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arXiv · 2608.00708

Hypergraphs on Riemannian manifolds and their double homology

Shiquan Ren
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Abstract

In this paper, by constructing double homology for graded submanifolds of $Δ$-manifolds, we study the double homology of hyper(di)graphs on Riemannian manifolds. With the help of the canonical projection $π$ from ordered sequences to their underlying unordered sets, we prove a commutative diagram of the double homology of hyper(di)graphs on Riemannian manifolds and a commutative diagram of the spaces of differential forms of hyper(di)graphs on Riemannian manifolds. Besides, we give some relations between the fundamental goup(oid)s of hyper(di)graphs on surfaces and the (pure) braid groups on surfaces by certain homomorphisms induced from $π$. As applications, we study the homotopy types of filtrations for packings and coverings of balls on Riemannian manifolds with hypergraph constraints, and we characterize regular maps on Riemannian manifolds by geometric realizations of the independence complexes and hypergraphs on the manifolds.

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Shiquan Ren. 2026-08-01. Hypergraphs on Riemannian manifolds and their double homology. https://arxiv.org/abs/2608.00708

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