The Embedded Homology of Hypergraph Pairs
Shiquan Ren, Jie Wu, Mengmeng Zhang

TL;DR
This paper extends the theory of embedded homology for hypergraphs by introducing relative homology groups, establishing key long exact sequences, and exploring potential applications in two-dimensional persistence analysis.
Contribution
It generalizes the embedded homology framework for hypergraph pairs and develops new theoretical tools like long exact and Mayer-Vietoris sequences.
Findings
Established long exact sequences for relative embedded homology.
Proved a Mayer-Vietoris sequence for hypergraph pairs.
Discussed two-dimensional persistence in this context.
Abstract
In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren, and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3)(2019) 479-500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer-Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology · Neuroinflammation and Neurodegeneration Mechanisms
