A finiteness theorem for foliated manifolds

Abstract

This theorem is proved in no. 6) below: the method of proof, which is of independent interest, consists in making a parametrix for the exterior derivative by averaging over flows and then employing the theory of compact operators.In no. 1) we define the filtration and in no. 2) by using the theory of exact couples reduce the finiteness problem to an equivalent simpler form (Lemma 3). In no. 3) the relevant results from functional analysis are recalled,

On riemannian foliations with minimal leaves

“…Let ^ = H{8^. Clearly we have Ef = £f and E^ = £'^ , and we also obtain the associated long exact sequence in cohomology If T is Riemannian and M compact, then £2,^2, and ff(Oi) are of finite dimension [I], [2], [22], [23] (see also [7], [8], [13]). In particular, Ej'°i s of dimension zero or one [7], [8].…”

“…When T is Riemannian [20], the associated spectral sequence verifies some properties offiniteness and duality [I], [2], [3], [7], [8], [13], [14], [15], [22], [23]. Thus, in this case, the cohomological study of the minimality of the leaves has special interest.…”

“…Then we have the associated spectral sequence (Ei,di) (defined for example in [22]). It carries the vector space topology induced by the C°° -topology on the de Rham complex.…”

A decomposition theorem for the spectral sequence of Lie foliations

“…The spectral sequence (Ei' dd associated to a smooth foliation gr on a manifold M is defined for example in [Sa,KT2,and Se2]. The ~OO-topology in the space of differential forms induces in EI a topology which in general is not Hausdorff [Hal], obtaining two new bigraded differential algebras: The closure OE, of the trivial subspace in EI and the quotient g'J = EdOE, [He and A2].…”

“…If gr is Riemannian and M is compact there are several papers studying the finite-dimensional character and the duality in E2, S2 = H(g'J) and H(OE,) , or in parts of them: [Sa,EH,ESH,Sel,Se2,He,Al and A2]. For this type of foliations, with M oriented, it is proved in [A2] that the de Rham duality map induces in H ( OE,) and S2 different types of duality, so the possibility of obtaining duality in E2 depends on the properties of the canonical long exact sequence which relates these three homologies.…”