A review of some properties of persistent homology

Md. Safik Ullah, Mst. Sima Akhter Mou, Md. Morshed Bin Shiraj, Md. Mizanur Rahman, Md. Masum Murshed, Nasima Akhter


Every day, enormous complex geometric data are accumulating rapidly, and qualitative analysis is needed, which cannot be done properly without studying the shapes of those data. Persistent homology describes the homology of data sets of arbitrary size, producing state-of-the art results in data analysis across a significant number of fields and sparking a rigorous study of persistence in homology theory. In this study, persistent homology has been demonstrated as a homology theory by satisfying the Eilenberg-Steenrod axioms. A brief background on persistent homology groups has been written to understand their construction. Then other definitions of persistent homology based on functors and graded modules have also been reviewed. The Mayer-Vietoris-Vietorisfor persistent homology has been derived as a property of persistent homology. Subsequently, a long, exact sequence for persistent homology has been constructed. Furthermore, the stability of persistent homology has been examined carefully. Finally, the Diamond principle of persistent homology has been explained briefly. This study can be used to investigate new properties of persistent homology, among other benefits.


Persistent Homology; Eilenberg-Steenrod axioms; Homology theory; Diamond principle; Mayer-Vietoris exact sequence; Long exact sequence

Full Text:



Aktas, M. E., Akbas, E., & Fatmaoui, A. El. (2019). Persistence homology of networks: Methods and applications. Applied Network Science, Vol. 4. https://doi.org/10.1007/s41109-019-0179-3

Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16.

Carlsson, G. (2020). Persistent homology and applied homotopy theory. In Handbook of Homotopy Theory. https://doi.org/10.1201/9781351251624-8

Chazal, F., Silva, V. de, Glisse, M., & Oudot, S. (2016). The structure and stability of persistence modules. In SpringerBriefs in Mathematics.

Davies, T. (2019). The persistent homology of complexes from point data sets.

Dawson, R. J. M. G. (1988). A simplification of the eilenberg-steenrod axioms for finite simplicial complexes. Journal of Pure and Applied Algebra, 53(3). https://doi.org/10.1016/0022-4049(88)90126-0

Fabio, B. Di, & Landi, C. (2011). A mayer–vietoris formula for persistent homology with an application to shape recognition in the presence of occlusion. Foundations of Computational Mathematics, 11(5). https://doi.org/10.1007/s10208-011-9100-x

Edelsbrunner, H., & Harer, J. (2008). Persistent homology—a survey. https://doi.org/10.1090/conm/453/08802

Fugacci, U., Scaramuccia, S., Iuricich, F., & de Floriani, L. (2016). Persistent homology: A step-by-step introduction for newcomers. Italian Chapter Conference 2016 - Smart Tools and Apps in Computer Graphics, STAG 2016. https://doi.org/10.2312/stag.20161358

Gamble, J., & Heo, G. (2010). Exploring uses of persistent homology for statistical analysis of landmark-based shape data. Journal of Multivariate Analysis, 101(9). https://doi.org/10.1016/j.jmva.2010.04.016

Graff, G., Graff, B., Pilarczyk, P., Jabłoński, G., Gąsecki, D., & Narkiewicz, K. (2021). Persistent homology as a new method of the assessment of heart rate variability. PLoS ONE, 16(7 July). https://doi.org/10.1371/journal.pone.0253851

Koplik, G. (2019). Persistent homology: A non-mathy introduction with examples using topological data analysis (tda) tools in data science. Retrieved August 28, 2020, from Towards Data Science website: https://towardsdatascience.com/persistent-homology-with-examples-1974d4b9c3d0

Meng, Z., Anand, D. V., Lu, Y., Wu, J., & Xia, K. (2020). Weighted persistent homology for biomolecular data analysis. Scientific Reports, 10(1). https://doi.org/10.1038/s41598-019-55660-3

Munkres, J. R. (2018). Elements of algebraic topology. In Elements of Algebraic Topology. https://doi.org/10.1201/9780429493911

Nicolau, M., Levine, A. J., & Carlsson, G. (2011). Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences of the United States of America, 108(17). https://doi.org/10.1073/pnas.1102826108

Palser, M. (2019). An excision theorem for persistent homology.

Pike, J. A., Khan, A. O., Pallini, C., Thomas, S. G., Mund, M., Ries, J., … Styles, I. B. (2020). Topological data analysis quantifies biological nano-structure from single molecule localization microscopy. Bioinformatics, 36(5). https://doi.org/10.1093/bioinformatics/btz788

Seversky, L. M., Davis, S., & Berger, M. (2016). On time-series topological data analysis: New data and opportunities. IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops. https://doi.org/10.1109/CVPRW.2016.131

Sizemore, A. E., Phillips-Cremins, J. E., Ghrist, R., & Bassett, D. S. (2019). The importance of the whole: Topological data analysis for the network neuroscientist. Network Neuroscience, 3(3). https://doi.org/10.1162/netn_a_00073

Varli, H., Yilmaz, Y., & Pamuk, M. (2018). Homological properties of persistent homology. ArXiv Preprint ArXiv:1805.01274.

Zomorodian, A., & Carlsson, G. (2005). Computing persistent homology. Discrete and Computational Geometry, 33(2). https://doi.org/10.1007/s00454-004-1146-y

DOI: http://dx.doi.org/10.24042/djm.v6i2.17812


Copyright (c) 2023 Desimal: Jurnal Matematika

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

  Creative Commons License
Desimal: Jurnal Matematika is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.