Local structures on stratified spaces

TL;DR

This paper develops a comprehensive theory of conically smooth stratified spaces, characterizing their sheaves and invariants through tangential data and establishing foundational theorems for their geometric structure.

Contribution

It introduces a new framework for conically smooth stratified spaces, including classifying maps, and proves key theorems like handlebody decompositions and isotopy extension.

Findings

Characterization of sheaves via tangential data

Existence of open handlebody decompositions

Inverse function and tubular neighborhood theorems

Abstract

We develop a theory of conically smooth stratified spaces and their smooth moduli, including a notion of classifying maps for tangential structures. We characterize continuous space-valued sheaves on these conically smooth stratified spaces in terms of tangential data, and we similarly characterize 1-excisive invariants of stratified spaces. These results are based on the existence of open handlebody decompositions for conically smooth stratified spaces, an inverse function theorem, a tubular neighborhood theorem, an isotopy extension theorem, and functorial resolutions of singularities to smooth manifolds with corners.