Quantization of branched coverings

TL;DR

This paper establishes a mathematical framework linking branched coverings of topological spaces with Hilbert C*-modules and conditional expectations, enabling the definition of non-commutative analogues of coverings.

Contribution

It introduces a novel correspondence between branched coverings, Hilbert C*-modules, and conditional expectations, extending classical topology into non-commutative geometry.

Findings

Branched coverings are identified with Hilbert C*-modules over C(X).

Faithful unital positive conditional expectations correspond to branched coverings.

Non-commutative analogues of coverings are defined via this framework.

Abstract

We identify branched coverings (continuous open surjections p:Y->X of Hausdorff spaces with uniformly bounded number of pre-images) with Hilbert C*-modules C(Y) over C(X) and with faithful unital positive conditional expectations E:C(Y)->C(X) topologically of index-finite type. The case of non-branched coverings corresponds to projective finitely generated modules and expectations (algebraically) of index-finite type. This allows to define non-commutative analogues of (branched) coverings.