Approximate and discrete Euclidean vector bundles

• Published in: arXiv.org 2024-02
• Main Authors: Scoccola, Luis, Perea, Jose A
• Format: Article
• Language: English
• Subjects: Algorithms; Noise tolerance
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2104.07563

We introduce \(\varepsilon\)-approximate versions of the notion of Euclidean vector bundle for \(\varepsilon \geq 0\), which recover the classical notion of Euclidean vector bundle when \(\varepsilon = 0\). In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that \(\varepsilon\)-approximate vector bundles can be used to represent classical vector bundles when \(\varepsilon > 0\) is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data, and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.