A cohomology for foliated manifolds

Abstract

for a e T(v A V ) , X1? . . . , Xk + 1 e T(T). Since the curvature tensor of V restricted to T is identically zero we have that d o d = 0. Denote the homology of this complex by F*(T; V). This is the cohomology of the Lie algebra of vector fields tangent to the foliation with coefficients in sections of the normal bundle, the representation being given by the connection [GF]. In general the groups F\x ; v) are not finitely generated (the complex is not elliptic) but they satisfy the following. (i) F* is a functor from the category of foliated manifolds and transverse maps to the category of abelian groups and homomorphisms. (ii) If ƒ : N -» M is an embedded transverse submanifold, we can define relative cohomology groups F*(T; v, ƒ) and obtain the usual long exact sequence. (iii) F* is an invariant of the diffeomorphism type of the foliation. However, F* is not an invariant of the integrable homotopy type of the foliation when M is an open manifold.