The K-theory of Heegaard quantum lens spaces

Authors

Piotr M. Hajac, Uniwersytet Warszawski
Adam Rennie, Australian National UniversityFollow
Bartosz Zielinski, University Of Lodz

RIS ID

76331

Publication Details

Hajac, P. M., Rennie, A. & Zielinski, B. (2013). The K-theory of Heegaard quantum lens spaces. Journal of Noncommutative Geometry, 7 (4), 1185-1216.

Abstract

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