Alexander-Spanier cohomology of foliated manifolds

Abstract

For a smooth foliated manifold $(M,\mathcal F)$, the basic and the foliated cohomologies are defined by using the de Rham complex of $M$. These cohomologies are related with the cohomology of the manifold by the de Rham spectral sequence of $\mathcal F$. A foliated manifold is an example of a space with two topologies, one coarser than the other. For these spaces one can define a continuous cohomology that, for a foliated manifold, corresponds to the continuous foliated (or leafwise) cohomology. In this paper we introduce a construction for spaces with two topologies based upon the Alexander-Spanier continuous cochains. It allows us to define a spectral sequence, similar to the de Rham spectral sequence for a foliation. In particular, continuous basic and foliated cohomologies are defined and related with the cohomology of the space. For a smooth foliated manifold, we also consider Alexander-Spanier differentiable cochains. We compare the continuous and differentiable cohomologies, and the latter with the de Rham cohomology. We prove that all three spectral sequences are isomorphic from $E_2$ onwards if $\mathcal F\/$ is a Riemannian foliation. As a consequence, we conclude that this spectral sequence is a topological invariant of the Riemannian foliation. We also compute some examples. In particular, we give an isomorphism between the $E_2$ term for a $G$-Lie foliation and the reduced cohomology of $G$ (in the sense of S.-T. Hu) with coefficients in the reduced foliated cohomology of $\mathcal F$.