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Stratified Vector Bundles: Examples and Constructions

Ethan Ross

TL;DR

The paper develops stratified vector bundles to extend differential-geometric tools to singular spaces, addressing the variable-rank nature of stratified tangents. It provides an alternate monoid-action characterization and builds extensive examples from Whitney stratifications, singular foliations, and equivariant vector bundles, then extends smooth-vector-bundle functorial properties to the stratified setting. A key contribution is the framework for applying linear functors fiber-wise, including injective structures and orthogonalizable functors, while preserving Whitney A regularity. This work lays groundwork for further developments such as VB-groupoids, quotient constructions, and potential extensions of characteristic classes to stratified vector bundles, with implications for quantization in singular settings.

Abstract

A stratified space is a kind of topological space together with a partition into smooth manifolds. These kinds of spaces naturally arise in the study of singular algebraic varieties, symplectic reduction, and differentiable stacks. In this paper, we introduce a particular class of stratified spaces called stratified vector bundles, and provide an alternate characterization in terms of monoid actions. We will then provide large families of examples coming from the theory of Whitney stratified spaces, singular foliation theory, and equivariant vector bundle theory. Finally, we extend functorial properties of smooth vector bundles to the stratified case.

Stratified Vector Bundles: Examples and Constructions

TL;DR

The paper develops stratified vector bundles to extend differential-geometric tools to singular spaces, addressing the variable-rank nature of stratified tangents. It provides an alternate monoid-action characterization and builds extensive examples from Whitney stratifications, singular foliations, and equivariant vector bundles, then extends smooth-vector-bundle functorial properties to the stratified setting. A key contribution is the framework for applying linear functors fiber-wise, including injective structures and orthogonalizable functors, while preserving Whitney A regularity. This work lays groundwork for further developments such as VB-groupoids, quotient constructions, and potential extensions of characteristic classes to stratified vector bundles, with implications for quantization in singular settings.

Abstract

A stratified space is a kind of topological space together with a partition into smooth manifolds. These kinds of spaces naturally arise in the study of singular algebraic varieties, symplectic reduction, and differentiable stacks. In this paper, we introduce a particular class of stratified spaces called stratified vector bundles, and provide an alternate characterization in terms of monoid actions. We will then provide large families of examples coming from the theory of Whitney stratified spaces, singular foliation theory, and equivariant vector bundle theory. Finally, we extend functorial properties of smooth vector bundles to the stratified case.
Paper Structure (13 sections, 18 theorems, 109 equations)

This paper contains 13 sections, 18 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Stratified Vector Bundles
  3. Stratified Spaces
  4. Smooth Structures
  5. Stratified Vector Bundles
  6. Stratified Regular Monoid Actions
  7. Interesting Examples
  8. Singular Foliations
  9. Equivariant Vector Bundles
  10. Linear Functors and Stratified Vector Bundles
  11. Injective Stratified Vector Bundles
  12. Orthogonalizable Linear Functors and Grassmannians
  13. The Whitney A Condition

Key Result

Theorem 2.1

Let $H$ be a compact Lie group and $V$ and $H$ representation. With $p$ as in equation (hilbertmap), the induced map is a surjection.

Theorems & Definitions (82)

  • Definition 2.1
  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.4
  • Example 2.5
  • ...and 72 more