Coverings, correspondences, and noncommutative geometry

Understand

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.