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PreprintCC BY 4.0arXiv

Homological critical points for hypergraphs and simplicial complexes

  • Shiquan Ren
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Abstract

In this paper, we study the homological critical points for persistence hypergraphs and persistence simplicial complexes. With the help of the relative homology groups, we define the homological critical points for persistence morphisms between persistence hypergraphs and persistence simplicial maps between persistence simplicial complexes. We prove some commutative diagrams of subset relations for the homological critical points of persistence hypergraphs as well as persistence morphisms between them. We also prove some commutative diagrams of subset relations for the homological critical points of persistence simplicial complexes as well as persistence simplicial maps between them. As examples, we use the parametric configuration spaces to construct the parametric independence complexes as the space of parametric packings and construct the parametric dominating hypergraphs as the space of parametric coverings.

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Cite

BibTeXRIS
BibTeX
@misc{ren2026homological,
  title = {{Homological critical points for hypergraphs and simplicial complexes}},
  author = {Shiquan Ren},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23528},
  eprint = {2609.23528},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23528},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Homological critical points for hypergraphs and simplicial complexes
AU  - Shiquan Ren
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.23528
UR  - https://arxiv.org/abs/2609.23528
AB  - In this paper, we study the homological critical points for persistence hypergraphs and persistence simplicial complexes. With the help of the relative homology groups, we define the homological critical points for persistence morphisms between persistence hypergraphs and persistence simplicial maps between persistence simplicial complexes. We prove some commutative diagrams of subset relations for the homological critical points of persistence hypergraphs as well as persistence morphisms between them. We also prove some commutative diagrams of subset relations for the homological critical points of persistence simplicial complexes as well as persistence simplicial maps between them. As examples, we use the parametric configuration spaces to construct the parametric independence complexes as the space of parametric packings and construct the parametric dominating hypergraphs as the space of parametric coverings.
ER  -