La cohomologie basique d'un feuilletage riemannien est de dimension finie (Q791933)

Let f be a codimension n foliation on a compact connected manifold M. (i) If F is transversally parallelisable we construct a spectral sequence converging to the base-like cohomology \(H^*(M/F)\) of the foliated manifold (M,F) and show that this cohomology is finite dimensional. (ii) If F is Riemannian, the lifted foliation \(\hat F\) to the transverse orthonormal frame bundle \(SO(n)\to \hat M\to M\) is transversally parallelisable. To this fiber bundle we associate a spectral sequence connecting the base-like cohomology of (M,F), the cohomology of (\^M,\^F). Using this spectral sequence and (i) we show that \(H^*(M/F)\) is finite dimensional and \(H^ n(M/F)\) is isomorphic to 0 or \({\mathbb{R}}\).