GENERALISED WINDING NUMBERS WITH APPLICATIONS TO THE <i>K</i> -THEORY OF ADELIC AND RATIONAL GROUP <i>C</i> ^* -ALGEBRAS

Abstract We establish a two-part framework for analysing homotopy equivalence in periodic adelic functions. First, we introduce the concept of pre-periodic functions and define their homotopy invariant through the construction of a generalised winding number. Second, we establish a fundamental correspondence between periodic adelic functions and pre-periodic functions. By extending the generalised winding number to periodic adelic functions, we demonstrate that this invariant completely characterises homotopy equivalence classes within the space of periodic adelic functions. Building on this classification, we further obtain an explicit description of the $K_{1}$ -group of the rational group $C^\ast $ -algebra, $K_{1}(C^{*}(\mathbb {Q}))$ . Finally, we employ a similar strategy to derive the K -theory of $C^{\ast }(\mathbb {A})$ .