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The embedded homology of hypergraph pairs

Shiquan Ren

School of Mathematics and Statistics, Henan University, Kaifeng 475004, P. R. China

E-mail Address: renshiquan@henu.edu.cn

First author. The author(s) is/are the first author(s) contributed equally to the paper.

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Jie Wu

Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, P. R. China

E-mail Address: wujie@bimsa.cn

First author. The author(s) is/are the first author(s) contributed equally to the paper.

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, and

Mengmeng Zhang

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, P. R. China

Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, P. R. China

E-mail Address: mm_zhang00@163.com

First author. The author(s) is/are the first author(s) contributed equally to the paper.

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https://doi.org/10.1142/S0218216522501036Cited by:1 (Source: Crossref)

Abstract

In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3) (2019) 479–500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer–Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs.

Keywords:

AMSC: 55U10, 55U15, 68P05, 68P15

References

1. C. Berge, Graphs and Hypergraphs (North-Holland, New York, 1976). Google Scholar
2. S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3) (2019) 479–500. Crossref, Web of Science, Google Scholar
3. F. Cagliari, B. D. Fabio and M. Ferri, One-dimensional reduction of multidimensional persistent homology, Proc. Amer. Math. Soc. 138(8) (2010) 3003–3017. Crossref, Web of Science, Google Scholar
4. G. Carlsson and A. Zomorodian, The theory of multidimensional persistence, in Proc. Twenty-Third Annual Symp. Computational Geometry, Gyeongiu, South Korea (Association for Computing Machinery, New York, 2007), pp. 184–193. Google Scholar
5. A. Cerri and C. Landi, The persistence space in multidimensional persistent homology, in Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Vol. 7749 (Springer, Berlin, 2013), pp. 180–191. Crossref, Google Scholar
6. K.-C. Chang, Methods in Nonlinear Analysis (Springer-Verlag, Berlin, 2005). Google Scholar
7. A. Grigor’yan, R. Jimenez, Y. Muranov and S.-T. Yau, On the path homology theory of digraphs and Eilenberg-Steenrod axioms, Homology Homotopy Appl. 20(2) (2018) 179–205. Crossref, Web of Science, Google Scholar
8. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homotopy theory for digraphs, Pure Appl. Math. Q. 10(4) (2014) 619–674. Crossref, Web of Science, Google Scholar
9. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Path complexes and their homologies, J. Math. Sci. 248 (2020) 564–599. Crossref, Google Scholar
10. A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002). Google Scholar
11. J. R. Munkres, Elements of Algebraic Topology (Addison-Wesley Publishing Company, California, 1984). Google Scholar
12. A. D. Parks and S. L. Lipscomb, Homology and hypergraph acyclicity: A combinatorial invariant for hypergraphs, Final Report, Naval Surface Warfare Center (1991). Google Scholar
13. S. Ren, Persistent homology for hypergraphs and computational tools — A survey for users, J. Knot Theory Ramifications 29(13) (2020) 2043007. Link, Web of Science, Google Scholar
14. E. H. Spanier, Algebraic Topology (Springer-Verlag, New York, 1966). Google Scholar