Approximate and discrete Euclidean vector bundles

Luis Scoccola ; Jose A. Perea

Forum of Mathematics, Sigma, Tome 11 (2023)

Résumé

We introduce $\varepsilon $-approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$, which recover the classical notion of a 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.

Détail

DOI : 10.1017/fms.2023.16