Stratified Vector Bundles: Examples and Constructions

TL;DR

This paper introduces stratified vector bundles, a new class of stratified spaces, providing examples, characterizations, and extending properties of smooth vector bundles to this stratified setting.

Contribution

It defines stratified vector bundles, offers an alternative characterization via monoid actions, and extends functorial properties from smooth to stratified vector bundles.

Findings

Provides large families of examples from Whitney stratified spaces and singular foliation theory.

Characterizes stratified vector bundles using monoid actions.

Extends functorial properties of smooth vector bundles to stratified spaces.

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.