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Vector Bundles and Gromov–Hausdorff Distance

Published online by Cambridge University Press:  25 August 2009

Marc A. Rieffel
Marc A. Rieffel
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, U. S., rieffel@math.berkeley.edu
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Abstract

We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov–Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. Our approach employs the Lipschitz constants of projection-valued functions that determine vector bundles. We develop some computational techniques, and we illustrate our ideas with simple specific examples involving vector bundles on the circle, the two-torus, the two-sphere, and finite metric spaces. Our topic is motivated by statements concerning “monopole bundles” over matrix algebras in the literature of theoretical high-energy physics.

Keywords

vector bundlesGromov–Hausdorff distanceLipschitzprojectionsmonopole bundle

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Type
Research Article
Information
Journal of K-Theory , Volume 5 , Issue 1 , February 2010 , pp. 39 - 103
Copyright
Copyright © ISOPP 2010

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