In 1964, John Stallings established an important relationship between the low-dimensional homology of a group and its lower central series. We establish a similar relationship between the low-dimensional homology of a group and its derived series. We apply this to the study of homology cobordism of manifolds. Using the Cheeger-Gromov von Neumann rho invariant, for each n, we define a real valued homology cobordism invariant rho_n. We show that, under mild hypotheses, the rho_n take on a dense set of values and are independent for varying n.