AI 中文总结
本文通过为Δ-流形分次子流形构造双同调，结合典范投影π研究黎曼流形上超（有向）图的双同调，推导相关交换图、群关系，应用于球填充覆盖滤过同伦型及正则映射刻画。
AI 中文摘要
本文通过为Δ-流形的分次子流形构造双同调，研究黎曼流形上的超（有向）图的双同调。借助从有序序列到其基础无序集的典范投影π，我们证明了黎曼流形上超（有向）图的双同调的一个交换图，以及黎曼流形上超（有向）图的微分形式空间的一个交换图。此外，我们通过π诱导的某些同态，给出了曲面上超（有向）图的基本群（胚）与曲面上的（纯）辫群之间的一些关系。作为应用，我们研究了受超图约束的黎曼流形上球的填充与覆盖的滤过的同伦型，并通过流形上独立复形与超图的几何实现，刻画了黎曼流形上的正则映射。
英文摘要
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.
Comments49 pages
