[Paper Review] Natural Stratifications of Reeb Spaces and Higher Morse Functions.

Ryan E. Grady, Anna Schenfisch|arXiv (Cornell University)|Nov 17, 2020

Topological and Geometric Data Analysis16 references4 citations

TL;DR

This paper establishes natural stratifications on Reeb spaces and higher Morse functions by leveraging the Jacobi set of a map $ f: X \to \mathbb{R}^k $, showing that the induced structures on $ X $, $ \mathbb{R}^k $, and the Reeb space yield stratified maps under certain conditions. It further extends this framework to higher Morse functions by using the singular locus to define multi-parameter filtrations via sub-posets.

ABSTRACT

Both Reeb spaces and higher Morse functions induce natural stratifications. In the former, we show that the data of the Jacobi set of a function $f:X o \mathbb{R}^k$ induces stratifications on $X,\mathbb{R}^k$, and the associated Reeb space, and give conditions under which maps between these three spaces are stratified maps. We then extend this type of construction to the codomain of higher Morse functions, using the singular locus to induce a stratification of which sub-posets are equivalent to multi-parameter filtrations.

Motivation & Objective

• To understand how the Jacobi set of a map $ f: X \to \mathbb{R}^k $ induces stratifications on the domain $ X $, the codomain $ \mathbb{R}^k $, and the Reeb space.
• To identify conditions under which the maps between $ X $, $ \mathbb{R}^k $, and the Reeb space are stratified.
• To extend the stratification framework from Reeb spaces to higher Morse functions using the singular locus.
• To show that sub-posets derived from the singular locus correspond to multi-parameter filtrations in the higher Morse function setting.

Proposed method

• Utilize the Jacobi set of a smooth map $ f: X \to \mathbb{R}^k $ to define a stratification on the domain $ X $.
• Construct a stratification on the codomain $ \mathbb{R}^k $ by lifting the singular structure from the Jacobi set.
• Define a stratification on the Reeb space as the quotient of $ X $ under the equivalence relation induced by level sets of $ f $, compatible with the stratifications on $ X $ and $ \mathbb{R}^k $.
• Establish conditions under which the quotient map from $ X $ to the Reeb space and the map from the Reeb space to $ \mathbb{R}^k $ are stratified maps.
• Extend the construction to higher Morse functions by analyzing the singular locus of the function to generate a stratification on the codomain.
• Identify sub-posets of the stratified codomain that correspond to multi-parameter filtrations, using the structure of the singular locus.

Experimental results

Research questions

• RQ1 How does the Jacobi set of a map $ f: X \to \mathbb{R}^k $ induce a natural stratification on the domain $ X $?
• RQ2 Under what conditions are the maps from $ X $, $ \mathbb{R}^k $, and the Reeb space stratified?
• RQ3 Can the stratification framework used for Reeb spaces be generalized to higher Morse functions?
• RQ4 How does the singular locus of a higher Morse function give rise to a stratification of the codomain?
• RQ5 What is the relationship between sub-posets of the stratified codomain and multi-parameter filtrations in higher Morse theory?

Key findings

• The Jacobi set of a map $ f: X \to \mathbb{R}^k $ naturally induces a stratification on the domain $ X $, compatible with the smooth structure.
• The codomain $ \mathbb{R}^k $ inherits a stratification from the image of the Jacobi set under $ f $, forming a stratified subspace.
• The Reeb space inherits a stratification from the quotient of $ X $, and the quotient map becomes a stratified map under appropriate transversality conditions.
• Maps from the Reeb space to $ \mathbb{R}^k $ are stratified when the singularities of $ f $ satisfy certain regularity conditions.
• The singular locus of a higher Morse function induces a stratification on the codomain, enabling the definition of multi-parameter filtrations.
• Sub-posets of the stratified codomain derived from the singular locus are shown to be equivalent to multi-parameter filtrations in the higher Morse function setting.