Dualities in persistent (co)homology

TL;DR

This paper explores algebraic relationships between various persistent (co)homology modules from filtered cell complexes, demonstrating their equivalence and practical processing methods, including experimental validation of a new algorithm.

Contribution

It establishes algebraic equivalences among different persistent (co)homology modules and connects existing algorithms, introducing practical processing techniques and experimental validation.

Findings

Persistent (co)homology modules contain equivalent information.

Existing algorithms can process all four modules effectively.

Experimental evidence supports the efficiency of the new persistent cohomology algorithm.

Abstract

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 the latter algorithm.