Haddou, H A

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Available at: http://www.ictp.trieste.it/~pub−off IC/2004/8United Nations Educational Scientific and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSON THE LERAY-HIRSCH THEOREMFOR THE LICHNEROWICZ COHOMOLOGYHassan Ait Haddou1Institut de Mathe´matiques et de Sciences Physiques,B.P. 613, Porto-Novo, Be´ninandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.AbstractThe purpose of this paper is to prove the Leray-Hirsch theorem for the Lichnerowicz coho-mology with respect to basic and vertical closed 1-forms. This is a generalization of the Ku¨nneththeorem to fiber bundles.MIRAMARE – TRIESTEMarch 20041ait.haddou@imsp-uac.org, haithaddou@yahoo.com1 IntroductionThe Lichnerowicz cohomology, H∗ω(M) (i.e. cohomology of the complex of differential forms ona smooth manifold with the de Rham differential operator deformed by a closed 1-form ω) wasinitiated by A. Lichnerowicz [7] . Since its introduction, Lichnerowicz cohomology has attracteda lot of interest, see for instance [1, 2, 3, 8, 10]. Its importance comes from the fact that Lich-nerowicz cohomology is well adapted to locally conformal symplectic (lcs) geometry. Recall thata lcs manifold consists of manifold M , together with a non-degenerate 2-form Ω on M and aclosed 1-form ω such that dΩ = −ω ∧ Ω. See for example [1, 2, 4]. It was observed in [1] thatH∗ω(M) is an invariant of the lcs structure. This cohomology is very different from the de Rhamcohomology (see proposition 2.2, theorem 2.2).In this note, we show that many properties of the de Rham cohomology still have theiranalogues within the Lichnerowicz cohomology.In the second section, we first recall the definitions and some basic properties: Poincare´duality and Ku¨nneth formula hold for the Lichnerowicz cohomology.In the last section, we consider a generalization of Ku¨nneth formula; namely the Leray-HirschTheorem for basic and vertical forms.2 PreliminariesIn this section, we define the Lichnerowicz cohomology and give some basic theorems.Let M be a differentiable manifold and ω a closed 1-form on M .The Lichnerowicz cohomology is the cohomology of differential forms on M with the differ-ential operator dω (see [7]) defined bydω = d + e(ω),d being the usual exterior differential and e(ω) the operator given bye(ω)α = ω ∧ α for all α ∈ Ω∗(M).Obviously we have dω ◦dω = 0. Denote by H∗ω(M) the cohomology of the complex (Ω∗(M), dω).Similarly we define the Lichnerowicz cohomology with compact support H ∗ω(M)c.We consider 0 as a 1-form and define H∗0 (M) to be the de Rham cohomology of M .2For two closed 1-forms ω1, ω2 and α ∈ Ωp(M), β ∈ Ωq(M) an easy calculation shows thatdω1+ω2(α ∧ β) = dω1α ∧ β + (−1)pα ∧ dω2β.Hence the wedge product induces a bilinear mapping∧ : Hpω1(M)×Hqω2(M) −→ Hp+qω1+ω2(M)([α], [β]) 7−→ [α ∧ β].Proposition 2.1 (see [8])Let M be a differentiable manifold and ω a closed 1-form on M .(a) If ω is exact then H∗ω(M)∼= H∗dR(M);(b) Suppose [ω′] = [ω] ∈ H1(M): there exists a smooth positive function f such that ω ′ =ω + d(lnf). Then the following applicationϕ : H∗ω′(M) −→ H∗ω(M)[α] 7−→ [fα]is an isomorphism.The spaces Hpω(M) can also be obtained as the cohomology of M with coefficients in a sheaf. Infact, let us denote by Fω(M) the sheaf of germs of differentiable functions f : M −→ IR whichare dω-closed, i.e.dωf = df + fω = 0.Using the fact that dω satisfies a Poincare´ lemma and that0 −→ Fω(M) −→ A0 −→ A1 −→ ...is a fine resolution of Fω(M), where Ap is the sheaf of germs of differentiable p-forms on M ,one can proveProposition 2.2 [14]For every manifold M and a closed 1-form ω on M , one has an isomorphismHp(M,Fω(M) ≈ Hpω(M).Let us recall another interpretation of the Lichnerowicz cohomology given by Banyaga in [1].Let pi : M˜ −→ M be the minimum regular cover over which the 1-form ω is pulled back to anexact 1-form and let λ : M˜ −→ IR be a positive function on M˜ such thatpi∗ω = dλ.It is well known that the group of automorphisms A of the covering M˜ , is isomorphic to thegroup of periods of ω [6].3Lemma 2.1For any τ ∈ A, the function(λ ◦ τ)λis a constant, we denote cτ , independent of the choice of λ.The set F∗cA(M) of all differential forms α on M˜ such that τ∗α = cτα for all τ ∈ A, is asubcomplex of the de Rham complex of M˜ . We denote its cohomology by H∗cA(M). Thefollowing theorem is proved in [2]Theorem 2.1H∗cA(M) is isomorphic with H∗ω(M).Example 2.1Let M = S1 and ω be a generator of its first de Rham cohomology. We claim that H 0ω(S1) = 0.So let f ∈ Ω0(S1) such that df = 0 and τ ∗f = cτf . The condition df = 0 means that f = k isa constant. Then, the second condition k = foτ = cτk translates to (1 − cτ )k = 0, but if ω isnon-exact, cτ 6= 1, hence k = 0. So H0ω(S1) = 0 and by Poincare´ duality H1ω(S1) = 0.The following lemma proves the homotopy invariance of the Lichnerowicz cohomology. Fora smooth g : M −→ N we have an induced mapping g∗ : H∗ω(N) −→ H∗g∗ω(M).Lemma 2.2 (Homotopy invariance)Let ω be a closed 1-form on differentiable manifold N and let g : M × I −→ N be a smoothhomotopy. Define f ∈ C∞(M × I, IR) by ft := exp(∫ t0 φ∗siT g∗ωds) whereφs : M −→ M × I;x 7−→ (x, s) and iT is the contraction by the vector field T =∂∂t.Then f1g∗1 = f0g∗0 : H∗ω(N) −→ H∗g∗0ω(M).If g is proper the same holds with compact supports.For the proof, see [8].2.1 Hodge TheorySuppose that M is a compact differentiable manifold of dimension n, that ω is a closed 1-form on M and that g is a Riemannian metric. Consider the vector field T on M given byω(X) = g(X,T ), for all X ∈ X (M). Denote by δ the codifferential operator and by iT thecontraction by the vector field T (see for example [16]). Then one can define the operatorδω : Ωp(M) −→ Ωp−1(M) byδω = δ + iT .Now, one can consider the standard scalar product on the space Ω∗(M) given by〈α, β〉 =∫α∧ ∗β, for all α, β ∈ Ωp(M) [13]. Since M is compact and the operator δω is elliptic,4the Hodge decomposition into orthogonal parts holds:Ωp(M) = Hpω(M)⊕ dω(Ωp−1(M)) ⊕ δω(Ωp+1(M)),where Hpω(M) = {α ∈ Ωp(M)/dω(α = 0, δω(α) = 0}. Hence,Hpω(M)∼= Hpω(M).Using the Hodge theory, we get:Theorem 2.2 (Gue´dira and Lichnerowicz)[7].Let M a n-dimensional manifold and ω a closed 1-form not dω-exact on M . All n-forms on Mare dω-exact i.e. Hnω(M) = {0}.Remark 2.1It is shown in [10] that if ω is everywhere non zero and parallel with respect to some Riemannianmetric on M , then H∗ω(M) is trivial.2.2 Poincare´ dualityLet M be an oriented n-dimensional manifold and ω a closed 1-form on M . For any integerp ≤ n, we may define a pairing〈., .〉ω : Hp−ω(X)×Hn−pω (M)c∧−→ Hnc (M)∫M−→ IIRby〈[σ], [τ ]〉ω =∫M[σ] ∧ [τ ] =∫M[σ ∧ τ ],which can also be expressed in the guise of the linear maps;Dpω : Hp−ω(M) −→ (Hn−pω (M)c)∗[σ] −→ Dpω([σ])([τ ]) := 〈[σ], [τ ]〉ωTheorem 2.3 (Poincare´ Duality)For any n-dimensional oriented differentiable manifold M and any integer p ≤ n, the linear mapDpω : Hp−ω(M) −→ (Hn−pω (M)c)∗is an isomorphism.3 Ku¨nneth formulaIn this section, we will prove the Leray-Hirsch Theorem which is the generalization of theKu¨nneth formula.The purpose of Ku¨nneth formula is the computation of the cohomology of the Cartesian product5when the cohomologies of the factors are known.Suppose we have two differentiable manifolds M , N and two closed 1-forms ω1 and ω2 on Mand N respectively. Then ω = pr∗1ω1 + pr∗2ω2 is a closed 1-form in M × N where pr1 and pr2are the canonical projections.Let α be a p-form on M and β a q-form on N . We write α∧¯β =: pr∗1α∧ pr∗2β ∈ Ω(p+q)(M ×N).It is obvious thatdω(α∧¯β) = dω1α∧¯β + (−1)pα∧¯dω2βand hence we have an induced mapping in cohomology given by∪ : Hpω1(M)⊗Hqω2(N) −→ Hp+qω (M ×N)([α], [β]) 7−→ [α ∧ β] =: α ∪ β.Definition 3.1 [5]A covering U of manifold M is called good if for all m ∈ IN and U1, ..., Um ∈ U the intersectionU1 ∩ ... ∩ Um is either empty or contractible.Remark 3.1Every Riemannian manifold admits a good covering and these coverings are cofinal in the set ofall coverings [15].Theorem 3.1 (Ku¨nneth formula)Suppose that M and N have good covers and let ω1 respectively ω2 be a closed 1-form on Mrespectively N , then the map∪ : H∗ω1(M)⊗H∗ω2(N) −→ H∗ω(M ×N)is an isomorphism.Example 3.1Let (M,ω1) be a smooth manifold equipped with the closed 1-form ω1 and consider ζ thegenerator of the first de Rham cohomology of S1. Denote ω = pr∗1ω1 + pr∗2ζ. The Ku¨nnethformula shows thatHrω(M × S1) = ⊕p+q=rHpω1(M)⊗Hqζ (S1).Example 2.1 shows that H0ζ (S1) = 0 and hence H1ζ (S1) = 0.Thus Hrω(M × S1) = 0.We generalize the above example as follows.Example 3.2Let G be a Lie group with Lie algebra G, E −→ M a principal G-bundle on M , and ω aconnection 1-form on E. Thus ω ∈ Ω1(E,G), and its curvature is given byΩ = dω +12[ω, ω] ∈ Ω2(E,G).6A connection ω with Ω = 0 is a flat connection and a bundle with a flat connection is said tobe flat.If E is an S1-flat bundle, then [ω, ω] = 0 and therefore Ω = dω = 0. So ω is an ordinary closed1-form.Suppose that M has a good cover U and let U ∈ U . Take df an exact 1-form on U , then;H∗pr∗2df+ω|U×S1(U × S1) ∼= H∗df (U)⊗H∗ω|S1(S1)∼= H∗(U)⊗H∗ω|S1(S1)∼= 0since H∗ω|S1(S1) = 0.Similarly, for V ∈ U , we getH∗pr∗2df+ω|V ×S1(V × S1) ∼= H∗df (V )⊗H∗ω|S1(S1)∼= H∗(V )⊗H∗ω|S1(S1)∼= 0andH∗pr∗2df+ω|U∩V×S1(U ∩ V × S1) ∼= 0.Using the Mayer-Vietoris sequence inductively, we immediately obtainH∗ω(E)∼= 0.This is a particular case of the Leray-Hirsch theorem for vertical one form (see theorem 3.3).Theorem 3.2Let (E,M,F, pi) be a fiber bundle over M . Suppose that M has a finite good cover. If there areglobal cohomology classes (e1, e2, ..., er) on E which when restricted to each fiber freely generatethe cohomology of the fiber, then H∗pi∗ω(E) is a free module over H∗ω(M) with basis (e1, e2, ..., er),i.e.H∗pi∗ω(E)∼= H∗ω(M)⊗ IR{e1, ..., er}∼= H∗ω(M)⊗H∗(F ).ProofThe assumption that M has a finite good cover is necessary for the induction argument.Remark also that the assumption that there are global cohomology classes (e1, e2, ..., er) on Ewhich when restricted to each fiber, freely generate the cohomology of the fiber, is necessarybecause all fibers have the same cohomology but their generators may be different. For exam-ple take x, y in M and put Ex = pi−1(x) and Ey = pi−1(y). Denote by (u1, u2, ..., ur) resp.7(v1, v2, ..., vr) the generators of de Rham cohomology of Ex resp. Ey. We have H∗(Ex) ∼=H∗(Ey), but (ui) may have no relation with (vj). The assumption means that the generators ofthe cohomology of the fibers over intersection agree since they are restrictions of ei.Let U and V be open contractible sets in M .Since U is contractible, there exists a smooth function f nowhere vanish such that ω |U = df andthen H∗ω(U)∼= H∗DR(U).Using this and the classical Leray-Hirsch theorem [5] we show thatH∗pi∗ω(E|U )∼= H∗(E|U ) ∼= H∗(U)⊗ IR{e1|U , ..., er|U}∼= H∗(U)⊗H∗(F )∼= H∗ω(U)⊗H∗(F ).Similarly, we getH∗pi∗ω(E|V )∼= H∗ω(V )⊗ IR{e1|V , ..., er|V }∼= H∗ω(V )⊗H∗(F )andH∗pi∗ω(E|U∩V )∼= H∗ω(U ∩ V )⊗ IR{e1|U∩V , ..., er |U∩V }∼= H∗ω(U ∩ V )⊗H∗(F ).So the Leray-Hirsch theorem holds for U , V and U ∩ V .Suppose M is the union of two open subsets U , V and definer : Ωp(M) −→ Ωp(U)⊕ Ωp(V ),θ 7−→ (θ|U , θ|V );s : Ωp(U)⊕ Ωp(V ) −→ Ωp(U ∩ V )(α, β) 7−→ α|U∩V − β|U∩V .Then the following is a short exact sequence of cochain complexes0 −→ Ωp(M)r−→ Ωp(U)⊕ Ωp(V )s−→ Ωp(U ∩ V ),−→ 0.So we obtain the following Mayer-Vietoris sequence;...−→ Hpω(U ∪ V ) −→ Hpω|U(U)⊕Hpω|V(V ) −→ Hpω|U∩V(U ∩ V ) −→ Hp+1ω (U ∪ V ) −→...and then, we get an exact sequence by tensoring with Hn−p(F )...−→ Hpω(U ∪ V )⊗Hn−p(F ) −→ (Hpω|U(U)⊗Hn−p(F ))⊕ (Hpω|V(V )⊗Hn−p(F )) −→−→ Hpω|U∩V(U ∩ V )⊗Hn−p(F ) −→ Hp+1ω (U ∪ V )⊗Hn−p(F ) −→...since tensoring with a vector space preserve exactness.Summing over p = 0, ..., n, yields the exact sequence;...−→ ⊕n0Hpω(U ∪ V )⊗Hn−p(F ) −→ ⊕n0 (Hpω|U(U)⊗Hn−p(F )) ⊕ (Hpω|V(V )⊗Hn−p(F )) −→−→ ⊕n0Hpω|U∩V(U ∩ V )⊗Hn−p(F ) −→ ⊕n0Hp+1ω (U ∪ V )⊗Hn−p(F ) −→...8It was checked in [5] page 49-50, that the last square of the following diagram is commutativefor the de Rham cohomology. Since the Lichnerowicz cohomology and the de Rham cohomologyfor contractible sets are isomorphic (see proposition 2.1(b)), the following diagram is commuta-tive;... //	⊕n0Hpω(U ∪ V )⊗Hn−p(F ) //ψ	⊕n0 (Hpω|U(U)⊗Hn−p(F ))⊕ (Hpω|V(V )⊗Hn−p(F )) //ψ|U⊕ψ|V	... // Hnpi∗ω(E) // Hnpi∗ω(E|U )⊕Hnpi∗ω(E|V ) ////	⊕n0Hpω|U∩V(U ∩ V )⊗Hn−p(F ) //ψ|U∩V	...// Hnpi∗ω(E|U∩V )// ...So if the Leray-Hirsch theorem holds for U , V and U ∩ V it also holds for U ∪ V by the”five lemma”. Finally, one chooses a good covering U such that every U ∈ U does only intersectfinitely many other sets of U . Then we can write M = W1∪ ...∪Wn where every Wi is a disjointunion of open balls in U . Since the Leray-Hirsch theorem holds for Wi, Wj and Wi ∩Wj (thelatter is also an disjoint union of open balls) it holds also for Wi ∪Wj . Proceeding inductivelywe get the result  In the next paragraph, we will prove the Leray-Hirsch theorem for vertical forms.Theorem 3.3Let (E,M,F, pi) be a fiber bundle over M and ω a closed 1-form on E. Suppose that M hasa finite good cover. If there are global cohomology classes (e2, ..., er) on E such that ([ω] =e1, e2, ..., er) restricted to each fiber freely generate the cohomology of the fiber, thenH∗ω(E)∼= H∗(M)⊗H∗i∗ω(F )where the application i : F −→ E is the natural induction.ProofLet U and V be open sets in M . The following diagram is commutativeU × Fpr2//φ##HHHHHHHHHFi~~||||||||E|Uwhere E|U = pi−1(U) and φ : U × F −→ E|U is a smooth chart of the fiberation.Let ω|U be the restriction of ω to E|U and ω1 = i∗ω|U .Set ω˜ =: pr∗1df + pr∗2ω1 where f is a non zero function on U , then [ω˜] = [pr∗2ω1] .9The diffeomorphism φ induces an isomorphism φ∗ : H∗ω|U(E|U )∼= H∗φ∗ω|U(U×F ). By the previouscommutative diagram, pr∗2i∗(ω|U ) = φ∗(ω|U ). ThereforeH∗φ∗ω|U(U × F ) ∼= H∗pr∗2i∗ω(U × F ),then by proposition 2.1 (b)H∗pr∗2i∗ω(U × F )∼= H∗pr∗1df+pr∗2i∗ω(U × F )and hence by Ku¨nneth formulaH∗pr∗1df+pr∗2i∗ω(U × F )∼= H∗df (U)⊗H∗i∗ω(F )∼= H∗(U)⊗H∗i∗ω(F ).So we have shownH∗ω|U(E|U )∼= H∗(U)⊗H∗ω1(F )for all open set in M .Similarly, we get H∗ω|V(E|V )∼= H∗(V )⊗H∗ω1(F ) and H∗ω|U∩V(E|U∩V )∼= H∗(U ∩V )⊗H∗ω1(F ).Using the Mayer-Vietoris sequences and the ”five lemma”, the result holds for U ∪V . Again,one can choose a good covering and proceeding inductively we get the result  AcknowledgmentsI would like to thank the Abdus Salam International Centre for Theoretical Physics for its warmhospitality and support during the preparation of the last phase of this work. Also many thanksto Augustin Banyaga and Jean-Pierre Ezin for their permanent encouragement, many helpfuldiscussions, advice and interest in this work.10References[1] A. Banyaga, Some properties of locally conformal symplectic structures, Comment. Math.Helv. 77(2002)383-398.[2] A. Banyaga, A geometric integration of the extended Lee homomorphism, Journal of Ge-ometry and Physics, 39(2001) 30-40.[3] A. Banyaga, Quelques invariants des structures localement conforme´ment symplec-tiques,C.R.Acad. Sci. Paris, Serie 1 (2001) 29-32.[4] A. Banyaga, On the geometry of locally conformal symplectic manifolds, Infinite dimen-tional Lie groups In Geometry and representation theory, A. Banyaga, J.Leslie and T.Robart Ed. World Scientific 2002 pp 79-91.[5] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New YorkHeidelberg Berlin,(1982).[6] C. Godbillon, Ele´ments de topologie alge`brique, Hermann, Paris, (1971), 45-76.[7] F. Guedira and A. Lichnerowicz, Ge´ome´trie des alge`bres de Lie locales de Kirilov, J. Math.Pures et Appl. 63(1984), 407-484.[8] S. Haller and T. Rybicki, On the group of diffeomorphisms preserving a locally conformalsymplectic structure. Ann. Global Anal. and Geom. 17(1999), 475-502.[9] I. Madsen and J. Tornehave, Riemannian geometry and geometry analysis, CambridgeUniversity Press (1999), 78-89.[10] M. de Leo´n, B. Lo´pez, J. C. Marrero, E. Padro´n, On the computation of the Lichnerowicz-Jacobi Cohomology, Journal of Geometry and Physics, 44(2003) 507-522.[11] M. Karoubi and C. Leruste, Algebraic Topology via Differential Geometry, London Math-ematical Society, Lecture Note Series 99, Cambridge University Press.[12] N. Nakahara, Geometry, topology and Physics. (1990), 199-201.[13] S. I Goldberg, Curvature and homology, Academic Press, New York, 1962.[14] I. Vaisman, Remarkable operators and commutation formulas on locally conformal Kaehlermanifolds, Composition Math. , 40(1980) 227-259.[15] M. Spivak, Differential geometry I, Publish or Prish, BErkeley (1979).[16] F. Warner, Fondations of differentiable manifolds and Lie groups. Springer-Verlag. (1983),189-191 et 205-206.11

On the Leray-Hirsch theorem for the Lichnerowicz cohomology

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