Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911647357992960 |
|---|---|
| author | Torras-Casas, Álvaro Yim, Ka Man Pennig, Ulrich |
| author_facet | Torras-Casas, Álvaro Yim, Ka Man Pennig, Ulrich |
| contents | Given a poset-graded chain complex of vector spaces, a Conley complex is the minimal chain-homotopic reduction of the initial complex that respects the poset grading. A connection matrix is a matrix representing the differential of the Conley complex. In this work, we give an algebraic derivation of the Conley complex and its connection matrix using homological perturbation theory and algebraic Morse theory. Under this framework, we use a graded splitting of relative chain groups to determine the connection matrix, rather than Forman's acyclic partial matching in the usual discrete Morse theory setting. This splitting is obtained by means of the clearing optimisation, a commonly used technique in persistent homology. Finally, we show how this algebraic perspective yields an algorithm for computing the connection matrix via column reductions on the differential of the initial complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory Torras-Casas, Álvaro Yim, Ka Man Pennig, Ulrich Algebraic Topology 37B30, 55N31, 55U15, 57Q70 Given a poset-graded chain complex of vector spaces, a Conley complex is the minimal chain-homotopic reduction of the initial complex that respects the poset grading. A connection matrix is a matrix representing the differential of the Conley complex. In this work, we give an algebraic derivation of the Conley complex and its connection matrix using homological perturbation theory and algebraic Morse theory. Under this framework, we use a graded splitting of relative chain groups to determine the connection matrix, rather than Forman's acyclic partial matching in the usual discrete Morse theory setting. This splitting is obtained by means of the clearing optimisation, a commonly used technique in persistent homology. Finally, we show how this algebraic perspective yields an algorithm for computing the connection matrix via column reductions on the differential of the initial complex. |
| title | Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory |
| topic | Algebraic Topology 37B30, 55N31, 55U15, 57Q70 |
| url | https://arxiv.org/abs/2503.09301 |