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arXiv · 2610.08196

A cubical approach to homology theories for hypergraphs

Syed Hadi Ali Zaidi·Samira Sahar Jamil
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Abstract

We introduce three homology theories for hypergraphs, namely $Γ$-homology, $\Box$-homology, and $\times$-homology, and show that they are pairwise non-isomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that $\Box$-homology is invariant under this homotopy, whereas $Γ$-homology and $\times$-homology fail to satisfy homotopy invariance. Based on excision, we also identify a distinctive structural behavior exhibited by $Γ$-homology that further differentiates it from $\Box$-homology.

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Syed Hadi Ali Zaidi, Samira Sahar Jamil. 2026-10-06. A cubical approach to homology theories for hypergraphs. https://arxiv.org/abs/2610.08196

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