Journal of Noncommutative Geometry

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Volume 8, Issue 1, 2014, pp. 61–105
DOI: 10.4171/JNCG/149

Published online: 2014-03-20

Line bundles and the Thom construction in noncommutative geometry

Edwin Beggs[1] and Tomasz Brzeziński[2]

(1) University of Wales Swansea, UK
(2) University of Wales Swansea, UK

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct $\mathbb{Z}$- and $\mathbb{N}$-graded algebras, the $\mathbb{Z}$-graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the $\mathbb{N}$-graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.

Keywords: Morita context, C*-algebra, bimodules, line bundles, Thom construction, Hopf–Galois extension, Chern class

Beggs Edwin, Brzeziński Tomasz: Line bundles and the Thom construction in noncommutative geometry. J. Noncommut. Geom. 8 (2014), 61-105. doi: 10.4171/JNCG/149