Hitchin famously used index-theoretic arguments to show that the space
of positive scalar curvature metrics on some closed spin manifolds is not
connected, and his work has been extended by Crowley, Schick, and Steimle
who proved that the space has nontrivial homotopy groups in infinitely
many degrees. In the talk I shall explain how to adapt the technique to
the space of complete metrics of nonnegative sectional curvature on
certain open spin manifolds. A new ingredient of independent interest is
homotopy density of the subspace of metrics with cylindrical ends.