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Persistent Stiefel-Whitney Classes of Tangent Bundles

Dongwoo Gang

TL;DR

This work develops a rigorous framework to import classical characteristic classes into a persistent-topology setting. By proving the equivalence between persistent vector bundle classes and vector bundle filtrations and introducing a persistent Wu formula, it enables computation of persistent Stiefel–Whitney classes from point-cloud data via Čech/Alpha filtrations. The authors present concrete algorithms for cohomology operations and the Wu-based computation of $w_{(k,n)}$, along with probabilistic guarantees and a detailed time-complexity analysis. They validate the approach on structured manifolds, image patches, and molecular conformation spaces, demonstrating the practical extraction of obstruction-type invariants from data. The work thereby advances persistent TDA by equipping it with robust, computable tangent-bundle invariants with potential applications in embedding obstructions and manifold learning.

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 Čech 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.

Persistent Stiefel-Whitney Classes of Tangent Bundles

TL;DR

This work develops a rigorous framework to import classical characteristic classes into a persistent-topology setting. By proving the equivalence between persistent vector bundle classes and vector bundle filtrations and introducing a persistent Wu formula, it enables computation of persistent Stiefel–Whitney classes from point-cloud data via Čech/Alpha filtrations. The authors present concrete algorithms for cohomology operations and the Wu-based computation of , along with probabilistic guarantees and a detailed time-complexity analysis. They validate the approach on structured manifolds, image patches, and molecular conformation spaces, demonstrating the practical extraction of obstruction-type invariants from data. The work thereby advances persistent TDA by equipping it with robust, computable tangent-bundle invariants with potential applications in embedding obstructions and manifold learning.

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 Čech 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 -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.

Paper Structure

This paper contains 21 sections, 17 theorems, 56 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Cohomology operations
  4. Vector bundle and Stiefel–Whitney class
  5. Wu formula
  6. Persistence theory
  7. Persistent Stiefel–Whitney class
  8. persistent vector bundle class and vector bundle filtration
  9. Persistent Wu formula
  10. Čech estimation
  11. Filtrations from point clouds
  12. Probabilistic argument
  13. Algorithm
  14. Algorithm for computing persistent Stiefel–Whitney classes
  15. Algorithms for cohomology operations
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathbb{X} = \{X^r\}_{r \in [s,t]}$ be a filtration of spaces defined on a filtration interval $[s,t]$. Suppose that for every $s\leq r\leq t$, each space $X^r$ deformation retracts onto a smooth $n$-dimensional manifold $M$. Then, for $1\leq k\leq n$, the $k$-th persistent Stiefel–Whitney clas

Figures (4)

  • Figure 1: The second persistent cohomology barcodes and the second Stiefel–Whitney classes of type 4 at the maximum filtration scale computed for $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ and $S^2 \times S^2$. The horizontal axis represents the filtration scale, while the vertical axis indicates the persistent cohomology barcodes. Red barcodes correspond to cohomology classes whose sum yields the second Stiefel–Whitney class of type 4.
  • Figure 2: Persistent cohomology and persistent Stiefel–Whitney classes of type $2$ for the LINES dataset. The leftmost plot shows the persistent cohomology diagram over $\mathbb{Z}/2$. The middle and rightmost plots illustrate the first and second persistent barcodes and persistent Stiefel–Whitney classes of type $2$. The shaded regions indicate the filtration interval $[0.1,0.2]$ over which these classes satisfy the persistent Wu criterion.
  • Figure 3: Molecular structure of cyclooctane ($\text{C}_8\text{H}_{16}$) and a point cloud sampled from its conformation space projected into $\mathbb{R}^3$. The conformation space consists of a Klein bottle and a 2-sphere intersecting along two disjoint circles.
  • Figure 4: Persistent cohomology and persistent Stiefel–Whitney classes of type 2 for the point cloud sampled from the Klein bottle in the conformation space of cyclooctane. The leftmost plot shows the persistent cohomology diagram over $\mathbb{Z}/2$. The middle and rightmost plots illustrate the first and second persistent cohomology barcodes and the persistent Stiefel–Whitney classes of type 2. The shaded regions indicate the filtration interval $[0.7,1.2]$ over which these classes satisfy the persistent Wu criterion.

Theorems & Definitions (42)

  • Theorem 1: Theorem \ref{['swequal']}
  • Definition 1: medina2023new, Cup-$i$ Coproduct
  • Theorem 2: medina2023new
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 1
  • Definition 2
  • Definition 3: ren2025persistent, Definition 3
  • ...and 32 more