On the Nielsen-Thomsen sequence

The Nielsen-Thomsen sequence plays a pivotal role in refining invariants for C$^*$-algebras beyond the Elliott classification framework. This paper revisits the sequence, introducing the concepts of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms, to better understand its unnatural splitting. These insights enable novel comparison methods for *-homomorphisms at the level of the Hausdorffized algebraic K$_1$-groups, and subsequently the Hausdorffized unitary Cuntz group. We apply our methods to classification via the Hausdorffized unitary Cuntz semigroup. In particular, we present a new proof of the non-isomorphism between two A$\mathbb{T}$-algebras constructed by Gong, Jiang and Li. We also exhibit several pairs of non-unitarily equivalent *-homomorphisms with domain C($\mathbb{T}$).