Poincaré Duality for Generalized Persistence Diagrams of (co)Filtrations

TL;DR

This work develops a dual theory of generalized persistence diagrams by introducing cofiltrations as the dual to filtrations and defining their birth-death functions and persistent diagrams via Möbius inversion, yielding the diagrams $\partial ZB^dF$ and $\partial ZB_dF$. It establishes functoriality of these generalized diagrams through Rota's Galois Connection Theorem, enabling a coherent theory over arbitrary finite posets and for both filtrations and cofiltrations. When the ambient space is a manifold, the authors prove a Poincaré duality linking the persistent cohomology diagram of a cofiltration with the persistent homology diagram of its dual filtration, via $H^i_{\mathrm{c}}$ and $H_{m-i}$ and the deformation retraction between $|F(a)|$ and $|G(a)|$. The paper also proves that two independent constructions of persistence diagrams—the cycles/boundaries viewpoint and the persistence-module viewpoint—are equivalent, and extends this equivalence to the cofiltration setting, providing a robust duality framework for generalized, multi-parameter persistence.

Abstract

We dualize previous work on generalized persistence diagrams for filtrations to cofiltrations. When the underlying space is a manifold, we express this duality as a Poincaré duality between their generalized persistence diagrams. A heavy emphasis is placed on the recent discovery of functoriality of the generalized persistence diagram and its connection to Rota's Galois Connection Theorem.

Figures (8)

• Figure 1: Hasse diagrams of a lattice $P$ (left) and its corresponding interval lattice ${\text{Int}}\,{P}$ (right)
• Figure 2: A filtration $F: P \to \Delta K$ (left), its corresponding birth-death function ${{ZB}}_1 F: {\text{Int}}\,{P} \to {\mathbb Z}$ (middle), and the persistent homology diagram $\partial {{ZB}}_1 F : {\text{Int}}\,{P} \to {\mathbb Z}$ (right)
• Figure 3: A cofiltration $F: P \to \nabla K$ (left), its corresponding birth-death function ${{ZB}}^1 F: {\text{Int}}\, P \to \mathbb{Z}$ (middle), and the persistent cohomology diagram $\partial {{ZB}}^1 F: {\text{Int}}\, P \to \mathbb{Z}$ (right)
• Figure 4: A persistence module $M$ (right) and a free module $F$ (left). A free presentation of $M$ is given by the natural transformation $\phi : F \Rightarrow M$ shown in grey.
• Figure 5: Birth-death function ${{ZB}} \phi: {\text{Int}}\,{P} \to {\mathbb Z}$ (left) associated to the free presentation from Figure \ref{['fig:presentation']} and its persistence diagram $\partial {{ZB}} \phi: {\text{Int}}\,{P} \to {\mathbb Z}$ (right)

Theorems & Definitions (38)

• Lemma 2.1
• proof
• Remark 2.2
• Theorem 2.3: Rota's Galois Connection Theorem GulenMcCleary
• Theorem 2.4: Poincaré Duality
• Definition 3.1
• Proposition 3.2
• proof
• Definition 3.3
• Proposition 3.4