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Computing persistent Stiefel–Whitney classes of line bundles

Computing persistent Stiefel–Whitney classes of line bundles We propose a definition of persistent Stiefel–Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its first persistent Stiefel–Whitney class. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2). We illustrate our method on several datasets inspired by image analysis. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

Computing persistent Stiefel–Whitney classes of line bundles

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References (27)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-021-00080-4
Publisher site
See Article on Publisher Site

Abstract

We propose a definition of persistent Stiefel–Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its first persistent Stiefel–Whitney class. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2). We illustrate our method on several datasets inspired by image analysis.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: Mar 1, 2022

Keywords: Persistent homology; Vector bundles; Stiefel–Whitney classes; Simplicial approximation; 55N31; 55R40; 05E45

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