  We study relative differential and integral forms and their cohomology on
families of supermanifolds and prove a relative version of Poincar\'e duality
relating the cohomology of differential and integral forms in different
geometric categories. In the smooth category, we complement Poincar\'e duality
by proving the compactly supported Poincar\'e lemmas for both differential and
integral forms, filling a gap in the literature. We then use our results to
study supergravity. In particular, we describe supergravity in dimension three
via higher Cartan structures, which can be regarded as certain classes of
connections taking values in $L_\infty$-superalgebras. We then use relative
Poincar\'e duality to state a general form of the action principle in a
mathematically rigorous manner. This interpolates in a unified fashion between
two equivalent formulations of supergravity, one based on the superspace
approach and the other on the group manifold approach in the physics
literature.
