Ana Rodríguez-Fernández

Xosé M. Masa

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 342 (2006) 601–604http://france.elsevier.com/direct/CRASS1/TopologyCohomology of singular Riemannian foliationsXosé M. Masa 1, Ana Rodríguez-FernándezDepartamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, SpainReceived 13 January 2004; accepted 21 February 2006Available online 20 March 2006Presented by Alain ConnesAbstractAssociated to a smooth foliation (M,F ) are defined the basic and the foliated cohomologies. These cohomologies are relatedto the de Rham cohomology by the de Rham spectral sequence of F , Es,tr,dR(M), constructed by filtering the de Rham complexof the manifold. For a Riemannian foliation on a compact manifold the second term of this spectral sequence, Es,t2,dR(M), is finitedimensional and a topological invariant.In this Note we prove these two results, fitness dimension and topological invariance, for the cohomology of singular Riemannianfoliations. The proof uses the previous theorems for the regular case and the structure of singular Riemannian foliations describedby P. Molino. For the basic cohomology these results have been proved by R. Wolak. To cite this article: X.M. Masa, A. Rodríguez-Fernández, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméCohomologie des feuilletages riemanniens singuliers. La filtration du complexe de De Rham d’une variété feuilletée (M,F )définit une suite spectrale Es,tr,dR(M) reliant les cohomologies basique et feuilletée du feuilletage F à la cohomologie de De Rhamde la variété ambiance. Pour un feuilletage riemannien d’une variété compacte le deuxième terme de cette suite spectrale est uninvariant topologique de dimension finie.Dans cette Note, nous prouvons la finitude et l’invariance topologique de ce terme pour les feuilletages riemanniens singuliers.La preuve combine les résultats du cas régulier avec la description de la structure des feuilletages riemanniens singuliers faitepar P. Molino. Pour la cohomologie basique, ces résultats ont été prouvés par R. Wolak. Pour citer cet article : X.M. Masa,A. Rodríguez-Fernández, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.E-mail address: xosemasa@usc.es (X.M. Masa).URL: http://xtsunxet.usc.es/xmasa/ (X.M. Masa).1 Partially supported by grant DGICYT BFM2001-3280.1631-073X/$ – see front matter  2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2006.02.022602 X.M. Masa, A. Rodríguez-Fernández / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 601–6041. Preliminaries1.1. Singular Riemannian foliations (see [3, Chapter 6])We assume that the manifold M is compact. A singular Riemannian foliation on M is a partition F of M byconnected immersed submanifolds, the leaves, such that the module of smooth fields tangent to the leaves is transi-tive on each leaf and there exists a Riemannian metric that is adapted to F in the sense that every geodesic that isperpendicular at one point to a leaf remains perpendicular to every leaf it meets.We denote by F the partition of M by leaf closures and by M/F the corresponding quotient space. The mapp :M → M/F has the following good local property. Let x be a point of M/F , X = p−1(x). The foliation F defineson X a regular Riemannian foliation FX with dense leaves and X has a base of saturated tubular neighborhoods,X̃ ×π V , quotient of X̃ × V by the diagonal action of π = π1(X), where X̃ is the universal covering space of X andV is an open set of Rk , k the codimension of X in M , and where π acts on V by linear isometries.The homothetic lemma [3] states that X and X̃ ×π V are homotopically equivalent by homotopies that preserve thefoliation, in the sense that each leaf goes into a leaf for each parameter of the homotopy. The homotopy equivalencesare the protection p : X̃ ×π V → X and the zero section s0 :X → X̃ ×π V .The open sets p(X̃ ×π V ) will be called adapted open sets in M/F . A cover by adapted open sets is said to begood if any finite nonempty intersection is also adapted (for its existence, see [4, Proposition 1]).1.2. The Čech complexLet W be a topological space, L a sheaf on W and U = {Uα; α ∈ I } an open cover of W . We shall use σn to denotean ordered n-simplex 〈α0, . . . , αn〉 of the nerve N(U) of U, and Uσn to denote Uα0,...,αn = Uαo ∩ · · · ∩ Uαn . We definethe Čech complex of U with coefficients in L asČn(U;L) =∏σnL(Uσn),with the usual differentialδ :Cn(U,L) −→ Cn+1(U,L),(δw)α0···αn+1 =n+1∑i=0(−1)ij∗i wα0···α̂i ···αn+1,where ji denotes the inclusion U〈α0,...,αn+1〉 → U〈α0,...,α̂i ,...,αn+1〉.If we start with a differential sheaf (L∗, d), we get a double complex Č∗(U;L∗) and two associated spectralsequences,IEs,t2 = Ȟ s(U,Ht(L∗))and IIEs,t2 = Hs(Ht(U,L∗)).We are interested in the case where L∗ is acyclic, then Ht(U,L∗) = Ht(W,L∗) = 0 for t > 0 and we haveIEs,t2 = Ȟ s(U,Ht(L∗)) ⇒ Hs+t(Γ (L∗)). (1)1.3. Alexander–Spanier spectral sequence [2]Let X be a topological space, and X′ the same set as X with a finer topology. Let U be an open set in X. A mapϕ :Up+1 → Ris said to be a basic Alexander–Spanier p-cochain in U if it is locally constant when one considers in Up+1 thetopology induced by X′.For each U , the vector space of basic Alexander–Spanier cochains in U , with the obvious restriction maps, definesa presheaf, which generates the sheaf of basic Alexander–Spanier cochains AS∗(X′|X). With the usual differentialδϕ(x0, . . . , xp) =p∑(−1)iϕ(x0, . . . , x̂i , . . . , xp) (2)i=0X.M. Masa, A. Rodríguez-Fernández / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 601–604 603we have a resolution of the constant sheaf RX and the Alexander–Spanier spectral sequence of (X′|X),Ep,q2 (X′ | X) = HpHq(X,AS(X′|X)) ⇒ Hp+q(X,R). (3)One can consider only continuous or differentiable Alexander–Spanier cochains, to get a continuous or differentiableresolution, respectively, and the corresponding spectral sequences. Let (M,F) be a foliated manifold, M = ⋃x∈M Lx .Denote by MF the set M with the leaf topology: a basis is formed by the connected components of intersections ofopen sets of M with leaves. The Alexander–Spanier sheaf and the spectral sequence of the foliated manifold will bethe one associated to (MF | M). We use the notationcontAS∗F , diffAS∗Ffor the continuous and differentiable Alexander–Spanier sheaves, respectively, andEp,qr,cont(M) and Ep,qr,diff(M)for the corresponding spectral sequences.2. Finiteness of E2,dR(M)The de Rham spectral sequence of F , Ep,qr,dR(M), can be obtained from the resolution of the constant sheaf RMgiven by the sheaves of basic forms, (AF , d). In fact, Ep,qr,dR(M) ∼= HpHq(M,A∗F ).Let Hq(p,AtF ) be the sheaf in M/F defined byHq(p,AtF)(U) = Hq(p−1(U),AtF)for an open set U ⊂ M/F . It is the derived sheaf of AtF . With the differential induced by d , Hq(p,A∗F ) is a differentialsheaf. As H0(p,A0F ) is soft, and the sheaves Hq(p,AtF ) are H0(p,A0F )-modules, they are acyclic.Theorem 2.1. The second term of the de Rham spectral sequence of a singular Riemannian foliation is finite dimen-sional.We consider a finite good cover U of M/F . We consider the double complex Čs(U,Hq(p,AtF )) and the spectralsequenceEs,t2 = Ȟ s(U,Ht(Hq(p,A∗F))) ⇒ Es+t,q2,dR (M).Now, for any Uσ , σ ∈ N(U), denote by Xσ the leaf closure such that p−1(Uσ ) is homotopically equivalent to it. WehaveHt(Hq(p,A∗F))(U) = Et,q2,dR(p−1(U)) ∼= Et,q2,dR(Xσ ).The foliation on Xσ being regular, Et,q2,dR(X) is finite dimensional [1]. As a consequence, already the complexČs(U,Ht (Hq(p,A∗F ))) is finite dimensional, so Ȟ s(U,Ht (Hq(p,A∗F ))) and Es+t,q2 (F) are finite dimensional.3. Topological invariance of E2(F)Theorem 3.1. The de Rham spectral sequence of a singular Riemannian foliation is a topological invariant from E2onwards.For each differential sheaf A∗F , diffAS∗F and contAS∗F , and each q  0, we have three spectral sequences ofthe type (1), constructed with the Čech complex associated to a good cover U of M/F and to the derived sheavesHq(p,A∗F ), Hq(p, diffAS∗F ) and Hq(p, contAS∗F ). These spectral sequences areȞ s(U,Ht(Hq(p,A∗F))) ⇒ Es+t,q2,dR (M),Ȟ s(U,Ht(Hq(p, diffAS∗))) ⇒ Es+t,q(M),F 2,diff604 X.M. Masa, A. Rodríguez-Fernández / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 601–604andȞ s(U,Ht(Hq(p, contAS∗F))) ⇒ Es+t,q2,cont(M).We are going to prove that all of them are isomorphic to each other from E1 onwards. But the latter, constructed withcontAS∗F , is obviously invariant by homeomorphisms, and so the theorem follows.Let I : diffAS∗F → contAS∗F be the inclusion map. We define another homomorphism of differential sheavesΛ : diffAS∗F →A∗F in a standard way: for an open set U of M , for x ∈ U and Z1, . . . ,Zp ∈ TxM ,Λ(ϕ)x(Z1, . . . ,Zp) = 1p!∑τ∈Spsgn(τ )∂∂ε1· · · ∂∂εpϕ(x, expx ε1Zτ(1), . . . , expx εpZτ(p))|εi=0,where εi ∈ R,1  i  p.The homomorphisms I and Λ induced isomorphisms between the sheaf Ht (Hq(p, diffAS∗F ) and the sheavesHt (Hq(p,A∗F ) and Ht (Hq(p, contAS∗F ), respectively. To prove that, we compute their stacks. Let U be an adaptedopen set such that p−1(U) is a tubular neighborhood of X = p−1(x). By the same argument of the proof of Theorem 1,we see that the space of sections over U of these sheaves areEt,q2,diff(p−1(U)) ∼= Et,q2,diff(X), Et,q2,dR(p−1(U)) ∼= Et,q2,dR(X),andEt,q2,cont(p−1(U)) ∼= Et,q2,cont(X).But on X the foliation is Riemannian regular, so the homomorphisms I and Λ induce isomorphisms between thesecond terms of the spectral sequences [2].References[1] J.A. Álvarez López, A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois J. Math. 33 (1989) 79–92.[2] X.M. Masa, Alexander–Spanier cohomology of foliated manifolds, Illinois J. Math. 46 (2002) 979–998.[3] P. Molino, Riemannian Foliations, Progr. Math., Birkhäuser, 1988.[4] R. Wolak, Basic cohomology for singular Riemannian foliations, Monatsh. Math. 128 (1999) 159–163.

Cohomology of singular Riemannian foliations

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Cohomology of singular Riemannian foliations

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