Moduli Spaces of Morse Functions for Persistence

TL;DR

This paper introduces new invariants for Morse functions on spheres that are more discerning than existing tools, providing insights into their equivalence classes and moduli spaces within persistent homology.

Contribution

It proposes novel invariants for Morse functions, relates Morse--Smale vector fields via fundamental moves, and explores the combinatorial structure of height-equivalent Morse functions.

Findings

New invariants are simpler yet more discerning than persistence barcodes.

A method to relate Morse--Smale vector fields via fundamental moves.

Analysis of the poset structure of level sets for height-equivalent Morse functions.

Abstract

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse--Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in $\mathbf R^3$. Their level-set invariant, a poset generated by nested disks and annuli from levels sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode.