Twisted topological correspondences and Cartan subalgebras

We introduce twisted topological correspondences, which generalize both Katsura's topological correspondences as well as the twisted topological graphs introduced by Li. We show that, up to isomorphism, they are in bijection with certain principal bundles. This makes it possible to study topological correspondences using the machinery of principal and fiber bundles. We show how to associate a $C^*$-correspondence to a twisted topological correspondence, and give two different characterizations of the $C^*$-correspondences arising that way. The first one is the existence of an atlas of the vector bundle associated to the $C^*$-correspondence whose transition functions take values in a certain subgroup of the unitary group $U(n)$, and which is in some sense compatible with the left action. The other characterization is in terms of Cartan subalgebras in the compact operators on the $C^*$-correspondence. We use our findings to prove rigidity results of the $C^*$-correspondences associated to twisted topological correspondences.