Some properties of a cohomology group associated to a totally geodesic foliation (Q1064618)

Let \({\mathcal F}\) be an orientable totally geodesic foliation of dimension p on a compact connected manifold M. Associated to \({\mathcal F}\), there is a natural cohomology group \(H^*_{tg}\). We prove that \(H^*_{tg}\) has finite dimension and that \(H^ p_{tg}\cong {\mathbb{R}}\) or 0. Further, we construct an element \(c\in H^ 1_{tg}\) such that \(H^ p_{tg}\cong {\mathbb{R}}\) if and only if \(c=0\). We show that when \(H^ p_{tg}\cong {\mathbb{R}}\), then \(H^*_{tg}\cong H_{tg}^{*^{-p}}\). Moreover, if \({\mathcal F}\) has a compact leaf L, then there is an injection of \(H^*_{tg}\) into \(H^*(L)\).