  We study global gravitational anomalies in type IIB string theory with
nontrivial middle cohomology. This requires the study of the action of
diffeomorphisms on this group. Several results and constructions, including
some recent vanishing results via elliptic genera, make it possible to consider
this problem. Along the way, we describe in detail the intersection pairing and
the action of diffeomorphisms, and highlight the appearance of various
structures, including the Rochlin invariant and its variants on the mapping
torus.
