Morse-based Modular Homology for Evolving Simplicial Complexes

TL;DR

This paper presents MMHM, a Morse-based framework that efficiently updates homology groups of evolving simplicial complexes through localized modifications, avoiding costly full recomputations.

Contribution

It introduces a novel Morse-based modular approach for maintaining homology under local complex modifications, significantly improving efficiency over traditional methods.

Findings

Achieves significant amortized performance gains.

Guarantees homology preservation during local updates.

Enables fast, exact updates for evolving complexes.

Abstract

The computation of homology groups for evolving simplicial complexes often requires repeated reconstruction of boundary operators, resulting in prohibitive costs for large-scale or frequently updated data. This work introduces MMHM, a Morse-based Modular Homology Maintenance framework that preserves homological invariants under local complex modifications. An initial discrete Morse reduction produces a critical cell complex chain-homotopy equivalent to the input; subsequent edits trigger localized updates to the affected part of the reduced boundary operators over a chosen coefficient ring. By restricting recomputation to the affected critical cells and applying localized matrix reductions, the approach achieves significant amortized performance gains while guaranteeing homology preservation. A periodic recompression policy together with topology-aware gating and a column-oriented sparse boundary representation with a pivot-ownership map confines elimination to the affected columns and can bypass linear algebra when invariants are decidable combinatorially. The framework offers a drop-in upgrade for topology pipelines, turning costly rebuilds into fast, exact updates that track homology through local edits. Reframing dynamic homology as a locality-bounded maintenance task provides an exact alternative to global recomputation for evolving meshes and complexes.