A finiteness theorem for transitive foliations and flat vector bundles
Let \({\mathcal F}\) be a foliation on a manifold \(M,{\mathcal X} (M)\) the Lie algebra of vector fields on \(M\). Let \({\mathcal X} (M,{\mathcal F})\) be the Lie algebra of infinitesimal transformations of \((M, {\mathcal F})\). The foliation \({\mathcal F}\) is called transitive if \({\mathcal X}(M,{\mathcal F})\) is a transitive space. Let \(E\) be a flat vector bundle over \(M\) with a flat connection \(\nabla\) and \((A(M,E),d\nabla)\) the complex of forms on \(M\) with values in \(E\). \((A(M,E), d\nabla)\) together with a filtration \(F^kA (M,E)\) by \(A(M)\)-modules of \(A(M,E)\) is a filtered complex; so there is a spectral sequence \((E_i(\nabla), (d\nabla)_i)\) which converges to the real cohomology \(H(M,E)\) of \((A(M,E), d\nabla)\) after a finite number of steps. The aim of this paper is to study the spectral sequence \((E_i(\nabla), (d\nabla)_i)\). The author gives a finiteness theorem for this spectral sequence. The proof is based on the Riesz theory of compact operators. Theorem: Let \({\mathcal F}\) be a transitive foliation on a compact manifold \(M\), and let \(\nabla\) be a flat connection on a smooth vector bundle \(E\) over \(M\). Then the spectral sequence \((E_i(\nabla), (d\nabla)_i)\) of the filtered complex \((A(M,E), d\nabla)\) satisfies: (1) There exists a finite dimensional topological filtered subcomplex \(K\subset A(M,E)\) with spectral sequence \(E_i(K)\) such that the induced linear maps \(E_2(K) \to E_2(\nabla)\) and \(E_2(K)\to \mathbb{E}_2 (\nabla)\) are topological isomorphisms, where \(\mathbb{E}_2 (\nabla) =H(\mathbb{E}_1 (\nabla))\), \(\mathbb{E}_1 (\nabla)= E_1(\nabla)/ \overline O\). (2) \(E_2 (\nabla)\) and \(\mathbb{E}_2(\nabla)\) are finite dimensional Hausdorff, and \(E_2(\nabla) \cong\mathbb{E}_2 (\nabla)\) canonically and topologically, so that \(H(\overline O)=0\). (3) Each \(E_i (\nabla) \cong E_i(K)\), \(2\leq i\leq \infty\), and \(H(M,E) \cong H(K)\) is finite dimensional and its topology is the Euclidean topology. The identities \(E_{i+1} (\nabla) \equiv H(E_i(\nabla))\), \(0\leq i<\infty\), are also topological. Some examples of homogeneous Lie foliations are discussed in more detail.
scientific article; zbMATH DE number 1239667
scientific article