Coverings, correspondences, and noncommutative geometry

TL;DR

This paper develops a new categorical framework using embedded graphs and 3-manifolds as branched covers to explore spectral correspondences in noncommutative geometry, linking geometric and algebraic structures.

Contribution

It introduces an additive category with morphisms as geometric correspondences, connecting branched cover cobordisms to noncommutative spectral problems.

Findings

Constructed a category of embedded graphs and 3-manifolds.

Linked convolution algebras with geometric time evolutions.

Explored applications to spectral correspondences in noncommutative geometry.

Abstract

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.