A compact manifold with infinite-dimensional co-invariant cohomology
aa r X i v : . [ m a t h . DG ] J a n A COMPACT MANIFOLD WITH INFINITE-DIMENSIONALCO-INVARIANT COHOMOLOGY
MEHDI NABIL
Abstract.
Let M be a smooth manifold. When Γ is a group acting on M by diffeo-morphisms one can define the Γ -co-invariant cohomology of M to be the cohomologyof the complex Ω c ( M ) Γ = span { ω − γ ∗ ω, ω ∈ Ω c ( M ) , γ ∈ Γ } . For a Lie algebra G acting on the manifold M , one defines the cohomology of G -divergence forms tobe the cohomology of the complex C G ( M ) = span { L X ω, ω ∈ Ω c ( M ) , X ∈ G} . Inthis short paper we present a situation where these two cohomologies are infinitedimensional. Mathematics Subject Classification 2010:
Keywords:
Cohomology, Transformation Groups.1.
Introduction
In [1], the authors have introduced the concept of co-invariant cohomology. Inbasic terms it is the cohomology of a subcomplex of the de Rham complex generatedby the action of a group on a smooth manifold. The authors showed that under niceenough hypotheses on the nature of the action, there is an interplay between the deRham cohomology of the manifold, the cohomology of invariant forms and the co-invariant cohomology, and this relationship can be exhibited either by vector spacedecompositions or through long exact sequences depending on the case of study(Theorems . and . in [1]). Among the various consequences that can be derivedfrom this inspection, it is evident that the dimension of de Rham cohomology hassome control over the dimension of the co-invariant cohomology, and in most casespresented in [1] the latter is finite whenever the former is. This occurs for instancein the case of a finite action on a compact manifold or more generally in the caseof an isometric action on a compact oriented Riemannian manifold, and this factholds as well for a non-compact manifold as long as one requires the action to befree and properly discontinuous with compact orbit space. A concept closely relatedto co-invariant cohomology is the cohomology of divergence forms, which is definedby means of a Lie algebra action on a smooth manifold and was introduced by A.Abouqateb in [2]. In the course of his study, the author gave many examples wherethe cohomology of divergence forms is finite-dimensional.The goal of this paper is to show that this phenomenon heavily depends on thenature of the action in play, and that without underlying hypotheses, co-invariantcohomology and cohomology of divergence forms are not generally well-behaved.This is illustrated by an example of a vector field action on a smooth compactmanifold giving rise to infinite-dimensional cohomology of divergence forms and Date : January 5, 2021. whose discrete flow induces an infinite-dimensional co-invariant cohomology as op-posed to the de Rham cohomology of the manifold. This shows in particular thatmany results obtained in [1] and [2] cannot be easily generalized and brings intoperspective the necessity to look for finer finiteness conditions of co-invariant co-homology in a future study which would put the present paper in a broader context.The general outline of the paper is as follows : In the first paragraph, we brieflyrecall the notions of co-invariant forms and divergence forms, then we define an ho-momorphism of the de Rham complex that is induced by a complete vector field onthe manifold, and which maps divergence forms relative to the action of the vectorfield onto the complex of co-invariant differential forms associated to its discreteflow (see (1) and Proposition 2.0.1). The next paragraph is concerned with the set-ting on which our cohomology computations will take place, it comprises a smoothcompact manifold, the -dimensional hyperbolic torus, which can be obtained asthe quotient of a solvable Lie group by a uniform lattice (the construction givenhere is that of A. EL Kacimi in [3]), the Lie algebra action considered is by meansof a left-invariant vector field. We then use a number of results to prove Theorem3.0.1 which states that the operator defined in (1) is an isomorphism between thecomplex of divergence forms and the complex of co-invariant forms, hence allowingto only consider the cohomology of co-invariant forms for computation. Finally, thelast paragraph is dedicated to the main computation in which we prove that thediscrete flow of the vector field in question on the hyperbolic torus gives infinite-dimensional co-invariant cohomology. Acknowledgement.
The author would like to thank Abdelhak Abouqateb for hishelpful discussions and advice concerning this paper.
COMPACT MANIFOLD WITH INFINITE-DIMENSIONAL CO-INVARIANT COHOMOLOGY3 Preliminaries
Let M be a smooth n -dimensional manifold and denote Diff( M ) the group ofdiffeomorphisms of M and χ ( M ) the Lie algebra of smooth vector fields on M .Let ρ : Γ −→ Diff( M ) be an action of a group Γ on M by diffeomorphisms. Foran r -form ω on M and element γ ∈ Γ , we denote γ ∗ ω the pull-back of ω by thediffeomorphism ρ ( γ ) : M −→ M . Let Ω c ( M ) = ⊕ p Ω pc ( M ) denote the de Rhamcomplex of forms with compact support on M and put: Ω pc ( M ) ρ := span { ω − γ ∗ ω, γ ∈ Γ , ω ∈ Ω pc ( M ) } . Any element of Ω pc ( M ) ρ is called a ρ -co-invariant or just a ( Γ -)co-invariant whenthere is no ambiguity. The graded vector space Ω c ( M ) ρ := ⊕ p Ω pc ( M ) ρ is a dif-ferential subcomplex of the de Rham complex Ω c ( M ) , it is called the complex ofco-invariant differential forms on M . When M is compact this complex is sim-ply denoted Ω( M ) ρ . In the case where ρ : Z −→ M is the action induced by adiffeomorphism γ : M −→ M , i.e ρ ( n ) := γ n , then we get that: Ω pc ( M ) = { ω − γ ∗ ω, ω ∈ Ω pc ( M ) } . Let τ : G −→ χ ( M ) be a Lie algebra homomorphism and denote ˆ X := τ ( X ) forany X ∈ G then define: C pτ ( M ) := span { L ˆ X ω, X ∈ G , ω ∈ Ω pc ( M ) } . Any element of C pτ ( M ) is called a τ -divergence p -form or simply G -divergence form.The graded vector space C τ ( M ) := ⊕ p C pτ ( M ) is a differential subcomplex of thede Rham complex. If X is any vector field on M , with corresponding Lie algebrahomomorphism τ : R −→ χ ( M ) , τ (1) := X then: C pτ ( M ) = span { L X ω, ω ∈ Ω pc ( M ) } . In what follows, X ∈ χ ( M ) is a complete vector field and φ : M × [0 , −→ M is the flow φ X of the vector field X restricted to M × [0 , . We define the linearoperator I : Ω( M ) −→ Ω( M ) by the expression: I ( η ) := − Z φ ∗ η ∧ pr ∗ ( ds ) (1)where − R : Ω ∗ ( M × [0 , − → Ω ∗− ( M ) is the fiberwise integration operator of thetrivial bundle M × [0 , pr −→ M (see [4]) and ds the usual volume form on [0 , .Let τ : R −→ χ ( M ) be the Lie algebra homomorphism induced by X and let ρ : Z −→ Diff( M ) be the discrete flow of X i.e the group action given by ρ ( n ) := φ Xn . Proposition 2.0.1.
The operator I : Ω( M ) −→ Ω( M ) defined by (1) is a differen-tial complex homomorphism i.e I ◦ d = − d ◦ I . Moreover I ( C τ ( M )) ⊂ Ω c ( M ) ρ andthe restriction of I : C τ ( M ) −→ Ω c ( M ) ρ is surjective.Proof. Let η ∈ Ω( M ) and denote ι s : M −→ M × { s } ֒ → M × [0 , be the naturalinclusion, then using Stokes formula for fiberwise integration we get that: I ( dη ) = − Z φ ∗ ( dη ) ∧ pr ∗ ( ds ) = − Z d ( φ ∗ η ∧ pr ∗ ( ds )) = − dI ( η ) + (cid:2) ι ∗ s ( φ ∗ η ∧ pr ∗ ds ) (cid:3) , and since ι ∗ s pr ∗ ( ds ) = 0 then I ( dη ) = − dI ( η ) . For the second claim we start byshowing that I ( η ) has compact support whenever η does. Indeed assume η ∈ Ω c ( M ) MEHDI NABIL and denote K := supp( η ) , next consider the map: f : M × R −→ M, ( x, s ) φ − s ( x ) := φ ( x, − s ) , Then f is continuous and therefore L := f ( K × [0 , is compact. For any y ∈ M \ L and any s ∈ [0 , we get that φ s ( y ) / ∈ K and therefore ( φ ∗ η ) ( y,s ) = 0 , this impliesthat I ( η ) y = 0 . We conclude that supp I ( η ) ⊂ L i.e I ( η ) ∈ Ω c ( M ) .From the relation T ( x,t ) φ (0 ,
1) = X φ t ( x ) one gets that φ ∗ ◦ i X = i (0 , ∂∂s ) ◦ φ ∗ andtherefore φ ∗ ◦ L X = L (0 , ∂∂s ) ◦ φ ∗ . Moreover we have that L (0 , ∂∂s ) pr ∗ ( ds ) = 0 and − Z ◦ i (0 , ∂∂s ) = 0 . If we write η = L X ω for some ω ∈ Ω c ( M ) then we get that: I ( L X ω ) = − Z φ ∗ ( L X ω ) ∧ pr ∗ ( ds )= − Z L (0 , ∂∂s ) ( φ ∗ ω ) ∧ pr ∗ ( ds )= − Z L (0 , ∂∂s ) ( φ ∗ ω ∧ pr ∗ ( ds ))= − Z d ◦ i (0 , ∂∂s ) ( φ ∗ ω ∧ pr ∗ ( ds )) + − Z i (0 , ∂∂s ) d ( φ ∗ ω ∧ pr ∗ ( ds ))= d (cid:18) − Z i (0 , ∂∂s ) ( φ ∗ ω ∧ pr ∗ ( ds )) (cid:19) + [ ι ∗ s i (0 , ∂∂s ) ( φ ∗ ω ∧ pr ∗ ( ds ))] = [ φ ∗ s ω ] = φ ∗ ω − φ ∗ ω = φ ∗ ω − ω. It follows that I ( C τ ( M )) ⊂ Ω c ( M ) ρ . This also shows that I : C τ ( M ) −→ Ω c ( M ) ρ is surjective. (cid:3) Remark 2.0.1.
Note that φ X -invariant forms on M are fixed by I i.e if ω ∈ Ω( M ) such that L X ω = 0 then I ( ω ) = ω . The hyperbolic torus
Consider A ∈ SL(2 , Z ) with tr( A ) > . It is easy to check that A = P DP − forsome P ∈ GL(2 , R ) and D = diag( λ, λ − ) . Clearly λ > and λ = 1 . Hence itmakes sense to set D t = diag( λ t , λ − t ) and define A t = P D t P − for any t ∈ R .Next we define the Lie group homomorphism: φ : R −→ Aut( R ) , t A t The hyperbolic torus T A is the smooth manifold defined as the quotient Γ \ G where G := R ⋊ φ R and Γ := Z ⋊ φ Z . The natural projection R ⋊ φ R p −→ R induces a fiber bundle structure T A p −→ S with fiber type T and p [ x, y, t ] = [ t ] . COMPACT MANIFOLD WITH INFINITE-DIMENSIONAL CO-INVARIANT COHOMOLOGY5 If (1 , a ) and (1 , b ) are the eigenvectors of A respectively associated to the eigenvalues λ and λ − then: v = (1 , a, , w = (1 , b, and e = (0 , , − log( λ ) − ) , forms a basis of g = Lie( G ) , and we can check that: [ v, w ] g = 0 , [ e, v ] g = − v, and [ e, w ] g = w. (2)Denote X , Y and Z the left invariant vector fields on R ⋊ φ R associated to v , w and e respectively, then { X, Y, Z } defines a parallelism on T A , a direct calculationleads to: X = λ t (cid:18) ∂∂x + a ∂∂y (cid:19) , Y = λ − t (cid:18) ∂∂x + b ∂∂y (cid:19) and Z = − log( λ ) − ∂∂t . (3)Now denote α , β and θ the dual forms associated to X , Y and Z respectively. Itis clear that the vector fields X and Y of T A are tangent to the fibers of the fiberbundle T A p −→ S , and that θ = − (log λ ) p ∗ ( σ ) where σ is the invariant volumeform on S satisfying R S σ = 1 . Assume in what follows that the eigenvalue λ of A is irrational, then from the relation: A (cid:18) a (cid:19) = λ (cid:18) a (cid:19) we deduce that a ∈ R \ Q . This remark leads to: Proposition 3.0.1.
The orbits of the vector field X defined in (3) are dense in thefibers of the fiber bundle T A p −→ S . In particular for any f ∈ C ∞ ( T A ) , X ( f ) = 0 is equivalent to f = p ∗ ( φ ) for some φ ∈ C ∞ ( S ) .Proof. We shall identify S with R / Z . Fix [ t ] ∈ S and consider the diffeomorphism: Φ t : T −→ p − [ t ] , [ x, y ] [ x, y, t ] . Then define the vector field ˆ X on T given by: ˆ X [ x,y ] = T [ x,y,t ] φ − t ( X [ x,y,t ] ) = λ t (cid:18) ∂∂x + a ∂∂y (cid:19) . Since a is irrational we get that the family { , a } is Q -linearly independent and thusthe orbits of ˆ X are dense in T , consequently the orbits of X are dense in p − [ t ] ,this proves the assertion since t is arbitrary. (cid:3) The following Lemma is of central importance for the development of this para-graph and for the computations of the next section:
Lemma 3.0.1.
Let f ∈ C ∞ ( T A ) then for every s ∈ R we have the followingformula: Z (cid:0) ( φ Xs ) ∗ ( f ) (cid:1) = − s ( φ Xs ) ∗ (cid:0) X ( f ) (cid:1) + ( φ Xs ) ∗ (cid:0) Z ( f ) (cid:1) . (4) In particular Z ( γ ∗ f ) = − X ( γ ∗ f ) + γ ∗ ( Z ( f )) and i Z ◦ γ ∗ = − γ ∗ ◦ i X + γ ∗ ◦ i Z ,where γ := φ X . MEHDI NABIL
Proof.
For any ( x, y, t ) ∈ R , a straightforward computation gives that: Z (cid:0) ( φ Xs ) ∗ ( f ) (cid:1) ( x, y, t ) = − λ d ( f ◦ φ Xs ) ( x,y,t ) (0 , , − λ ddu | u =0 ( f ◦ φ Xs )( x, y, t + u )= − λ ddu | u =0 f ( sλ t + u + x, asλ t + u + y, t + u )= − λ ( df ) φ Xs ( x,y,t ) ( s log( λ ) λ t , as log( λ ) λ t , − s ( df ) φ Xs ( x,y,t ) ( λ t , aλ t , − λ ( df ) φ Xs ( x,y,t ) (0 , , − s ( X ( f ) ◦ φ Xs )( x, y, t ) + ( Z ( f ) ◦ φ Xs )( x, y, t ) . Which achieves the proof. (cid:3)
Corollary 3.0.1.
Let f ∈ C ∞ ( T A ) such that f = γ ∗ f with γ := φ X . Then X ( f ) = 0 and consequently f = p ∗ ψ with ψ ∈ C ∞ ( S ) .Proof. Since f = γ ∗ f we get that for every n ∈ Z , f = ( γ n ) ∗ ( f ) thus the precedinglemma gives that: Z ( f ) = − nX ( f ) + ( γ n ) ∗ ( Z ( f )) . Consequently we obtain that for every n ∈ Z : | X ( f ) | ≤ n (cid:0) k Z ( f ) k ∞ + k ( γ n ) ∗ ( Z ( f )) k ∞ (cid:1) ≤ n k Z ( f ) k ∞ , which leads to X ( f ) = 0 and achieves the proof. (cid:3) In what follows we denote M := T A . It is straightforward to check that: dα = − α ∧ θ, dβ = β ∧ θ, dθ = 0 . and that L X α = − θ and L X β = L X θ = 0 thus L X ( α ∧ β ∧ θ ) = 0 .Let τ : R −→ χ ( M ) be the Lie algebra homomorphism corresponding to the vectorfield X and ρ : Z −→ Diff( M ) the discrete action generated by γ := φ X where φ X is the flow of X , that is, ρ ( n )( x ) = φ Xn ( x ) for any n ∈ Z . Theorem 3.0.1.
The homomorphism I : C τ ( M ) −→ Ω( M ) ρ defined in (1) is anisomorphism.Proof. In view of Proposition 2.0.1 it only remains to prove that I is injective.Choose η ∈ C τ ( M ) and write η = L X ω for some ω ∈ Ω( M ) . Assume that I ( η ) = 0 ,in view of the previous computation this is equivalent to ω = γ ∗ ω .If η ∈ C τ ( M ) then Corollary 3.0.1 gives that ω = p ∗ φ for some φ ∈ C ∞ ( S ) ,thus η = 0 . On the other hand if η ∈ C τ ( M ) we can write η = X ( f ) α ∧ β ∧ θ forsome f ∈ C ∞ ( M ) satisfying f = γ ∗ f and so by Corollary 3.0.1 we get η = 0 .Now for η ∈ C τ ( M ) we can write: ω = f α + gβ + hθ, f, g, h ∈ C ∞ ( M ) . COMPACT MANIFOLD WITH INFINITE-DIMENSIONAL CO-INVARIANT COHOMOLOGY7
Applying I to L X α = − θ leads to γ ∗ α = α − θ . Moreover since θ and β are φ X -invariant we get that β = γ ∗ β and θ = γ ∗ θ thus: γ ∗ ω = ( γ ∗ f ) α + ( γ ∗ g ) β + ( γ ∗ h − γ ∗ f ) θ, hence ω = γ ∗ ω is equivalent to f = γ ∗ f , g = γ ∗ g and h = γ ∗ h − f . The last relationthen implies that h − ( γ n ) ∗ h = nf for all n ∈ Z and thus: k f k ∞ ≤ n k h − ( γ n ) ∗ h k ∞ ≤ n k h k ∞ −→ n → + ∞ . Therefore f = 0 and h = γ ∗ h , g = γ ∗ g which according to Corollary 3.0.1 givesthat X ( g ) = 0 and X ( h ) = 0 , and so using that L X β = 0 and L X θ = 0 it followsthat η = L X ω = 0 . Finally let η ∈ C τ ( M ) and write: ω = f α ∧ β + gα ∧ θ + hβ ∧ θ, f, g, h ∈ C ∞ ( M ) . Then using γ ∗ α = α − θ we obtain that: γ ∗ ω = ( γ ∗ f ) α ∧ β + ( γ ∗ g ) α ∧ θ + ( γ ∗ h + γ ∗ f ) β ∧ θ, and so ω = γ ∗ ω is equivalent in this case to f = γ ∗ f , g = γ ∗ g and h = γ ∗ h + γ ∗ f .As before, this leads to f = 0 , X ( g ) = 0 and X ( h ) = 0 and so: η = L X ω = L X ( gα ∧ θ + hβ ∧ θ ) = gL X ( α ∧ θ ) = − gθ ∧ θ = 0 . Thus I : C τ ( M ) −→ Ω( M ) ρ is an isomorphism. (cid:3) This result gives in particular that H( C τ ( M )) ≃ H(Ω( M ) ρ ) and therefore weonly need to compute the cohomology of ρ -co-invariant forms in this case.4. Cohomology computation
We now have all the necessary ingredients to perform our computation. Let M denote the hyperbolic torus T A defined in the previous section with A having irra-tional eigenvalues and let X, Y, Z ∈ χ ( M ) be the vector fields defined in (3) withrespective dual -forms α , β and θ . Define the action ρ : Z −→ Diff( M ) to be thediscrete flow of the vector field X with γ := ρ (1) . The main goal is to prove thatfirst and second co-invariant cohomology groups are infinite-dimension, however weshall compute the whole cohomology in order get a global picture. Calculating H (Ω( M ) ρ ) : Choose f ∈ Ω ( M ) ρ such that df = 0 , then f is aconstant function equal to g − γ ∗ g for some g ∈ C ∞ ( M ) . Consequently we obtainthat: Z M f α ∧ β ∧ θ = Z M ( g − γ ∗ g ) α ∧ β ∧ θ = Z M gα ∧ β ∧ θ − Z M γ ∗ ( gα ∧ β ∧ θ ) = 0 . Thus f = 0 and we conclude that H (Ω( M ) ρ ) = 0 . Calculating H (Ω( M ) ρ ) : We prove that H (Ω( M ) ρ ) is infinite dimensional.In order to do so, we prove that the map p ∗ : Ω ( S ) −→ H (Ω( M ) ρ ) is well-defined and injective or equivalently we can show that p ∗ (Ω ( S )) ⊂ Z (Ω( M ) ρ ) and p ∗ (Ω ( S )) ∩ B (Ω( M ) ρ ) = 0 .An element η ∈ p ∗ (Ω ( S )) can always be written as η = p ∗ ( φ ) θ where φ ∈ C ∞ ( S ) . MEHDI NABIL
Since L X θ = 0 and L X α = − θ , then by applying I to L X α we get that θ = α − γ ∗ α ,therefore: η = p ∗ ( φ ) θ = p ∗ ( φ ) α − γ ∗ ( p ∗ ( φ ) α ) . Moreover observe that dη = 0 , hence we deduce that p ∗ (Ω ( S )) ⊂ Z (Ω( M ) ρ ) .Now suppose η = d ( g − γ ∗ g ) then clearly X ( g − γ ∗ g ) = 0 and Z ( g − γ ∗ g ) = p ∗ ( φ ) ,thus according to Proposition 3.0.1, g − γ ∗ g = p ∗ ψ for some ψ ∈ C ∞ ( S ) . Byinduction we can show that for any n ∈ N , g = ρ ( n ) ∗ g + np ∗ ψ which then leads to: | p ∗ ψ | ≤ n | g − ρ ( n ) ∗ g | ≤ n k g k ∞ −→ n → + ∞ . (5)Hence p ∗ ψ = g − γ ∗ g = 0 and so η = 0 . Thus p ∗ (Ω ( S )) ∩ B (Ω( M ) Z ) = 0 . Calculating H (Ω( M ) ρ ) : We will show that p ∗ (Ω ( S )) ∧ β ⊂ H (Ω( M ) ρ ) . Todo this, we fix a -form η = p ∗ ( φ ) θ ∧ β such that φ ∈ C ∞ ( S ) . We can easily checkthat dη = 0 , moreover from the previous calculations and the fact that L X β = 0 we get that β = γ ∗ β , therefore: p ∗ ( φ ) θ ∧ β = ( p ∗ φα ∧ β ) − γ ∗ ( p ∗ ( φ ) α ∧ β ) . Hence p ∗ (Ω ( S )) ∧ β ⊂ Z (Ω( M ) ρ ) . Now assume that η = d ( ω − γ ∗ ω ) , then usingthat [ X, Y ] = 0 we get i Y i X ( dω ) = γ ∗ ( i Y i X dω ) hence according to Corollary 3.0.1we can write i X i Y dω = p ∗ ψ for some ψ ∈ C ∞ ( S ) . On the other hand we get fromLemma 3.0.1 that: p ∗ φ − i Y i X ( dω ) = γ ∗ ( i Z i Y dω ) − i Z i Y dω. It follows from these remarks that p ∗ ( φ − ψ ) = γ ∗ ( i Z i Y dω ) − i Z i Y dω , and as in (5)we once again prove that p ∗ ( φ − ψ ) = 0 , so we deduce that p ∗ φ = p ∗ ψ = i Y i X ( dω ) .Now if we write ω = f α + gβ + hθ , then we get that p ∗ φ = X ( g ) − Y ( f ) . Moreover,from X ( p ∗ φ ) = Y ( p ∗ φ ) = 0 we get that for every s ∈ R : s p ∗ φ = Z s Z s ( φ Xt ) ∗ ( φ Yu ) ∗ ( p ∗ φ ) dudt = Z s Z s ( φ Xt ) ∗ ( φ Yu ) ∗ ( X ( g )) dudt − Z s Z s ( φ Xt ) ∗ ( φ Yu ) ∗ ( Y ( f )) dudt = Z s ( φ Xt ) ∗ X (cid:18)Z s ( φ Yu ) ∗ ( g ) du (cid:19) dt − Z s ( φ Yu ) ∗ Y (cid:18)Z s ( φ Xt ) ∗ ( f ) dt (cid:19) du = ( φ Xs ) ∗ (cid:18)Z s ( φ Yu ) ∗ ( g ) du (cid:19) − Z s ( φ Yu ) ∗ ( g ) du − ( φ Ys ) ∗ (cid:18)Z s ( φ Xt ) ∗ ( f ) dt (cid:19) + Z s ( φ Xt ) ∗ ( f ) dt. It follows that: s | p ∗ φ | ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z s ( φ Yu ) ∗ ( g ) du (cid:13)(cid:13)(cid:13)(cid:13) ∞ + 2 (cid:13)(cid:13)(cid:13)(cid:13)Z s ( φ Xt ) ∗ ( f ) dt (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ | s | ( k g k ∞ + k f k ∞ ) Hence | p ∗ φ | ≤ | s | ( k g k ∞ + k f k ∞ ) −→ s → + ∞ .We conclude that η = 0 and p ∗ (Ω ( S )) ∧ β ∩ B (Ω( M ) ρ ) = 0 , in particular thisproves that H (Ω( M ) ρ ) is infinite dimensional. Calculating H (Ω( M ) ρ ) : The elements of Ω ( M ) ρ are of the form: ( f − γ ∗ f ) α ∧ β ∧ θ, COMPACT MANIFOLD WITH INFINITE-DIMENSIONAL CO-INVARIANT COHOMOLOGY9 for some f ∈ C ∞ ( M ) . Put: c = R M f α ∧ β ∧ θα ∧ β ∧ θ , then Z M ( f − c ) α ∧ β ∧ θ = 0 . Thus ( f − c ) α ∧ β ∧ θ = dω and since L X ( α ∧ β ∧ θ ) = 0 then ( γ ∗ f − c ) α ∧ β ∧ θ = d ( γ ∗ ω ) and therefore it follows that: ( f − γ ∗ f ) α ∧ β ∧ θ = d ( ω − γ ∗ ω ) , i.e H (Ω( M ) ρ ) = 0 . References [1] A. Abouqateb, M. Boucetta and M. Nabil,
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