# Dualities in persistent (co)homology

@article{Silva2011DualitiesIP, title={Dualities in persistent (co)homology}, author={Vin de Silva and Dmitriy Morozov and Mikael Vejdemo-Johansson}, journal={ArXiv}, year={2011}, volume={abs/1107.5665} }

We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of…

## 129 Citations

The Structure of Morphisms in Persistent Homology, I. Functorial Dualities

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- 2020

Duality results for absolute and relative versions of persistent (co)homology are proved, generalizing previous results in terms of barcodes and laying the groundwork for the efficient computation of bar codes for images, kernels, and cokernels of such morphisms.

Distributing Persistent Homology via Spectral Sequences

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- 2019

We set up the theory for a distributive algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and…

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- 2013

A number of facts about persistence modules are presented; ranging from the well-known but under-utilized to the reconstruction of techniques to work in a purely algebraic approach to persistent homology.

Computing Persistent Homology with Various Coefficient Fields in a Single Pass

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- 2014

An algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction allows us to infer the prime divisors of the torsion coefficients of the integral homology groups of the topological space at any scale, hence furnishing a more informative description of topology than persistence in asingle coefficient field.

Fast computation of persistent homology representatives with involuted persistent homology

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- 2021

Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives.…

Parallel Computation of Persistent Homology using the Blowup Complex

- Mathematics, Computer ScienceSPAA
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A parallel algorithm that computes persistent homology, an algebraic descriptor of a filtered topological space, by operating on a spatial decomposition of the domain, as opposed to a decomposition with respect to the filtration.

Efficient Computation of Image Persistence

- MathematicsArXiv
- 2022

An algorithm for computing the barcode of the image of a morphism in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes is presented and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes.

Künneth Formulae in Persistent Homology

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Künneth-type theorems for the persistent homology of the categorical and tensor product of filtered spaces are proved and two applications are presented: one towards more efficient algorithms for product metric spaces with the maximum metric, and the other recovering persistence homology calculations on the n-torus.

Kunneth formuale in persistent homology

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- 2018

The classical Kunneth formula in algebraic topology describes the homology of a product space in terms of that of its factors. In this paper, we prove Kunneth-type theorems for the persistent…

Persistent Homology Computation with a Twist

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An output-sensitive complexity analysis is given for the new algorithm which yields to sub-cubic asymptotic bounds under certain assumptions and completely avoids reduction on roughly half of the columns.

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