JournalsjncgVol. 8, No. 1pp. 61–105

Line bundles and the Thom construction in noncommutative geometry

  • Edwin Beggs

    University of Wales Swansea, UK
  • Tomasz Brzeziński

    University of Wales Swansea, UK
Line bundles and the Thom construction in noncommutative geometry cover
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Abstract

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 Z\mathbb{Z}- and N\mathbb{N}-graded algebras, the Z\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 N\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.

Cite this article

Edwin Beggs, Tomasz Brzeziński, Line bundles and the Thom construction in noncommutative geometry. J. Noncommut. Geom. 8 (2014), no. 1, pp. 61–105

DOI 10.4171/JNCG/149