The fundamental group of a noncommutative space

TL;DR

This paper defines a fundamental group for noncommutative spaces using differential graded algebras and flat connections, generalizing classical notions and establishing key invariance properties.

Contribution

It introduces a new noncommutative fundamental group via Tannakian categories of flat connections, extending classical topology to noncommutative geometry.

Findings

Fundamental group of noncommutative torus described as algebraic hull of a topological group

Category of flat connections forms a Tannakian category, linking to affine group schemes

Fundamental group is functorial, homotopy, and Morita invariant

Abstract

We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme $G$, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define $G$ to be the fundamental group of the noncommutative space under consideration. The needed assumptions on the differential graded algebra are rather mild and completely natural in the context of noncommutative differential geometry. We establish the appropriate functorial properties, homotopy and Morita invariance of this fundamental group. As an example we find that the fundamental group of the noncommutative torus can be described as the algebraic hull of the topological group $(\mathbb Z+\theta \mathbb Z)^2$.