  We show that the moduli space of nonnegatively curved metrics on each member
of a large class of 2-connected 7-manifolds, including each smooth manifold
homeomorphic to $S^7$, has infinitely many connected components. The components
are distinguished using the Kreck-Stolz $s$-invariant computed for metrics
constructed by Goette, Kerin and Shankar. The invariant is computed by
extending each metric to the total space of an orbifold disc bundle and
applying generalizations of the Atiyah-Patodi-Singer index theorem for
orbifolds with boundary. We develop methods for computing characteristic
classes and integrals of characteristic forms appearing in index theorems for
orbifolds, in particular orbifolds constructed using Lie group actions of
cohomogeneity one.
