Vector bundles and Gromov-Hausdorff distance

Details Vector bundles and Gromov-Hausdorff distance Vector bundles and Gromov-Hausdorff distance

2006-08-10

arxiv.org/abs/math/0608266v4

Journal of K-Theory, 5 (2010), 39-103

Mathematics

Metric Geometry

66 pages; revised to reflect the new paper arXiv:0810.4695 of Hanfeng Li, which answers a question in my previous versions, an

Scientific paper

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.

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