# Morse-based Modular Homology for Evolving Simplicial Complexes

arXiv: https://arxiv.org/abs/2508.14429 · HTML: https://mlsanity.com/paper/2508.14429/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2025.

## Built on by (most cited first, 1 of 1)

- [Monitoring Neural Training with Topology: A Footprint-Predictable Collapse Index](https://mlsanity.com/paper/2604.26984/) (2026)

## Abstract (arXiv)

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.