Complexes of strong discrete Morse matchings
Abstract
Using the strong discrete Morse theory developed by Fernández, we define the complex 𝒮ℳ(K) of strong discrete Morse matchings on a simplicial complex K, as well as the pure subcomplex 𝒮ℳpure(K) generated by the facets in 𝒮ℳ(K) of maximal dimension. For most complexes K these objects are proper subcomplexes of the complexes M(K) and Mpure(K) defined by Chari--Joswig using all Morse matchings on K. The homotopy types of the latter are not well-understood in general, but they are known when K is the path Pn with n edges, the cycle Cn with n edges, the star Sn with n leaves, the n-simplex Δn (n≤ 3), and the boundary ∂Δn (n≤ 3). in this paper we compute the homotopy types of 𝒮ℳpure(Cn) and 𝒮ℳ(K) for K=Pn,Sn,Δn,∂Δn for all n. We also compute the homology of 𝒮ℳ(Cn) for n≤ 21.
Related
- Morse-Bott inequalities on Lefschetz complexes
- Discrete Morse Theory for Computing Filtered Cosheaf Homology
- Updating a Discrete Morse Vector Field for a Lower Star Filtration Vineyard
- Morse-Smale dynamics in discrete Huygens networks
- Graph Representation via Elements of Discrete Morse and Cobordism Theories
BibTeX
@misc{knudson2026complexes,
title = {{Complexes of strong discrete Morse matchings}},
author = {Kevin Knudson and Abigail Owens-White},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06844},
eprint = {2609.06844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06844},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complexes of strong discrete Morse matchings
AU - Kevin Knudson
AU - Abigail Owens-White
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06844
UR - https://arxiv.org/abs/2609.06844
AB - Using the strong discrete Morse theory developed by Fernández, we define the complex 𝒮ℳ(K) of strong discrete Morse matchings on a simplicial complex K, as well as the pure subcomplex 𝒮ℳ_pure(K) generated by the facets in 𝒮ℳ(K) of maximal dimension. For most complexes K these objects are proper subcomplexes of the complexes M(K) and M_pure(K) defined by Chari--Joswig using all Morse matchings on K. The homotopy types of the latter are not well-understood in general, but they are known when K is the path P_n with n edges, the cycle C_n with n edges, the star S_n with n leaves, the n-simplex Δ^n (n≤ 3), and the boundary ∂Δ^n (n≤ 3). in this paper we compute the homotopy types of 𝒮ℳ_pure(C_n) and 𝒮ℳ(K) for K=P_n,S_n,Δ^n,∂Δ^n for all n. We also compute the homology of 𝒮ℳ(C_n) for n≤ 21.
ER -