  We prove a rigidity property for mapping tori associated to minimal
topological dynamical systems using tools from noncommutative geometry. More
precisely, we show that under mild geometric assumptions, an
orientation-preserving leafwise homotopy equivalence of two mapping tori
associated to $\mathbb{Z}^d$-actions on a compact space can be lifted to an
isomorphism of their foliation $C^*$-algebras. This property is a
noncommutative analogue of topological rigidity in the context of foliated
spaces whose space of leaves is singular, where isomorphism type of the
$C^*$-algebra replaces homeomorphism type. Our technique is to develop a
geometric approach to the Elliott invariant that relies on topological and
index-theoretic data from the mapping torus. We also discuss how our
construction can be extended to slightly more general homotopy quotients
arising from actions of discrete cocompact subgroups of simply connected
solvable Lie groups, as well as how the theory can be applied to the magnetic
gap-labelling problem for certain Cantor minimal systems.
