Adelic solenoid I: Structure and topology

Abstract: Topologically the adelic Riemann sphere is the suspension of the adelic solenoid and because of this relation, here we study the adelic solenoid by studying the adelic Riemann sphere topology. The main result is the Birkhoff-Grothendieck Theorem: A holomorphic vector bundle splits as a sum of holomorphic line bundles whose Chern character is now a rational number. As a consequence, the Picard group is isomorphic to the additive group of rational numbers and the $K$--ring has new elements that factor the tautological class.