morphisms of Lie algebroids F∆:A∆−→ Bf(∆) over f/∆: ∆ −→ f(∆) and G∆0:B∆0−→ Cg(∆0)over g/∆0: ∆0−→ g(∆0) are given such that the families λ=(F∆)∆∈K, f:A−→ B and b λ=(G∆0)∆0∈L, g:B −→ C are morphisms of complexes of Lie algebroids. Then, the following properties hold: a) (b λ◦λ)∗ω=λ∗(b λ∗ω). b) (idA)∗ζ=ζ. c) λ∗(ξ+η) = λ∗ξ+λ∗ηand λ∗(ϕω) = (ϕ◦f)λ∗ω. d) λ∗(ξ∧η) = λ∗ξ∧λ∗η. for each ζ∈Ωp ps(A;K), ω∈Ωp ps(C;T), ξ,η∈Ωp ps(B;L) and ϕ:|L| −→ Rcontinuous. In order to obtain a complex of cochains, especially important is the analogues of exterior derivative. This operator also is obtained by the corresponding exterior derivative on each simplex. Namely, if A={A∆}∆∈Kis a complex of Lie algebroids on a simplicial complex K, we can define the mapping dp: Ωp ps(A;K)−→ Ωp+1 ps (A;K) setting dp((ω∆)∆∈K) = (dp ∆ω∆)∆∈K for each ω= (ω∆)∆∈K∈Ωp ps(A;K). For p= 0, the algebra Ω0 ps(A;K) is the vector space of all families (ϕ∆)∆∈K∈Q∆∈KC∞(∆) such that (ϕ∆)∆∈Kis compatible with the 46 restrictions to faces. So, the exterior derivative in degree zero is the usual derivative. We list below the main properties of the exterior derivative. Proposition 2.2.3. Let A={A∆}∆∈Kbe a complex of Lie algebroids on a simplicial complex Kand ω= (ω∆)∆∈K∈Ωp ps(A;K). Then, the followings properties hold: •dpis linear for any p≥0. •dp+1 ◦dp= 0 for any p≥0. •For each ξ= (ξ∆)∆∈K∈Ωp ps(A;K) and η= (η∆)∆∈K∈Ωq ps(A;K), dp+q(ξ∧η) = (dpξ)∧η+ (−1)pξ∧(dqη) Such as in the case of smooth forms on a Lie algebroid, the space Ω∗ ps(A;K), with the operations and differentiation above, becomes a commutative differential graded algebra, which is defined over R. Definition (Piecewise smooth cohomology). Keeping the same hypothesis and notation as above, the piecewise smooth cohomology space of Ais the cohomology space of the algebra Ωp ∗(A;K) equipped with the structures defined above. Its cohomology, H(Ω∗ ps(A;K)), will be denoted by H∗ ps(A;K) or simply by H∗(A;K). Proposition 2.2.4. Let A={A∆}∆∈K,B={B∆0}∆0∈Ltwo complexes of Lie algebroids on the simplicial complexes KeLrespectively. Let f:K−→ Lbe a simplicial map and suppose that, for each ∆ ∈K, a morphism of Lie algebroids F∆:A∆−→ Bf(∆) over f/∆: ∆ −→ f(∆) is given such that the family λ=(F∆)∆∈K, fis a complex of Lie algebroids morphism. Then, for each piecewise smooth form ω= (ω∆)∆∈K∈Ωp ps(A;K), the equality d(λ∗ω) = λ∗(dω) hold. Proof. For each simplex ∆ ∈K, the equality d(λ∗ω∆) = λ∗(dω∆) and so, the result follows.  47 piecewise smooth form of degree pdefined on Lcan be piecewise smoothly extended to a piecewise smooth form of degree pdefined on the whole K. We conclude from this proposition that the map rpK L: Ωp psA;K−→ Ωp psAL;Lfrom proposition 2.2.6 is surjective. Proposition 2.3.5. Let Kbe a simplicial complex and A={Aα}α∈Ka complex of Lie algebroids on K. Let K0and K1be two simplicial subcomplexes of Ksuch that K=K0∪K1and set L=K0∩K1. Consider the complexes of Lie algebroids A0={Aα}α∈K0,A1={Aα}α∈K1and A0,1={Aα}α∈Lgiven by restriction of Ato the simplicial subcomplexes K0,K1and L. Then, it holds a exact short sequence of cochain complexes {0} −→ Ω∗ ps(A;K)λ∗ −→Ω∗ ps(A0;K0)⊕Ω∗ ps(A1;K1)µ∗ −→Ω∗ ps(A0,1;L)−→ {0} in which the linear maps λp: Ω∗ ps(A;K)−→ Ω∗ ps(A0;K0)⊕Ω∗ ps(A1;K1) µp: Ω∗ ps(A0;K0)⊕Ω∗ ps(A1;K1)−→ Ω∗ ps(A0,1;L) are defined by λp(ω) = (ω/K0, ω/K1) and µp(ξ, η) = η/L −ξ/L. Proof. As in the case of smooth forms on a transitive Lie algebroid over a smooth manifold, the exterior derivative commutes with the restrictions to a simplicial subcomplexes (proposition 2.2.5) and, since dp(ξ, η)=(dp(ξ), dp(η)), one deduces immediately that λ∗ and µ∗are effectively cochain complex morphisms. Obviously, the linear map λpis injective. Since, for each piecewise smooth form ω∈Ωp ps(A;K), the forms ω/K0and ω/K1have the same restriction ωLto L, we conclude that µp◦λp= 0, and hence the image of the linear map λpis contained in the kernel of the linear map µp. Reciprocally, if µp(ξ, η) = 0, we have ξα=ηα, for each α∈L, and this equality allows to define a piecewise smooth form ω∈Ωp ps(A;K) by the condition ωα=ξα, for each α∈K0, and ωα=ηα, for each α∈K1. We have then λp(ω) = µp(ξ, η). We want now to prove that µpis surjective. 54 Let γ∈Ωp ps(A0,1;L) be a piecewise smooth form and consider the piecewise smooth form −1 2γ∈Ωp ps(A0,1;L). By the extension lemma, we can consider a piecewise smooth form α∈Ωp ps(A0;K0) such that α/L =−1 2γ. Analogously, we can consider a piecewise smooth form β∈Ωp ps(A1;K1) such that β/L =1 2γ. We have then that µp(α, β) = γ. By applying the zig-zag lemma to the sequence above, we obtain the long exact sequence in cohomology Hp−1 ps (A0,1;L)∂p−1//Hp ps(A;K)Hp(λ∗) //Hp ps(A0;K0)⊕Hp ps(A1;K1) Hp ps(A0;K0)⊕Hp ps(A1;K1) Hp(µ∗)//Hp ps(A0,1;L)∂p//Hp+1 ps (A;K) which is the Mayer-Vietoris sequence for piecewise smooth cohomology. 2.4 Generalization of piecewise smooth context It should be remarked that, in previous sections, specific properties of the simplices were not required neither in the formulation of the piecewise smooth context nor in the statement of some properties. Indeed, these notions and properties can be extended to more general spaces. Moreover, for the proof of the Mishchenko’s theorem given in next section, we are going to need a slightly modification of the concept of piecewise smooth cohomology given in the previous section. We shall notice, in this section, a general notion of piecewise smooth cohomology to other spaces which may not be simplicial complexes. A sheaf of the piecewise smooth forms on a complex of Lie algebroids will be constructed. As remarked in previous section, all simplicial complexes considered are geometric and finite and simplex means always closed simplex. Definition. Let K={N1, . . . , Ns}be a finite collection of submanifolds in an ambient 55 space such that, for any j1,. . . ,je∈ {1, . . . , s}, the intersection Nj1∩ · · · ∩ Njeis a submanifold. A complex of Lie algebroids on Kis a family A={Aj}j∈Jsuch that the following conditions hold. •For each j∈ {1, . . . , s},Ajis a transitive Lie algebroid on Nj. •For each i, j ∈ {1, . . . , s}, one has (Aj)!! Nj∩Ni= (Ai)!! Nj∩Ni. It is obvious that, by transitivity of restrictions of Lie algebroids, for each subset e Jof {1, . . . , s}and any partition {{j1, . . . , jr},{i1, . . . , it}} of e J, we have (A!! Nr)!! Nt= (A!! Nt)!! Nr, where Nr=Nj1∩ · · · ∩ Njrand Nt=Ni1∩ · · · ∩ Nit Definition. Let K={N1, . . . , Ns}be a finite collection of submanifolds in an ambient space such that, for any j1,. . . ,je∈ {1, . . . , s}, the intersection Nj1∩ · · · ∩ Njeis a submanifolds. Assume that a complex of Lie algebroids A={Aj}j∈Jon Kis given. A piecewise smooth form of degree p(p≥0) on Ais a family ω= (ω1...,ωs) such that, for each j∈ {1, . . . , s}, •For each j∈ {1, . . . , s},ωj∈Ωp(Aj;Nj) is a smooth form on Aj. •For each i, j ∈ {1, . . . , s}, one has ϕAj Nj∩Ni,Nj(ωj) = ϕAi Nj∩Ni,Ni(ωi) where ϕAj Nj∩Ni,Nj: Ω∗(Aj;Nj)−→ Ω∗((Aj)!! Ni∩Nj;Ni∩Nj) ϕAi Nj∩Ni,Ni: Ω∗(Ai;Ni)−→ Ω∗((Ai)!! Ni∩Nj;Ni∩Nj) denote the homomorphisms of cochain algebras generated by the inclusions maps ϕNj∩Ni,Nj:Ni∩Nj−→ Njand ϕNj∩Ni,Ni:Ni∩Nj−→ Nirespectively. The family of all such forms obtained in this way will be denoted by Ωp ps(A;K). This set is a real vector subspace of the product vector space Ωp(A1;N1)× · · · × Ωp(As;Ns). A 56 wedge product and an exterior derivative can be defined on Ω∗ ps(A;K) = Lp≥0Ωp ps(A;K) by the corresponding operations on each algebra Ω∗ ps(Aj;Nj) = Lp≥0Ωp(Aj;Nj), giving to Ωp ∗(A;K) a structure of cochain algebra defined over R. The cohomology space of this algebra will be denoted by H∗ ps(A;K). We notice that piecewise smooth cohomology of a complex of Lie algebroids defined on a simplicial complex is a particular case of this generalization. The reason for this generalization is that, as mentioned at the introduction of this section, we are going to deal with a complex of piecewise smooth forms that may not be defined over the family of all closed simplices of a simplicial complex. To illustrate this idea, let us briefly look at some cases of this construction. A first example of this generalization is take a simplicial complex and to fix our attention on an open star of one its vertex. In this case, we have the open star smoothly triangulated by a non-complete simplicial complex since each simplex of this triangulation does not contain the face opposite to the vertex. The family of submanifolds made by those simplices without the faces opposite to the vertex satisfies the conditions required in our definition of complex of Lie algebroids given at the beginning of this section. Any transitive Lie algebroid over the open star gives, by restriction, a complex of Lie algebroids. Another illustrative example consists of taking the family defined by intersections of open stars with any open subset of the polytope of a simplicial complex. The first example is obviously a particular example of this second case. The construction of a complex of Lie algebroids can be done in similar way. These two examples will be used in the proof of the main theorem of next section. Our third example extends the second one and and consists of taking intersections of generalized stars with open subsets of the polytope. This third example is not quite different of previous examples. Nevertheless, it enhances the construction of the sheaf of the piecewise smooth forms on a complex of Lie algebroids. We provide, in next section, a description of this third example as well of the corresponding sheaf of piecewise smooth forms. 57 2.5 Sheaves of piecewise smooth forms In this section, we describe a corresponding sheaf of piecewise smooth forms, which we can define on a special complex of Lie algebroids defined by using regular open subsets. Definitions and main properties of regular open subsets can be seen in [24]. The idea of construction of the sheaf of the piecewise smooth forms on a complex of Lie algebroids comes from [9] or [1]. Definition (Generalized star). Let Kbe a simplicial complex and aa point of the polyhedron |K|. The generalized star of a, denoted also by St a, is the union of the interiors of all simplices of Ksuch that abelongs to those simplices. Remark. When the point ais a vertex of K, it is obvious that the generalized star of ais the same as the star of a. Remark 2. For each a∈K, there is a unique simplex ∆aof Ksuch that abelongs to the interior of ∆a(see [24]). Proposition 2.5.1. Let Kbe a simplicial complex and aa point of the polytope |K|. Denote by ∆athe unique simplex of Ksuch that abelongs to the interior of ∆a. Then, the generalized star of acoincide with the star St ∆a. Consequently, the generalized star of ais an open subset of the polyhedron |K|. Proof. If ais one of the vertices of K, then ∆a={a}and the result is proved. Suppose now that ais different of any vertex of K. Then abelongs to the interior of ∆a. We shall see first that St a⊂St ∆a. Let ∆ be a simplex of Ksuch that a∈∆. Since ais different of any vertex of K, it follows that a∈◦ s, for some face sof ∆. But a∈ ◦ ∆aand so ◦ s= ◦ ∆a. Hence s= ∆aand therefore ∆ais a face of ∆. We conclude then ◦ ∆⊂St ∆a. Now, let e ∆ be a simplex of Ksuch that ∆ais a face of e ∆. Then, a∈e ∆ and so ◦ e ∆⊂St ∆a. The other inclusion is obvious. The second part of the proposition is immediate.  58 The collection of the stars of all vertices of a simplicial complex Kform an open covering of its polytope |K|and the base obtained from this covering is a contractible base for the topology of |K|. Below, we are going to deal with other coverings which are not obtained from its stars of its vertices, but from regular open subsets. We recall that a regular open subset of |K|is a star of some simplex of K(see [24]). Next definition is to generalize this notion in order to obtain other coverings of |K|which fit better for the sheaf of piecewise smooth forms on a complex of Lie algebroids. Definition (Regular open subset). Let Kbe a simplicial complex and |K|its polytope. Let a∈ |K|and Uan open subset of |K|with a∈U. The open subset Uis called regular open neighborhood of aif Uis the intersection of an open neighborhood of ain |K|with the generalized star of a. Given any open subset Vof |K|,Vis called a regular open subset of |K|, if there exists a point a∈ |K|such that Vis a regular open neighborhood of the point a. Remark. Obviously, a star of some simplex of a simplicial complex is a regular open subset of its polytope. Proposition 2.5.2. Let Kbe a simplicial complex and |K|its polytope. For each a∈ |K|, the set of all regular open neighborhoods of ais a fundamental system of neighborhoods of aand the set of all regular open subsets of |K|is a base for the topology of the space |K|. Proof. Standard arguments. The proposition 2.2.6 still remains true in the piecewise context obtained by using the set of all generalized regular open subsets of the polytope of a simplicial complex. We notice those facts below, beginning first to describe a special construction of a complex of Lie algebroids based in regular open subsets. Derived complex corresponding to regular open subsets. Let Kbe a simplicial 59 complex and A={A∆}∆∈Ka complex of Lie algebroids on K. Let Ube a regular open subset of |K|and consider a∈ |K|such that U=Z∩St a, in which Zis an open neighborhood of ain |K|. Consider the unique simplex ∆aof Ksuch that abelongs to the interior of ∆a. For each simplex ∆ ∈Ksuch that ∆ais a face of ∆, denote by ∆Uthe set ∆U=U∩∆. We have that ∆Uis an open submanifold of ∆. If ∆0is other simplex of K such that ∆ais a face of ∆0, then ∆ais a face of ∆ ∩∆0and the intersection U∩(∆ ∩∆) is a submanifold. The collection KU, made by the manifolds ∆U=U∩∆ such that ∆ais a face of ∆, satisfies the conditions required for the definition of complex of Lie algebroids given at the beginning of this section. The Lie algebroid A∆is transitive and so we can take the Lie algebroid restriction (A∆)!! ∆Uto ∆U. Therefore, we can consider the family AU={(A∆)!! ∆U: ∆ ∈K, ∆a≺∆} We claim that the family AUis a complex of Lie algebroids defined over the set of manifolds KU. Before proving this statement, we are going to check that the triangulation obtained in Udoes not depend on the point achosen, that is, if Z∩St a=e Z∩St b, then St a=St b. To see this, denote by ∆aand ∆bthe unique simplices of Kwhich contain aand bin its interior respectively. Then, St a=St ∆aand St b=St ∆b. Since b∈V∩St a, there exists a simplex ∆0∈Ksuch that ∆ais a face of ∆0and bbelongs to the interior of ∆0. Hence, ∆b= ∆0by uniqueness of ∆b, and so ∆ais a face of ∆b. Analogously, we conclude can that ∆bis a face of ∆aand so it holds that ∆b= ∆a. It remains to check that the family AUis indeed a complex of Lie algebroids. For that, fix two simplices ∆ and ∆0of Ksuch that ∆ais a common face of ∆ and ∆0. Let s= ∆ ∩∆0. We have (A∆)!! ∆U!! U∩s= (A∆)!! U∩s=(A∆)!! s!! U∩s= (As)!! U∩s and analogously (A∆0)!! ∆0 U!! U∩s= (As)!! U∩s 60 Hence, the family AUis a complex of Lie algebroids over the set of manifolds KU={U∩∆ : ∆ ∈K, ∆a≺∆} The complex AUis called the derived complex of the complex Acorresponding to the regular open subset U. The cochain algebra of the piecewise smooth forms on the derived complex of a complex of Lie algebroids will be denoted simply by Ω∗ ps(AU), dropping the letter that represents the family of submanifolds which the derived complex of Lie algebroids is defined on. Keeping the same hypothesis and notations as above, let Uand Vtwo regular open subsets of |K|such that V⊂U. Let aand b∈ |K|such that Uand Vare regular open neighborhoods of aand brespectively. Denote by ∆aand ∆bthe unique simplices of K which contain aand bin its interior respectively. Since b∈U, there exists a simplex ∆ ∈K such that ∆ais a face of ∆ and the point bbelongs to the interior of ∆. Hence, ∆b= ∆ and so ∆ais a face of ∆b. If ∆0is a simplex of Ksuch that ∆bis a face ∆0, then ∆ais a face of ∆0and, consequently, every element of KV={∆V: ∆ ∈K, ∆b≺∆}is a submanifold of the respective element of KU={∆U: ∆ ∈K, ∆a≺∆}. Let ω= (ω∆U)∆U∈KU∈Ω∗ ps(AU) be a piecewise sooth form on the complex of Lie algebroids AU. For each simplex ∆ ∈K such that ∆bis a face of ∆, we have that (A∆)!! ∆U!! ∆V= (A∆)!! ∆Vand we can restrict the smooth form ω∆U∈Ω∗(A∆)!! ∆U; ∆Uto the submanifold ∆V, obtaining the smooth form ω∆V= (ω∆U)!! ∆V∈Ω∗(A∆)!! ∆V; ∆V. Therefore, we obtain the differential form (ω∆V)∆V∈KV. Similar arguments given in the proof of the proposition 3.2 can be used to show that the form (ω∆V)∆V∈KVis a piecewise smooth form and so it belongs to ∈Ω∗ ps(AV). As done in the definition following proposition 3.4, the piecewise smooth form (ω∆V)∆V∈KV is denoted by ω/V or simply by ωV. Proposition 2.5.3. Let Kbe a simplicial complex and A={A∆}∆∈Ka complex of Lie algebroids on K. Let Uand Vbe two regular open subsets of |K|such that U⊂V and consider the derived complexes of Lie algebroids AUand AVcorresponding to Uand 61 Vrespectively. For each p≥0, denote by rpK L: Ωp ps(AU)−→ Ωp ps(AV) the map induced by restriction, that is, for each ω∈Ωp ps(AU), rpU V(ω) = ω/V •For each p≥0, rpU U=idΩp ps(AU). •r∗U V: Ω∗ ps(AU)−→ Ω∗ ps(AV) is a morphism of graded algebras. •If Wis other generalized regular open subset of |K|with W⊂Vand AWis the derived complex of Lie algebroids corresponding to W, then the diagram below is a commutative diagram of cochain complexes Ω∗ ps(AU) r∗U W&& L L L L L L L L L L r∗U V//Ω∗ ps(AV) r∗V W xxrrrrrrrrrr Ω∗ ps(AW) Consequently, for each p≥0, the correspondence which associates, to each regular open subset Uof |K|the real vector space Ωp ps(AU) of the piecewise smooth forms defined on U and, to each pair of regular open subsets Uand Vof |K|with V⊂Uthe homomorphism rpU V, is a presheaf, which is called the presheaf of the piecewise smooth forms of degree p of the complex A. Proof. Standard arguments. The last proposition leads us to the following definition (see [9]). Definition. Let Kbe a simplicial complex and A= (Aα)α∈Ka sheaf of Lie algebroids on K. For each p≥0, the sheaf of the piecewise smooth forms of degree pon the sheaf of Lie algebroids Ais the sheaf constructed canonically from the presheaf of the piecewise smooth forms of degree pon (Aα)α∈K. 62 Proposition 2.5.4. Let Kbe a simplicial complex and A= (Aα)α∈Ka sheaf of Lie algebroids on K. Then the sheaf Sof the piecewise smooth forms of degree pon the sheaf of Lie algebroids Ais fine. Proof. Let U={Uj}j∈Jbe a locally finite open covering of |K|by regular open subsets of |K|. Since the set of all regular open subsets of |K|is a base for the topology of |K|, we can assume that each open subset U∈Uis a regular open subset. If {ϕj}j∈Jis piecewise smooth partition of unity subordinated to the covering U, the homomorphisms of presheaves hj: Ωp ps(AU)−→ Ωp ps(AU) defined by hj(ω) = ϕj/U ωfor each ω∈Ωp ps(AU) induce homomorphisms from Apto Apsatisfying the conditions which characterize the definition of fine sheaf. Therefore, the result is proved if we find a piecewise smooth partition of unity subordinated to the covering U. By lemma shrinking, there is an open covering V={Vj}j∈Jsuch that, for each j∈J,Vj⊂Uj. Let U∈Uand V∈Vsuch that V⊂U. Consider a∈ |K|such that Uis a regular open neighborhood of ain |K|. For each simplex ∆ ∈Ksuch that ∆ais a face of ∆, consider the closed subset V∩∆ of ∆. Take the union of all V∩∆ such that ∆ais a face of ∆ and denote that union by W. Since |K|is compact, the topology of |K|coincide with the topology induced from the Euclidian space. We have that Wa closed subset of the Euclidian space. The open star St ∆ais open in |K|and so there is an open subset Zof the Euclidian space such that St ∆a=Z∩ |K|. The closed subset is contained in the open subset Z. Hence, we can fix a smooth function ϕ:Z−→ Rsuch that ϕdoes not vanish on W. By restriction to each submanifold ∆U=U∩∆, we have a piecewise smooth function on Uwhich does not vanish on each V∩∆. Take the sum of these functions and consider the quotient of each function by the sum. This defines a partition of unity made by piecewise smooth functions. 63 Proposition 3.1.1. Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex Kand Aa transitive Lie algebroid on M. Then, the map Ψ∗: Ω∗(A;M)−→ Ω∗ ps(A;K) given by Ψ(ω)=(ω∆)∆∈Kis natural. Proof. Let Mand Nbe two smooth manifolds, smoothly triangulated the the simplicial complexes Kand Lrespectively. Let Aand Bbe transitive Lie algebroids on Mand on Nrespectively and (F, f) a morphism of Lie algebroids from Ainto B. We shall see that the following diagram Hp(B;N)(F,f)∗ −→ Hp(A;M)   yΨ  yΨ Hp ps(B;L)(F,f)∗ −→ Hp ps(A;K) commutes. Let ω∈Ωp(B;N) a smooth form. For each simplex ∆ ∈K, the equality (F∗ω)/∆=F∗(ω/f(∆)) holds, and therefore, Ψ(F∗ω) = F∗(ω/f(∆))∆∈K. On the other side, Ψ(ω)=(ω∆0)∆0∈Land so, by definition of inverse image of differential form for the piecewise case, we have F∗(Ψ(ω)) = F∗(ω∆0)∆0∈L= (F∗ω/f(∆))∆∈K From here, the commutativity of the diagram above can be readily derived.  Next proposition is concerning the Mishchenko’s theorem for trivial Lie algebroids. Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K, and Aa transitive Lie algebroid on M. Take a simplex sof Kand let U=St(s). Assume that g is a real Lie algebra and consider the trivial Lie algebroid A=TU ⊕(M×g) on U, which is identified to the Lie algebroid A=TU ×gby a strong homomorphism of Lie algebroids over U. For each p≥0, one has Ωp(g) = Vpg. Consider the Lie algebroids morphisms γ:TU ×g−→ TU 70 and π:TU ×g−→ g For each simplex ∆ ∈Ksuch that sis a face of ∆, we denote ∆U=U∩∆ as done before. Consider the Lie algebroids morphisms γ∆U:T∆U×g−→ T∆U and π∆U:T∆U×g−→ g given by the projections on the first and second factors respectively. Proposition 3.1.2. Keeping the same hypothesis and notations as above, the morphism Ψ:Ω∗(A;U)−→ Ω∗ ps(A;U) ω−→ (ω∆)∆U⊂U induces an isomorphism in cohomology. Proof. By the K¨unneth theorem [6], we have that H(A;U)≃HdR(U)⊗H(g) Next, we wish to check that Hps(A;U)≃Hps(U)⊗H(g) For that, we are going to divide the proof in three parts. Part 1. We are going to check that Ω∗ ps(U)⊗Ω∗(g)≃Ω∗ ps(A;U) Let ξ= (ξ∆)∆U⊂U∈Ω∗ ps(U) and η∈Ω∗(g). If ∆0and ∆ are two simplices of Ksuch that s≺∆0≺∆, denote by (ϕT∆U×g ∆,∆0)!! and (ϕT∆U ∆,∆0)!! the canonical map induced from the diagrams 71 T∆U×gT∆U   yγ  yγ ∆0 U ϕ∆,∆0 −→ ∆U∆0 U ϕ∆,∆0 −→ ∆U It is obvious that γ∆U◦(ϕT∆U×g ∆,∆0)!! = (ϕT∆U ∆,∆0)!! ◦γ∆0 Uand π∆U◦(ϕT∆U×g ∆,∆0)!! =π∆0 U so the equalities (γ∗ ∆Uξ∆)∆0 U=γ∗ ∆0 Uξ∆0 and (π∗ ∆Uη)∆0 U=π∗ ∆0 Uη hold. These equalities show that the differential form (γ∗ ∆Uξ∆∧π∗ ∆Uη)∆U⊂U belongs to Ω∗ ps(A;U). Hence, we can consider a map k: Ω∗ ps(U)⊗Ω∗(g)−→ Ω∗ ps(A;U) such that k(ξ⊗η) = (γ∗ ∆Uξ∆∧π∗ ∆Uη)∆U⊂U where ξ= (ξ∆)∆U⊂U. This map is well defined. Now, we will see that the map kis an isomorphism of differential graded algebras. Obviously, the map kis a morphism of graded algebras. For each ∆ ∈Ksuch that s≺∆, let k∆: Ω∗(∆U)⊗Ω(g)−→ Ω∗(T∆U×g) be the K¨unneth isomorphism (see theorem [6]?. We have that, (k(ξ⊗η))∆U=γ∗ ∆Uξ∆∧π∗ ∆Uη=k∆(ξ∆⊗η) 72 Therefore, if ω=Pξ⊗η∈Ω∗ ps(U)⊗Ω∗(g) and k(ω) = 0, then k(ω)∆U= 0 and so 0 = k(Xξ⊗η)∆U=k∆(X(ξ∆⊗η)) Hence ω=P(ξ∆⊗η) = 0 and, with this, we have checked that kis injective. Take now λ= (λ∆)∆U⊂U∈Ω∗ ps(A;U). We want to find ω∈Ω∗ ps(U)⊗Ω∗(g) such that k(ω) = λ. Since k∆is surjective, we can consider smooth forms ξj∆∈Ω∗(∆U) and η∈Ω∗(g) such that k∆(X j (ξj∆⊗η)) = λ∆U Take then the form ω∆=Pj(ξj∆⊗η). If ∆0and ∆ are simplices of Kwith s≺∆0≺∆, we have the equalities k∆0X j (ξj∆)∆0 U⊗η=X j k∆0((ξj∆)∆0 U⊗η=X jγ∗ ∆0 U(ξj∆)∆0 U∧π∗ ∆0η= (∗) and k∆0X j (ξj∆0⊗η)=λ∆0= (λ∆)/∆0 U=k∆(X j (ξj∆⊗η)/∆0 U= =X j (γ∗ ∆U(ξj∆)∧π∗ ∆η)/∆0 U=X j (γ∗ ∆U(ξj∆)∧π∗ ∆η)/∆0 U= =X j (γ∗ ∆U(ξj∆/∆0 U )∧π∗ ∆0η)) = X jγ∗ ∆0 U(ξj∆)∆0 U∧π∗ ∆0η= (∗) Hence, k∆0X j (ξj∆)∆0 U⊗η=k∆0X j (ξj∆0⊗η) and, since k∆0is bijective, Pj(ξj∆)∆0 U⊗η=Pj(ξj∆0⊗η). Therefore, we can conclude that ξj∆/∆0 U=ξj∆0. Then, the form ω= (ω∆)∆U⊂Uwhere, for each ∆U⊂U,ω∆=Pj(ξj∆⊗η) belongs to Ω∗ ps(U)⊗Ω∗(g). Obviously k(ω) = λand then it is checked that kis an isomorphism of graded algebras. Part 2. In next part we are going to check that kcommutes with differential, being then proved that kis an isomorphism of differential graded algebras. For each ∆ ∈Ksuch 73 that s≺∆, denoting the differentials on the complexes Ω∗ ps(A;U) and Ω∗ ps(U) by dA ps and dU ps respectively, we have (dA ps ◦k)(ξ⊗η) = dA ps(γ∗ξ∧π∗η) = =dA ps(γ∗ξ)∧π∗η+ (−1)degξγ∗ξ∧dA ps(π∗η) = =γ∗(dU psξ)∧π∗η+ (−1)degωγ∗ξ∧π∗(dgη) = =k((dU psξ)⊗η)+(−1)degξk(ξ⊗dgη) = k◦δ(ξ⊗η) Part 3. The isomorphism kabove induces an isomorphism in cohomology. By applying the K¨unneth theorem, we obtain H∗ ps(A;U)≃H∗(Ω∗ ps(U)⊗Ω∗(g)) ≃H∗ ps(U)⊗H∗(g) Now, we shall see that Ψ induces an isomorphism in cohomology. Take the diagram Ω∗(U)⊗Ω∗(g) k  λ//Ω∗ ps(U)⊗Ω∗(g) kps  Ω∗(A;U)Ψ//Ω∗ ps(A;U) where kps =k: Ω∗ ps(U)⊗Ω∗(g)−→ Ω∗ ps(A;U) is the isomorphism defined above, kis the K¨unneth isomorphism ([6]) and λ= Φ ⊗Id in which Φ is the restriction map given on the Rham-Sullivan theorem for smooth manifolds. Obviously, the diagram is commutative and, by the de Rham-Sullivan theorem, Φ is an isomorphism in cohomology. Therefore, in cohomology, we have the commutative diagram H∗ dR(U)⊗H∗(g) ≃  H(λ)//H∗ ps(U)⊗H∗(g) ≃  H∗(A;U)H(Ψ) //H∗ ps(A;U) Hence, H(Ψ) is an isomorphism and the result is proved.  74 Next, we want to show that Ψ induces an isomorphism in cohomology, not only for the trivial Lie algebroid defined over a regular open subset but for any arbitrary transitive Lie algebroid over a regular open subset. For that, we state first a basic result needed for the statement. This result is a basic consequence of the functor homology. Proposition 3.1.3. Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K,sa simplex of Kand U=St (s). Let Aand Bbe two transitive Lie algebroids on Mand suppose there is an isomorphism of Lie algebroids between them. Then, the cohomology spaces Hps(A;U) and Hps(B;U) are isomorphic. Proposition 3.1.4. Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K,sa simplex of Kand U=St (s). Let Abe a transitive Lie algebroid on U. Then, the morphism Ψ:Ω∗(A;U)−→ Ω∗ ps(A;U) ω−→ (ω∆)∆U⊂U induces an isomorphism in cohomology. Proof. Since Uis contractible, Ais isomorphic to the trivial Lie algebroid B=TU×gon U, in which gis the fibre type of K=Ker γ. We conclude the result by the commutativity of the diagram Ωp(A;U)−→ Ωp ps(A;U)   y  y Ωp(TU ×g)Ψ −→ Ωp ps(TU ×g) and applying the propositions 3.1.2 and 3.1.3.  Proposition 3.1.5. Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K, and Aa transitive Lie algebroid on M. Let s1,. . . ,skbe simplices of the simplicial complex Kand consider the regular open subsets Uj=St (sj). For l∈ {1, . . . , k} fixed, consider the open subsets U=U1∪· · ·∪Uland V=Ul+1 ∪· · ·∪Ukof Mand assume 75 that M=U∪V. Denote by KUthe set of all submanifolds ∆U=U∩∆ such that ∆ ∈K and sjis a face of ∆ for some j∈ {1, . . . , l},KVthe set of all submanifolds ∆V=V∩∆ such that ∆ ∈Kand siis a face of ∆ for some i∈ {l+ 1, . . . , k}, and KU∩Vthe set of all submanifolds ∆U∩V= (U∩V)∩∆ such that ∆ ∈Kand sjand siare faces of ∆ for some j∈ {1, . . . , l}and i∈ {l+ 1, . . . , k}. Then, we have a commutative diagram of short exact sequences {0}//Ωp(A;M) Ψ  λ//Ωp(AU;U)⊕Ωp(AV;V) Ψ  µ//Ωp(AU∩V;U∩V) Ψ  //{0} {0}//Ωp ps(A;K)δ//Ωp ps(AU;U)⊕Ωp ps(AV;V)π//Ωp ps(AU∩V;U∩V)//{0} in which the maps λand µare the canonical maps given by the restriction and the difference and the maps δ: Ωp ps(A;K)−→ Ωp ps(AU;U)×Ωp ps(AV;V) π: Ωp ps(AU;U)×Ωp ps(AV;V)−→ Ωp ps(AU∩V;U∩V) are defined by δ(ω∆)∆∈K=(ω∆U)∆U∈KU,(ω∆V)∆V∈KV) and π(ξ∆U)∆U∈KU,(η∆V)∆V∈KV=(η∆U∩V−ξ∆U∩V)∆U∩V∈KU∩V Proof. In the section 1.2 of the first chapter, it was stated that the first arrow is exact. Regarding the second arrow, the proof is similar to the proof of proposition 2.3.5, section 2.3 of the second chapter. We shall only check that the map πis surjective. Since the set {U, V }is an open covering of Mwe can fix two smooth maps ϕ, ψ :M−→ [0,1] such that supp ϕ⊂U,supp ψ⊂Vand ϕ(x) + ψ(x)=1 ∀x∈M. Let (γ∆U∩V)∆U∩V∈KU∩V∈Ωp ps(AU∩V;U∩V) be a piecewise smooth form. We shall define a differential form (ξ∆U)∆U∈KU∈Ωp ps(AU;U) 76 as follows. For each ∆U∈KU, set ξ∆U(x) =              −ψ(x)γ∆U(x) if x∈∆U∩V 0x∈(A∆U)xif x∈∆U∩(M\supp ψ) The sets ∆U∩Vand ∆U∩(M\supp ψ) are open in ∆Uwith union equal to ∆U. Obviously, the restrictions of ξ∆Uto ∆U∩Vand to ∆U∩(M\supp ψ) are smooth. Therefore, we conclude that ξ∆U∈Ωp(A∆U). In order to obtain a piecewise smooth form belonging to Ωp ps(AU) it remains to check that (ξ∆U)∆U⊂Uis compatible with restrictions to faces. Let ∆ and ∆0be two simplices of Ksuch that sj≺∆≺∆0for some j∈ {1, . . . , e}. Then, one has ∆U∩V⊂∆0 U∩V⊂U∩Vand, since γis piecewise smooth, we have γ∆U(x)=(γ∆0 U)/∆U(x) for each x∈∆U. Hence, if x∈∆U∩V, ξ∆U(x) = −ψ(x)γ∆U(x) = −ψ(x)(γ∆0 U)/∆U(x) = (ξ∆0 U)/∆U(x) If x∈∆U∩(M\supp ψ) we have that ξ∆U(x)=(ξ∆0 U)∆U(x) = 0. Hence, the differential form (ξ∆U)∆U∈KUis a piecewise smooth form belonging to Ωp ps(AU;U). Analogously, we define a piecewise smooth form (η∆V)∆V∈KV∈Ωp ps(AV;V) by η∆V(x) =              −ϕ(x)γ∆V(x) if x∈∆V∩U 0x∈(A∆V)xif x∈∆V∩(M\supp ϕ) and we have that, for each x∈∆U∩V∈KU∩V, η∆U∩V(x)−ξ∆U∩V(x) = γ∆U∩V(x) and so (η∆U∩V−ξ∆U∩V)∆U∩V∈KU∩V= (γ∆U∩V)∆U∩V∈KU∩V Hence, the result is proved.  77 Proof of the Mishchenko’s theorem. We will prove the result by induction on the number of vertices of the simplicial complex K. Suppose then that v0,. . . ,vNis the family of all vertices of K. If Khas only one vertex, the result is trivial. Suppose we have established the result for all l < N. We know that M=Sj=N j=0 St vj. Taking the open subsets U=SN−1 j=0 St vjand V=St vNof M, we have that U∩V=N−1 [ j=0 St (vj)∩St (vN) = N−1 [ j=0 St (vj)∩St (vN)= N−1 [ j=0 St [vj, vN] where vj, vNdenotes the closed simplex generated by the vertices vjand vN. By last proposition, we have a commutative diagram of short exact sequences {0}//Ωp(A;M) Ψ  λ//Ωp(AU;U)⊕Ωp(AV;V) Ψ  µ//Ωp(AU∩V;U∩V) Ψ  //{0} {0}//Ωp ps(A;K)δ//Ωp ps(AU;U)⊕Ωp ps(AV;V)π//Ωp ps(AU∩V;U∩V)//{0} The map Ψ on the right side is quasi-isomorphism by induction. The map Ψ on the middle is quasi-isomorphism by induction and the proposition 3.1.4. By the Steenrod lemma, the map Ψ on the left side is also a quasi-isomorphism.  From Mishchenko’s theorem we easily conclude that the piecewise smooth cohomology of a combinatorial compact manifold does not depend on the triangulation used, that is, for any simplicial division of the simplicial complex, the piecewise smooth cohomology spaces of both combinatorial manifolds remains isomorphic. Precisely, this statement is our next proposition. Proposition 3.1.6. Let Mbe a smooth manifold smoothly triangulated by a simplicial complex Kand Aa transitive Lie algebroid on M. Let Lbe other simplicial complex and assume that La subdivision of K. Then, the piecewise smooth cohomology of the complex {A!! ∆}∆∈Kis isomorphic to the one of the complex {A!! ∆}∆∈L. Thus, the morphism from Ωp ps(A;K) to Ωp ps(A;L) which induces that isomorphism in cohomology is given by restriction of forms. 78 Proof. The result follows from the commutativity of the following diagram Ωp(A;M) Ψ wwpppppppppppΨ && N N N N N N N N N N N Ωp ps(A;K)Φ//Ωp ps(A;L) where Φ is also given by restriction.  3.2 Piecewise invariant cohomology In this section, we shall note a consequence of the Mishchenko’s theorem in piecewise invariant cohomology of transitive Lie algebroids equipped with an action of a Lie group. We recall basic definitions and the main result regarding invariant cohomology, following the paper [4] by Kubarski. As in the previous section all simplicial complexes are finite. We begin by stating a general result concerning natural transformations between functors. For next proposition, consider the category Cof all transitive Lie algebroids over combinatorial compact manifolds and the category Dof all cochain algebras. Suppose that F and Gare two functors from Cto D. Let tbe a natural transformation between the functors Fand G. For each transitive Lie algebroid Aover a combinatorial compact manifold, denote by tA:F(A)−→ G(A) the corresponding cochain algebra morphism. Our next proposition is the following. Proposition 3.2.1. Keeping the same hypothesis and notations as above, suppose yet that the following conditions hold. •For each finite dimensional real Lie algebra gand each contractible combinatorial compact manifold M, H(F(TM ×g)) ≃H(G(TM ×g)) 79 β:G−→ M. Let ϕ:N ,→Mbe a submanifold. Denote by GN Nthe set α−1(N)∩β−1(N). Then, if Nis transversal to G, the set GN Nis a Lie subgroupoid of Gwith base Nand GN N is called the Lie groupoid restriction of Gto N. The main proposition for our piecewise context is the following. The proof is a direct consequence of the definition of Lie algebroid of a Lie groupoid. Proposition 3.3.2 (Restrictions of Lie groupoids to simplices). Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K, and Ga Lie groupoid on M. For each simplex ∆ of K, one has A(G∆ ∆) = A(G)!! ∆ We recall now the definition of de Rham cohomology of left invariants forms of a Lie groupoid. Let Mbe a smooth manifold and Ga Lie groupoid on Mwith source projection αand target projection β. For each p≥0, let Ωp α(G;M) be the C∞(G)-module of the smooth sections of the vector bundle of all alternating p-linear maps from the vector bundle Fg∈GTGα(g)(disjoint union) to the trivial vector bundle RM. A smooth α-form of degree pon the Lie groupoid Gis, by definition, an element of Ωp α(G;M). Thus, a smooth α-form ω∈Ωp α(G;M) is a family defined on Gsuch that, for each g∈G, one has ωg∈ΛpG g∈G T∗ gGα(g);R The usual exterior derivative along the α-fibres is defined by (dp αω)(X1, X2,···, Xp+1) = p+1 X j=1 (−1)j+1Xj·(ω(X1,···,c Xj,···, Xp+1)) + +X i<k (−1)i+kω([Xi, Xk], X1,···,c Xi,···,c Xk,···, Xp+1) in which ω∈Ωp α(G;M) and X1,X2,···,Xp+1 are smooth vector α-fields on G. The complex (Ω∗ α(G;M), d∗ α) is a commutative cochain algebra defined on R. The set Ωp α,L(G;M) 86 consisting of all α-forms on Gwhich are invariant under the groupoid left translations is a subcomplex of (Ω∗ α(G;M), d∗ α). Its cohomology is denoted by H∗ α,L(G;M). Denote by 1 : M−→ Gthe object inclusion map of G. There is an isomorphism ψ: Ω∗ α,L(G:M)−→ Ω∗(A(G); M) of cochain algebras defined by ψ(ω)x=ω1x. Consequently, we have the following proposition. Proposition 3.3.3. Keeping the same hypothesis and notations as above, we have H∗ α,L(G;M)≃H∗(A(G); M) Let us to introduce the notion of piecewise smooth cohomology of Lie groupoids. Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K, and Ga locally trivial Lie groupoid on Mwith source projection αand target projection β. For each simplex ∆ ∈K, let G∆ ∆be the Lie groupoid restriction of Gto ∆ and A(G∆ ∆) its Lie algebroid. Since Gis locally trivial its Lie algebroid A(G) is transitive and we know that A(G∆ ∆)≃ A!! ∆. Similarly to piecewise smooth forms on Lie algebroids, we give now the notion of piecewise smooth form on G. Definition (Piecewise smooth form). Keeping the same hypothesis and notations as above, a piecewise smooth α-form of degree p(p≥0) on Gis a family ω= (ω∆)∆∈K such that the following conditions are satisfied. •For each ∆ ∈K,ω∆∈Ωp α,L(G∆ ∆; ∆) is a α-smooth form of degree pon G∆ ∆. •If ∆ and ∆0are two simplices of Ksuch that ∆0≺∆, one has (ω∆)/∆0=ω∆0. The C∞(G)-module of all piecewise α-smooth forms of degree pon Gis denoted by Ωp α,L,ps(G;M) or Ωp α,L,ps(G;K). As done in previous sections, a wedge product and an 87 exterior derivative can be defined on Ω∗ α,L,ps(G;K) by the corresponding operations on each submanifold G∆ ∆, giving to Ω∗ α,L,ps(G;K) a structure of cochain algebra defined over R. The cohomology space of this complex is denoted by H∗ α,L,ps(G;M) or H∗ α,L,ps(G;K). Our aim is to relate the cohomology space H∗ α,L,ps(G;K) of Gto the cohomology space H∗ ps(A(G); K) of its Lie algebroid A(G). For that, we have to consider a map φfrom the complex Ω∗ α,L,ps(G;K) to the complex Ω∗ ps(A(G); K). In order to obtain such map φ, we recall that, for each simplex ∆ of K, we have an isomorphism ψ∆: Ωp α,L(G∆ ∆; ∆) −→ Ωp(A(G∆ ∆); ∆) given by ψ(ω)x=ω1x. Consider now a piecewise smooth α-form ω= (ω∆)∆∈K∈Ωp α,L,ps(G;K) For each simplex ∆ ∈K, take the smooth form ξ∆=ψ∆(ω∆)∈Ωp(A(G∆ ∆); ∆). Next proposition says that this process gives a piecewise smooth form on A(G). Proposition 3.3.4. Keeping the same hypothesis and notations as above, if ∆ and ∆0 are two simplices of Ksuch that ∆0is a face of ∆, then (ξ∆)!! ∆0=ξ∆0and so ξ= (ξ∆)∆∈K is a piecewise smooth on A(G). Consequently, the map Φ:Ω∗ α,L,ps(G;K)−→ Ω∗ ps(A(G); K) defined by Φ((ω∆)∆∈K) = (ψ∆(ω∆))∆∈Kis an isomorphism of cochain algebras. We can state now the main proposition of this section. Let rGbe the restriction map rG: Ω∗ α,L(G;M)−→ Ω∗ α,L,ps(G;K) defined rG(ω)=(ω/∆)∆∈K. Our proposition is the following. Proposition 3.3.5. Let Mbe a smooth manifold, smoothly triangulated by a simplicial complex K, and Ga locally trivial Lie groupoid on Mwith source projection α 88 and target projection β. Then, the map rG: Ω∗ α,L(G;M)−→ Ω∗ α,L,ps(G;K) induces an isomorphism in cohomology. Consequently, the Rham cohomology of Gis isomorphic to the piecewise Rham cohomology of G. Proof. The diagram Ω∗ α,L(G;M) rG  iso //Ω∗(A;M) rA  Ω∗ α,L,ps(G;K)iso //Ω∗ ps(A;K) is commutative, where rAis the restriction map given at the Mishchenko’s theorem. We apply the Mishchenko’s theorem in cohomology and the proof is done.  Our last proposition says that the piecewise de Rham cohomology of a locally trivial Lie groupoid over a combinatorial manifold doesn’t depend on the triangulation. Proposition 3.3.6. Let Mbe a smooth manifold smoothly triangulated by a simplicial complex Kand Lother simplicial complex which a subdivision of K. Let Gbe a locally trivial Lie groupoid on M. Then, the piecewise de Rham cohomology of Gobtained by the triangulation corresponding Kis isomorphic to the the piecewise de Rham cohomology of Gobtained by the y the triangulation corresponding to L. Thus, this isomorphism is induced by the restriction map. Proof. Denote by φ: Ω∗ α,L,ps(G;K)−→ Ω∗ α,L,ps(G;L) the map given by restriction. The diagram Ωp α,L(G;M) wwnnnnnnnnnnnn '' P P P P P P P P P P P P Ω∗ α,L,ps(G;K)φ//  Ω∗ α,L,ps(G;L)  Ωp ps(A;K)//Ωp ps(A;L) 89 is commutative. By propositions 3.1.6, 3.3.4 and 3.3.5, the maps non labeled are isomorphisms in cohomology and so the map φalso is isomorphism in cohomology.  90 Conclusion H. Whitney started the study of cohomologies of cell-like spaces by taking different notions of differential form. Roughly speaking, Whitney used notions of forms such as piecewise smooth forms, elementary forms, polyhedral forms and flat forms. The relationship of these constructions is described in the Whitney’s book “Geometric Integration theory” and it is the genesis of the use of differential forms to solve the commutative cochain problem. Similar constructions were found out by Sullivan in the study of the rational homotopy type of a space. Namely, Sullivan considered the algebra of the polynomial forms on a cell space and proved that this algebra is quasi-isomorphic to the classic algebra of smooth forms. The algebra of the polynomial forms originated the theory of models, which has revealed crucial in the development of homotopy and formality theories for cell spaces. Our work was written in the effort to understand those constructions for transitive Lie algebroids over combinatorial manifolds. Our first aim was to study piecewise smooth cohomology of Lie algebroids on combinatorial manifolds. Some methods used in the study of those constructions on cell-like spaces had to be changed in order to be applied to Lie algebroids, especially because we do not have the notion of cell in Lie algebroids. The notion of cell structure was changed to obtain what we called complex of Lie algebroids and to define a complex of piecewise differential forms and consequently the notion of piecewise smooth cohomology of Lie algebroids. We have seen that the restriction map induces an isomorphism in cohomology between piecewise smooth and Lie algebroid cohomology. This result is based in the Rham-Sullivan theorem well as in some results on non-abelian extensions of Lie algebroids, which lead us to the triviality of transitive Lie algebroids over 91 contractible manifolds. It is not known whether other cell constructions can be developed for Lie algebroids. For transitive Lie algebroids with commutative kernel, the transition functions are flat and this is a nice hypothesis for developing other kind of forms in Lie algebroids, especially polynomial forms in Lie algebroids, giving us an alternative way to study formality of Lie algebroids. 92 Resumen El teorema de de Rham es un resultado de gran importancia, ya que ha sido el principal v´ınculo de uni´on entre el an´alisis en variedades y la propiedades topol´ogicas de las variedades. En breves palabras, la homolog´ıa espacios mide el n´umero de agujeros de una variedad y su nivel de complejidad. El teorema de de Rham garantiza que los espacios de homolog´ıa de variedades se pueden expresar mediante el uso de formas diferenciales y sus m´etodos anal´ıticos. El estudio de los espacios de homolog´ıa en t´erminos de formas diferenciales abri´o un camino para el estudio de la estructura m´as profunda de variedades. Sullivan, en su art´ıculo “ C´alculos infinitesimales en topolog´ıa ”, dice que dentro del mundo de la topolog´ıa hay m´as informaci´on topol´ogica en el ´algebra de Rham de formas diferenciales que simplemente la cohomolog´ıa real. La teor´ıa de Rham r´apidamente origin´o un profundo desarrollo de la topolog´ıa de variedades. Hay tambi´en una gran cantidad de situaciones matem´aticas en las que el conocimiento de formas diferenciales tiene consecuencias importantes y por consiguiente otras teor´ıas matem´aticas se desarrollaron a partir del teorema de de Rham. El invariante de Hopf, el producto de Massey, el grado de una aplicaci´on y la cohomolog´ıa de grupos de Lie compactos son algunos ejemplos de la importancia del teorema de de Rham. Sullivan y otros matem´aticos han implementado varias estrategias en el estudio del ´algebra de Rham de todas las formas diferenciables. Entre ellos, est´a la teor´ıa de modelos. Esta teor´ıa consiste en encontrar otras ´algebras graduadas, dentro del ´algebra de de Rham de todas las formas defirenciables, de tal manera que la inclusi´on can´onica induce un isomorfismo en cohomolog´ıa. A partir de estos desarrollos, una conclusi´on importante surgi´o, que puede expresarse en el siguiente diagrama conmutativo: H∗ p.C∞(M) R  H∗ PL(M)⊗QR 66 m m m m m m m m m m m m R ∼ =(( Q Q Q Q Q Q Q Q Q Q Q QH∗ dR(M) ffM M M M M M M M M M M ∼ = R xxp p p p p p p p p p p H∗(M, R) 93 Este esquema incorpora una gran cantidad de construcciones y enunciados. El presente trabajo surge de los esfuerzos por extender esas construcciones a algebroides de Lie transitivos. Entre esas construcciones, estamos particularmente interesados en la que dice que la cohomolog´ıa de de Rham de una variedad diferenciable, triangulada diferenciablemente por un complejo simplicial, es isomorfo a la cohomolog´ıa diferenciable por partes del complejo simplicial. Este isomorfismo viene dado por la restricci´on de formas diferenciables a todos los s´ımplices. El estudio de esta construcci´on o de otras construcciones de Sullivan en variedades simpliciales se basa en el teorema de de Rham para c´elulas, as´ı como extensiones de las formas diferenciables. Algunas dificultades surgen de la utilizaci´on de la teor´ıa de de Rham en el estudio de la cohomolog´ıa de algebroide de Lie. Sin embargo, a pesar de todas las dificultades que surgen de la teor´ıa de de Rham de algebroides de Lie, durante los ´ultimos a˜nos, la teor´ıa de cohomolog´ıa de algebroides de Lie se ha desarrollado a partir de una colecci´on de grandes resultados con fuertes conexiones con muchas otras partes de las matem´aticas, en particular, con la teor´ıa de Chern-Weil. Estas mejoras han reducido varios obst´aculos en el desarrollo de nuestro trabajo. Las ideas clave relativas a la clase de obstrucci´on derivada de las extensiones no abelianas de algebroides de Lie han inspirado Mishchenko y le llev´o a conjeturar que, dado un algebroide de Lie transitivo en una variedad combinat´orica, el morfismo dado por la restricci´on, que lleva formas diferenciables del algebroide de Lie en formas diferenciables a trozos en el mismo algebroide de Lie, sigue siendo un isomorfismo en cohomolog´ıa. El objetivo del presente trabajo es demostrar la conjetura de Mishchenko. Para este prop´osito, se ha utilizado una estructura llamada complejo de algebroides de Lie. Esta estructura se inicia fijando una triangulaci´on diferenciable de la base de un algebroide de Lie transitivo por un complejo simplicial y tomando la restricci´on del algebroide de Lie a todos los s´ımplices de la triangulaci´on. Como el algebroide de Lie es transitivo, siempre existe la restricci´on del algebroide de Lie a cada s´ımplice. Dado un complejo de algebroides de Lie, se define la noci´on de forma diferenciable a trozos de manera similar a las formas de Whitney en un complejo simplicial y el conjunto de todas las formas diferenciables a 94 trozos definidas en un complejo de algebroides de Lie es, naturalmente, equipado con una diferencial, produciendo un ´algebra diferencial graduada conmutativa. Su cohomolog´ıa es, por definici´on, la cohomolog´ıa diferenciable a trozos del algebroide de Lie. Cada forma diferenciable definida en el algebroide Lie da una forma diferenciable a trozos definida en el correspondiente complejo de algebroides de Lie tomando la restricci´on de la forma a cada s´ımplice. Esta correspondencia es una aplicaci´on natural del ´algebra usual de las formas suaves del algebroide Lie al ´algebra de las formas diferenciables a trozos del complejo correspondiente de algebroides de Lie. Bas´andose en tres resultados importantes, a saber, la trivialidad de un algebroide de Lie transitivo sobre una variadad diferenciable contr´actil (Mackenzie, Weinstein), el teorema de K¨unneth para algebroides de Lie (Kubarski) y el teorema de Rham-Sullivan para variedades diferenciables, mostramos que esta aplicaci´on es un isomorfismo en cohomolog´ıa. 95