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Homotopy, homology, and persistent homology using closure spaces

Peter Bubenik, Nikola Milićević

TL;DR

The paper develops a unified framework for persistent homology using closure spaces, extending beyond metric spaces to filtrations of closure spaces, graphs, and directed graphs. It introduces multiple interval and product constructions to define six cubical and three simplicial homology theories, each producing persistence modules when applied to filtrations. By extending Gromov-Hausdorff distance to filtrations and establishing ε-correspondences, the authors prove broad stability results for all homotopy-invariant functors within this setting, thereby connecting metric, graph-based, and topological data analyses under a common theory. The work also builds a rich adjoint-descent chain between closures, graphs, hypergraphs, and simplicial complexes, enabling functorial Vietoris-Rips and Čech constructions and demonstrating robust, scalable stability for persistent invariants across diverse combinatorial and geometric contexts.

Abstract

We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable.

Homotopy, homology, and persistent homology using closure spaces

TL;DR

The paper develops a unified framework for persistent homology using closure spaces, extending beyond metric spaces to filtrations of closure spaces, graphs, and directed graphs. It introduces multiple interval and product constructions to define six cubical and three simplicial homology theories, each producing persistence modules when applied to filtrations. By extending Gromov-Hausdorff distance to filtrations and establishing ε-correspondences, the authors prove broad stability results for all homotopy-invariant functors within this setting, thereby connecting metric, graph-based, and topological data analyses under a common theory. The work also builds a rich adjoint-descent chain between closures, graphs, hypergraphs, and simplicial complexes, enabling functorial Vietoris-Rips and Čech constructions and demonstrating robust, scalable stability for persistent invariants across diverse combinatorial and geometric contexts.

Abstract

We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable.

Paper Structure

This paper contains 48 sections, 119 theorems, 45 equations, 4 figures.

Table of Contents

  1. Closure spaces
  2. Closures
  3. Interiors
  4. Neighborhoods
  5. Correspondence of closures and neighborhoods
  6. Topological spaces
  7. Covers and pasting
  8. Elementary examples
  9. Products and inductive products
  10. Coproducts and pushouts
  11. Pullbacks
  12. Monomorphisms and epimorphisms
  13. Symmetric closures and Alexandroff closures
  14. Simple graphs and simple directed graphs as closure spaces
  15. Graph products from closure space products
  16. ...and 33 more sections

Key Result

Theorem 1

There is a poset of intervals and product operations with distinct non-trivial homotopy relations given by the following Hasse diagram. \begin{tikzcd}[every arrow/.append style={dash},row sep=2ex] & (I_{\tau},\boxplus) \ar[dr] \ar[dl] \\ (I_{\tau},\times) \ar[d] & & (J_+,\boxplus) \ar[dl

Figures (4)

  • Figure 1: Representations of the closure spaces $J_+$ (left), $J_-$ (middle), and $J_{5,29}$ (right). The head of the arrow is contained in the closure of the tail of the arrow. Note that 29 in binary is 11101
  • Figure 2: For the Alexandroff closure space in \ref{['example:Vietoris_Rips_and_Cech_Generalization']}, we have its corresponding directed graph (left), its Vietoris-Rips complex (middle), and its Čech complex (right).
  • Figure 3: The closure space $J_+\times J_+$ (left) and its Čech complex (right).
  • Figure 4: A closure space given by a digraph (left), its source Čech complex (middle), and its sink Čech complex (right).

Theorems & Definitions (283)

  • Definition : \ref{['def:homotopy']}
  • Theorem : \ref{['cor:homotopy-poset', 'thm:homotopy-poset']}
  • Definition : \ref{['def:cube', 'def:singular_cubes']}
  • Theorem : \ref{['theorem:homotopy_invariance_of_homology']}
  • Definition : \ref{['def:simplices', 'def:singular_simplices']}
  • Proposition : \ref{['prop:embeddings-R']}
  • Theorem : \ref{['thm:gh-metric']}
  • Theorem : \ref{['thm:gh']}
  • Theorem : \ref{['thm:stability']}
  • Theorem : \ref{['thm:stability-new']}
  • ...and 273 more