  In this paper we study the homology of a random Cech complex generated by a
homogeneous Poisson process in a compact Riemannian manifold M. In particular,
we focus on the phase transition for "homological connectivity" where the
homology of the complex becomes isomorphic to that of M. The results presented
in this paper are an important generalization of [7], from the flat torus to
general compact Riemannian manifolds. In addition to proving the statements
related to homological connectivity, the methods we develop in this paper can
be used as a framework for translating results for random geometric graphs and
complexes from the Euclidean setting into the more general Riemannian one.
