  Often, high dimensional data lie close to a low-dimensional submanifold and
it is of interest to understand the geometry of these submanifolds. The
homology groups of a manifold are important topological invariants that provide
an algebraic summary of the manifold. These groups contain rich topological
information, for instance, about the connected components, holes, tunnels and
sometimes the dimension of the manifold. In this paper, we consider the
statistical problem of estimating the homology of a manifold from noisy samples
under several different noise models. We derive upper and lower bounds on the
minimax risk for this problem. Our upper bounds are based on estimators which
are constructed from a union of balls of appropriate radius around carefully
selected points. In each case we establish complementary lower bounds using Le
Cam's lemma.
