Journal of Noncommutative Geometry


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Volume 12, Issue 3, 2018, pp. 823–863
DOI: 10.4171/JNCG/292

Published online: 2018-07-27

The $K$-theory of twisted multipullback quantum odd spheres and complex projective spaces

Piotr M. Hajac[1], Ryszard Nest[2], David Pask[3], Aidan Sims[4] and Bartosz Zieliński[5]

(1) Polish Academy of Sciences, Warsaw, Poland
(2) University of Copenhagen, Denmark
(3) University of Wollongong, Australia
(4) University of Wollongong, Australia
(5) University of Lodz, Poland

We find multipullback quantum odd-dimensional spheres equipped with natural $U(1)$-actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non-isomorphic, and that the $K$-groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts. We show that these $K$-groups remain the same for more general twisted versions of our quantum odd spheres and complex projective spaces.

Keywords: Free action on $C$*-algebras, associated noncommutative line bundle, multipullback and higher-rank graph $C$*-algebras, noncommutative deformation

Hajac Piotr, Nest Ryszard, Pask David, Sims Aidan, Zieliński Bartosz: The $K$-theory of twisted multipullback quantum odd spheres and complex projective spaces. J. Noncommut. Geom. 12 (2018), 823-863. doi: 10.4171/JNCG/292