Covering Dimension of C*-Algebras and 2-Coloured Classification

We introduce the concept of finitely coloured equivalence for unital ∗ ^* -homomorphisms between C ∗ C^* -algebras, for which unitary equivalence is the 1 1 -coloured case. We use this notion to classify ∗ ^* -homomorphisms from separable, unital, nuclear C ∗ C^* -algebras into ultrapowers of simple, unital, nuclear, Z Z -stable C ∗ C^* -algebras with compact extremal trace space up to 2 2 -coloured equivalence by their behaviour on traces; this is based on a 1 1 -coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z Z -stable C ∗ C^* -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a “homotopy equivalence implies isomorphism” result for large classes of C ∗ C* -algebras with finite nuclear dimension.