The K-theory of Heegaard quantum lens spaces

Abstract

Representing $\mathbb{Z}/N\mathbb{Z}$ as roots of unity, we restrict a natural $U(1)$-action on the Heegaard quantum sphere to $\mathbb{Z}/N\mathbb{Z}$, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of $\mathbb{Z}/N\mathbb{Z}$ to construct an associated complex line bundle. This paper proves the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer–Vietoris sequence, and an explicit form of the odd-to-even connecting homomorphism to prove the stable non-triviality of the bundles. On the algebraic side we prove the universality of the coordinate algebra of such a lens space for a particular set of generators and relations. We also prove the non-existence of non-trivial invertibles in the coordinate algebra of a lens space. Finally, we prolongate the $\mathbb{Z}/N\mathbb{Z}$-fibres of the Heegaard quantum sphere to $U(1)$, and determine the algebraic structure of such a $U(1)$-prolongation.