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Double complexes for configuration spaces and hypergraphs on manifolds

Shiquan Ren

TL;DR

This work develops a comprehensive framework of double complexes of differential forms on configuration spaces and hypergraphs lying on manifolds, organizing these objects within the theory of Δ-manifolds. It defines infimum and supremum chain complexes for graded subspaces and proves they are quasi-isomorphic, with exact equality when the hypergraph is a Δ-submanifold. The paper extends these constructions to associated Δ-manifolds and demonstrates how morphisms, automorphisms, and covering maps interact with the double complex structure. A key contribution is the application to obstructions for regular embeddings via characteristic classes of canonical bundles over hypergraph configuration spaces, linking higher-order network models to geometric topology. The results provide a rigorous, algebraic-topological foundation for analyzing configuration-space constrained motion planning and manifold-learning problems through hypergraphs and their differential-form complexes.

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 $Δ$-submanifold of the ambient configuration space.

Double complexes for configuration spaces and hypergraphs on manifolds

TL;DR

This work develops a comprehensive framework of double complexes of differential forms on configuration spaces and hypergraphs lying on manifolds, organizing these objects within the theory of Δ-manifolds. It defines infimum and supremum chain complexes for graded subspaces and proves they are quasi-isomorphic, with exact equality when the hypergraph is a Δ-submanifold. The paper extends these constructions to associated Δ-manifolds and demonstrates how morphisms, automorphisms, and covering maps interact with the double complex structure. A key contribution is the application to obstructions for regular embeddings via characteristic classes of canonical bundles over hypergraph configuration spaces, linking higher-order network models to geometric topology. The results provide a rigorous, algebraic-topological foundation for analyzing configuration-space constrained motion planning and manifold-learning problems through hypergraphs and their differential-form complexes.

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 -submanifold of the ambient configuration space.

Paper Structure

This paper contains 18 sections, 30 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Configuration spaces
  3. Chain complexes for hypergraphs
  4. Motivations
  5. Main results
  6. Organization
  7. The infimum and supremum chain complexes
  8. $\Delta$-manifolds and double complexes
  9. Double complex for $\Delta$-manifolds
  10. Double complexes for graded submanifolds of $\Delta$-manifolds
  11. Double complexes for associated $\Delta$-manifolds
  12. Configuration spaces and hypergraphs
  13. Morphisms of configuration spaces and hypergraphs
  14. Automorphism groups of configuration spaces and hypergraphs
  15. Covering maps of hypergraphs and order of the associated vector bundles
  16. ...and 3 more sections

Key Result

Theorem 1.1

We have a double complex with a graded group action of $\Sigma_n$ on $\Omega^\bullet({\rm Conf}_n(M))$ for each $n\geq 1$ which is commutative with $d$. In addition, if $M$ can be embedded in $\mathbb{R}$, then we have a sub-double complex consisting of the $\Sigma_\bullet$-invariant differential forms.

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2: Main result I
  • Theorem 1.3: Main result II
  • Lemma 2.1
  • proof
  • Definition 1
  • Remark 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 67 more