Given a sample of an abstract manifold immersed in some Euclidean space, we describe a way to recover the singular homology of the original manifold. It consists in estimating its tangent bundle---seen as subset of another Euclidean space---in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology. The construction we propose is consistent and stable, and does not involve the knowledge of the dimension of the manifold. In order to obtain quantitative results, we introduce the normal reach, which is a notion of reach suitable for an immersed manifold.