Double complexes for configuration spaces and hypergraphs on manifolds

TL;DR

This paper develops a new algebraic framework using double complexes of differential forms to study hypergraphs on manifolds, revealing their relationships and invariances within configuration spaces.

Contribution

It introduces double complexes for hypergraphs on manifolds and demonstrates their quasi-isomorphism properties, extending the algebraic tools for configuration space analysis.

Findings

Double complexes for hypergraphs are constructed as sub-double complexes of configuration space complexes.

Infimum and supremum double complexes are quasi-isomorphic under boundary maps.

All double complexes coincide when hypergraphs are $ ext{Δ}$-submanifolds of the configuration space.

Abstract

In this paper, we consider hypergraphs whose vertices are distinct points moving smoothly on a Riemannian manifold M. We take these hypergraphs as graded submanifolds of configuration spaces. We construct double complexes of differential forms on configuration spaces. Then we construct double complexes of differential forms on hypergraphs which are sub-double complexes of the double complex for the ambient configuration space. Among these double complexes for hypergraphs, the infimum double complex and the supremum double complex are quasi-isomorphic concerning the boundary maps induced from vertex deletion of the hyperedges. In particular, all the double complexes are identical if the hypergraph is a $\Delta$-submanifold of the ambient configuration space.