The K-theory of Heegaard quantum lens spaces

TL;DR

This paper investigates the K-theory and algebraic properties of Heegaard quantum lens spaces, demonstrating the stable non-triviality of associated line bundles and analyzing their algebraic structure using K-theory and homomorphisms.

Contribution

It introduces a new approach to compute K-theory of quantum lens spaces and proves the stable non-triviality of certain line bundles over these spaces.

Findings

Proves the stable non-triviality of line bundles over quantum lens spaces.

Computes the K-theory of quantum lens spaces using Mayer-Vietoris sequence.

Establishes the universality and non-existence of non-trivial invertibles in the coordinate algebra.

Abstract

Representing Z/N as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/N, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/N 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 Bass 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 Z/N-fibres of the Heegaard quantum sphere to U(1), and determine the algebraic structure of such a U(1)-prolongation.