Computing persistent Stiefel-Whitney classes of line bundles

Raphaël Tinarrage (1, 2)

1 DATASHAPE - Understanding the Shape of Data
2 LMO - Laboratoire de Mathématiques d'Orsay

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 persistent Stiefel-Whitney classes. 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).

Domaines: Géométrie algorithmique [cs.CG]; Topologie algébrique [math.AT]

Raphaël Tinarrage. Computing persistent Stiefel-Whitney classes of line bundles. 2020. ⟨hal-02619607v1⟩