Structured vector bundles define differential \(K\)-theory. (English) Zbl 1216.19009

Blanchard, Etienne (ed.) et al., Quanta of maths. Conference on non commutative geometry in honor of Alain Connes, Paris, France, March 29–April 6, 2007. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-5203-3/pbk). Clay Mathematics Proceedings 11, 579-599 (2010).
Summary: An equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called structured bundles, and their isomorphism classes form an abelian semiring. By applying the Grothendieck construction one obtains the ring \(\widehat K\), elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary \(K\) and the Chern-Weil form. This construction provides a simple model of differential \(K\)-theory, cf. M. J. Hopkins and I. M. Singer [J. Differ. Geom. 70, No. 3, 329–452 (2005; Zbl 1116.58018)], as well as a useful codification of vector bundles with connection.
For the entire collection see [Zbl 1206.00042].


19L50 Twisted \(K\)-theory; differential \(K\)-theory


Zbl 1116.58018
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