JournalsjncgVol. 7, No. 4pp. 1185–1216

The K-theory of Heegaard quantum lens spaces

  • Piotr M. Hajac

    IMPAN, Warsaw, Poland
  • Adam Rennie

    Australian National University, Canberra, Australia
  • Bartosz Zieliński

    University of Łódź, Poland
The K-theory of Heegaard quantum lens spaces cover
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Abstract

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

Cite this article

Piotr M. Hajac, Adam Rennie, Bartosz Zieliński, The K-theory of Heegaard quantum lens spaces. J. Noncommut. Geom. 7 (2013), no. 4, pp. 1185–1216

DOI 10.4171/JNCG/146