Section complexes of simplicial height functions

TL;DR

This paper develops a combinatorial theory of section complexes for simplicial height functions, using spectral sequences to compute homology and model topological flow of homology generators.

Contribution

It introduces the concept of section complexes and spectral sequences to analyze the topology of simplicial height functions in a combinatorial framework.

Findings

Section complexes encode flow lines and homotopies of height functions.

Spectral sequences compute the homology of the height function domain.

Reeb complexes provide an approximation of homology flow along height levels.

Abstract

A theory of sections of simplicial height functions is developed. At the core of this theory lies the section complex, which is assembled from higher section spaces. The latter encode flow lines along the height, as well as their homotopies, in a combinatorial way. The section complex has an associated spectral sequence, which computes the homology of the height functions domain. We extract Reeb complexes from the spectral sequence. These provide a first order approximation of how homology generators flow along height levels. Our theory in particular models topological section spaces of piecewise linear functions in a completely combinatorial way.