Persistent Stiefel-Whitney Classes of Tangent Bundles

Abstract

Stiefel-Whitney classes are topological invariants of vector bundles, and those of the tangent bundle capture essential features of a manifold, such as whether it is orientable and how it can be embedded in Euclidean space. We present an algorithm that computes these classes for the tangent bundle directly from a finite sample of points. Starting from the point cloud, we build a filtration of simplicial complexes and compute its persistent cohomology, and then apply the Wu formula, which recovers the Stiefel-Whitney classes from the cup product and the Steenrod squares alone, without estimating tangent spaces or a smooth structure. The key step, finding the Wu classes, reduces to solving a system of linear equations, so the computation runs in polynomial time in the number of simplices. We prove that whenever the sample recovers the shape of a closed manifold, the computed classes agree with the true Stiefel-Whitney classes of its tangent bundle, and that this remains true even when the data carry spurious topological features on which the Steenrod squares vanish, so the classes can be identified over a wide range of scales rather than only where the sample matches the manifold exactly. We illustrate the method on triangulated four-dimensional manifolds and on point clouds coming from image patches and from a molecular conformation space.