Approximate and discrete Euclidean vector bundles

TL;DR

This paper introduces approximate Euclidean vector bundles with a tolerance parameter, enabling finite, noise-tolerant representations and computational algorithms for characteristic classes, useful in applied and computational topology.

Contribution

It defines -approximate vector bundles, establishes their relation to classical bundles, and develops algorithms for computing characteristic classes from discrete data.

Findings

Approximate vector bundles can represent classical bundles when  is small.

Distances between approximate bundles determine when they represent the same classical bundle.

Algorithms enable computation of characteristic classes from finite, noisy data.

Abstract

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 \v{C}ech 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.