Vector bundles and Gromov-Hausdorff distance
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.
Cite
@article{arxiv.math/0608266,
title = {Vector bundles and Gromov-Hausdorff distance},
author = {Marc A. Rieffel},
journal= {arXiv preprint arXiv:math/0608266},
year = {2010}
}
Comments
66 pages; revised to reflect the new paper arXiv:0810.4695 of Hanfeng Li, which answers a question in my previous versions, and shows how to get better estimates in a number of my theorems. Also, a few small improvements elsewhere