Building on the work of von Neumann and Wigner, M. Berry showed that there are topologically protected level-crossings in the space of quantum systems. These level-crossings can be detected using the curvature of the Berry connection. In this talk I will describe analogs of this for interacting lattice systems in infinite volume. Although the Berry connection is ill-defined for such systems, one can construct analogs of the Berry curvature on the parameter space which turn out to be closed form of degree d+2, where d is the dimension of the lattice. Integrals of these forms probe the non-trivial topology of the space of gapped lattice systems and can be used to detect topologically protected gapless points in the phase diagram. I will also explain the relation between higher Berry curvatures and the occurrence of topologically protected gapless edge modes in families of gapped lattice systems.