Combinatorial models for topological Reeb spaces

TL;DR

This paper introduces a topological and combinatorial framework for studying Reeb spaces and functions, providing new tools like spectral sequences and simplicial models to analyze their homology and homotopy types.

Contribution

It develops a unified approach to Reeb spaces using topological categories and simplicial sets, bridging smooth and discrete Morse theory methods.

Findings

Spectral sequence for computing homology of Reeb spaces

Simplicial models for homotopy types of Reeb graphs

Unified framework for smooth and combinatorial Reeb functions

Abstract

There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all associated sections in a topological category. From this point of view, Reeb functions on stratified spaces are introduced, including both smooth and combinatorial examples. As a consequence of the simplicial approach taken, the theory comes with a spectral sequence for computing (generalized) homology. We also model the homotopy type of Reeb graphs/ topological Reeb spaces as simplicial sets, which are combinatorial in nature, as opposed to the typical description in terms of quotient spaces.