Persistent Stiefel-Whitney Classes of Tangent Bundles

Abstract: Stiefel-Whitney classes are invariants of the tangent bundle of a smooth manifold, represented as cohomology classes of the base manifold. These classes are essential in obstruction theory, embedding problems, and cobordism theory. In this work, we first reestablish an appropriate notion of vector bundles in a persistent setting, allowing characteristic classes to be interpreted through topological data analysis. Next, we propose a concrete algorithm to compute persistent cohomology classes that represent the Stiefel-Whitney classes of the tangent bundle of a smooth manifold. Given a point cloud, we construct a \v{C}ech or alpha filtration. By applying the Wu formula in this setting, we derive a sequence of persistent cohomology classes from the filtration. We show that if the filtration is homotopy equivalent to a smooth manifold, then one of these persistent cohomology classes corresponds to the $k$-th Stiefel-Whitney class of the tangent bundle of that manifold. To demonstrate the effectiveness of our approach, we present experiments on real-world datasets, including applications to complex manifolds, image patches, and molecular conformation space.