Approximate and discrete Euclidean vector bundles. (21st March 2023)

• Record Type: Journal Article
• Authors: Scoccola, Luis; Perea, Jose A.
• Is Part Of: Forum of mathematics. Volume 11(2023)
• Publication Date: 2023-03-21
• Subjects: 55R99 -- 55N31 -- 68W05 -- 55U99; Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2023.16
• Languages: English
• ISSNs: 2050-5094

Abstract

Abstract: 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.