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Dualities in Multiparameter Persistence

Ulrich Bauer, Fabian Lenzen, Michael Lesnick

Abstract

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the (co)homology of multiparameter Rips filtrations, we give a multiparameter generalization of this duality. Considering two duality functors on multiparameter persistence modules, the pointwise dual $(-)^*$ and the global dual $(-)^\dagger$, we show that $H_q(C)^* \cong H^{N+q}(C^\dagger)$ for chain complexes $C$ of free $N$-parameter persistence modules with acyclic colimit. We give an elementary and accessible proof based on a long exact sequence argument, and also give an alternate proof that casts the result as a special case of multigraded Grothendieck local duality. As a corollary, we recover a simple correspondence between minimal free resolutions of a persistence module $M$ and those of its pointwise dual $M^*$, a result previously obtained by Miller, 2000. These results form the foundation of a state-of-the-art algorithm for computing free resolutions of the homology of Vietoris--Rips bifiltrations, described in a forthcoming paper.

Dualities in Multiparameter Persistence

Abstract

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the (co)homology of multiparameter Rips filtrations, we give a multiparameter generalization of this duality. Considering two duality functors on multiparameter persistence modules, the pointwise dual and the global dual , we show that for chain complexes of free -parameter persistence modules with acyclic colimit. We give an elementary and accessible proof based on a long exact sequence argument, and also give an alternate proof that casts the result as a special case of multigraded Grothendieck local duality. As a corollary, we recover a simple correspondence between minimal free resolutions of a persistence module and those of its pointwise dual , a result previously obtained by Miller, 2000. These results form the foundation of a state-of-the-art algorithm for computing free resolutions of the homology of Vietoris--Rips bifiltrations, described in a forthcoming paper.
Paper Structure (21 sections, 20 theorems, 87 equations, 10 figures, 1 table)

This paper contains 21 sections, 20 theorems, 87 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Persistence modules
  4. Graded matrices
  5. Chain complexes and homology
  6. Resolutions
  7. Dualities for multiparameter persistence
  8. The global dual
  9. Dualities and matrices
  10. Main results
  11. Examples
  12. Relation to Grothendieck and Greenlees--May duality
  13. Derived functors
  14. Derived bifunctors of Hom
  15. The graded setting
  16. ...and 6 more sections

Key Result

Theorem 1.1

If $K$ is a finite $\mathbf{Z}$-/indexed simplicial filtration, then there is a bijective correspondence between persistence barcodes of $H_\bullet(K)$ and $H^\bullet(\cup K, K)$ such that

Figures (10)

  • Figure 1: Illustrations of $\mathbf{Z}^2$-/persistence modules $M$. Where necessary, a matrix representing the structure map $M_z \to M_{z+e_i}$ w.r.t. some basis is shown. If the structure map is equality, or a canonical inclusion or projection, then the matrix is omitted. In the diagrams, each grid point corresponds to an element $z \in \mathbf{Z}^2$. Intensity of shading indicates the value of $\dim M_z$, with no shading over points $z$ such that $\dim M_z=0$. Borders between regions indicate changes in pointwise dimension.
  • Figure 2: Free, flat (equivalently, projective), injective and cofree modules. Each quadrant, half plane or entire plane corresponds to one free, flat, injective or cofree indecomposable summand.
  • Figure 3: (\ref{['fig:filtration:K']}) A one-critical $\mathbf{Z}^2$-/indexed simplicial filtration $K$, (\ref{['fig:filtration:C']}) the associated absolute and relative filtered chain complexes $C_\bullet(K)$ and $C_\bullet(\cup K, K)$, (\ref{['fig:filtration:H']}) the only nonzero absolute/relative (reduced) persistent homology module of $K$.
  • Figure 4: Augmented minimal free resolution of the simple $\mathbf{Z}^2$-/persistence module $\mathbf{k}$.
  • Figure 5: A free module $M$, its pointwise dual $M^*$, and its global dual $M^\dagger$.
  • ...and 5 more figures

Theorems & Definitions (63)

  • Theorem 1.1: deSilvaMorozovEtAl:2011[Theorem 6.2]BauerSchmahl:2021a
  • Theorem 1.2: BauerSchmahl:2021a
  • Theorem 1.3: see proof:local-duality
  • Corollary 1.4: see proof:module-resolutions
  • Corollary 1.5
  • Corollary 1.6: see \ref{['proof:Combined_Result_Extended']}
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 53 more