The Homotopy 3-Type of Abelian C*-Algebras

TL;DR

This paper calculates the homotopy groups of unital abelian C*-algebras within a specific categorical framework, linking algebraic invariants to the topology of the underlying space.

Contribution

It provides explicit computations of homotopy groups for abelian C*-algebras in the Morita 3-category, connecting algebraic structures to topological data.

Findings

Homotopy groups are described in terms of the topological invariants of the space T.

Actions of the first homotopy group on higher groups are explicitly computed.

Results relate algebraic homotopy groups to the topology of the underlying compact space.

Abstract

We compute the homotopy groups at each unital abelian C*-algebra $C(T)$ in the Morita $3$-category of abelian C*-algebras, C*-algebras with central maps, C*-correspondences, and adjointable bimodule maps. We describe these groups in terms of the topological data of the underlying compact Hausdorff space $T$. We also compute the actions of the first homotopy group on the second and third homotopy groups in terms of these topological invariants of $T$.