Regularity via Links and Stein Factorization

TL;DR

This paper introduces a new notion of regularity for PL functions on combinatorial manifolds, stratifies their critical sets, and demonstrates that Stein factorization respects these stratifications, advancing the understanding of PL topology.

Contribution

It defines a novel regularity concept based on link restrictions, distinguishes it from existing definitions, and shows Stein factorization as a stratified space map.

Findings

New regularity definition for PL functions on manifolds

Stratification of the Jacobi set as a poset stratified space

Stein factorization preserves stratifications in this context

Abstract

Here, we introduce a new definition of regular point for piecewise-linear (PL) functions on combinatorial (PL triangulated) manifolds. This definition is given in terms of the restriction of the function to the link of the point. We show that our definition of regularity is distinct from other definitions that exist in the combinatorial topology literature. Next, we stratify the Jacobi set/critical locus of such a map as a poset stratified space. As an application, we consider the Reeb space of a PL function, stratify the Reeb space as well as the target of the function, and show that the Stein factorization is a map of stratified spaces.