  We construct an additive category where objects are embedded graphs in the
3-sphere and morphisms are geometric correspondences given by 3-manifolds
realized in different ways as branched covers of the 3-sphere, up to branched
cover cobordisms. We consider dynamical systems obtained from associated
convolution algebras endowed with time evolutions defined in terms of the
underlying geometries. We describe the relevance of our construction to the
problem of spectral correspondences in noncommutative geometry.
