A Künneth Formula for the Embedded Homology

Chong Wang

doi:doi:10.11648/j.ajam.20210901.15

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A Künneth Formula for the Embedded Homology

Chong Wang

Published in American Journal of Applied Mathematics (Volume 9, Issue 1)

Received: 8 March 2021 Published: 10 April 2021

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Abstract

Hypergraph is an important model for complex networks. A hypergraph can be regarded as a virtual simplicial complex with some faces missing and it is the key hub to connect the simplicial complex in topology and graph in combinatorics. The embedded homology groups of hypergraphs are new developments in mathematics in recent years, and the embedded homology groups of hypergraphs can reflect the topological and geometric characteristics of complex network which can not be reflected by the associated simplicial complex of hypergraphs. Künneth formulas describe the homology or cohomology of a product space in terms of the homology or cohomology of the factors. In this paper, we prove that the infimum chain complex of tensor products of free R-modules generated by hypergraphs is isomorphic to the tensor product of their respective infimum chain complexes, and give an analogues of Künneth formula for hypergraphs by classical algebraic Künneth formula based on the embedded homology groups of hypergraphs, which provides a theoretical basis for further study of cohomology theory of hypergraphs. In fact, the Künneth formula here can be extended to the Künneth formula of embedded homology of graded abelian groups of chain complexes, which can be used to extend the Künneth formula for digraphs with coefficients in a field.

Published in	American Journal of Applied Mathematics (Volume 9, Issue 1)
DOI	10.11648/j.ajam.20210901.15
Page(s)	31-37
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Hypergraphs, Embedded Homology, Associated Simplicial Complexes, Künneth Formula

References

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[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.
[3]	F. R. K. Chung and R. L. Graham, Cohomological aspects of hypergraphs. Trans. Amer. Math. Soc. 334 (1) (1992), 365-388.
[4]	E. Emtander, Betti numbers of hypergraphs. Commun. Algebra 37 (5), (2009), 1545-1571.
[5]	A. Grigor’yan, Y. Lin, Y. Muranov and S.T. Yau, Homologies of path complexes and digraphs. arXiv (2012). http://arxiv.org/abs/1207.2834.
[6]	A. Grigor’yan, Y. Muranov and S.T. Yau, Homologies of digraphs and K¨ unneth formulas. Commun. Anal. Geom. 25 (5) (2017), 969-1018.
[7]	A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2001.
[8]	J. Johnson, Hyper-networks of complex systems, in Complex Sciences, Series Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 4, (2009), 364-375.
[9]	J. P. May, Simplicial objects in algebraic topology. Van Nostrand, 1967 (reprinted by University of Chicago Press).
[10]	J. W. Milnor, The geometric realization of a semi- simplicial complex. Ann. of Math. 65 (2) (1957), 357- 362.
[11]	S. Klamt, U. Haus and F. Theis, Hypergraphs and cellular networks. PLoS Computational Biology 5(5) (2009), Article number e1000385.
[12]	R. Lung, N. Gaskó, M.A. Suciu, A hypergraph model for representing scientific output. Scientometrics 117(3) (2018), 1361-1379.
[13]	A. D. Parks and S. L. Lipscomb, Homology and hypergraph acyclicity: a combinatorial invariant for hypergraphs. Naval Surface Warfare Center, 1991.
[14]	Z. Meng and K. Xia, Persistent spectral based machine learning (PerSpect ML) for drug design. arXiv https://arxiv.org/abs/2002.00582, preprint.
[15]	Chong Wang, Shiquan Ren, Simplicial Descriptions for Digraphs and Their Path Homology From the Point of ∆- sets (in Chiniese). Mathematics in Practice and Theory, 49 (22) (2019), 238-247.

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APA Style

Chong Wang. (2021). A Künneth Formula for the Embedded Homology. American Journal of Applied Mathematics, 9(1), 31-37. https://doi.org/10.11648/j.ajam.20210901.15

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ACS Style

Chong Wang. A Künneth Formula for the Embedded Homology. Am. J. Appl. Math. 2021, 9(1), 31-37. doi: 10.11648/j.ajam.20210901.15

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AMA Style

Chong Wang. A Künneth Formula for the Embedded Homology. Am J Appl Math. 2021;9(1):31-37. doi: 10.11648/j.ajam.20210901.15

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@article{10.11648/j.ajam.20210901.15,
  author = {Chong Wang},
  title = {A Künneth Formula for the Embedded Homology},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {1},
  pages = {31-37},
  doi = {10.11648/j.ajam.20210901.15},
  url = {https://doi.org/10.11648/j.ajam.20210901.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210901.15},
  abstract = {Hypergraph is an important model for complex networks. A hypergraph can be regarded as a virtual simplicial complex with some faces missing and it is the key hub to connect the simplicial complex in topology and graph in combinatorics. The embedded homology groups of hypergraphs are new developments in mathematics in recent years, and the embedded homology groups of hypergraphs can reflect the topological and geometric characteristics of complex network which can not be reflected by the associated simplicial complex of hypergraphs. Künneth formulas describe the homology or cohomology of a product space in terms of the homology or cohomology of the factors. In this paper, we prove that the infimum chain complex of tensor products of free R-modules generated by hypergraphs is isomorphic to the tensor product of their respective infimum chain complexes, and give an analogues of Künneth formula for hypergraphs by classical algebraic Künneth formula based on the embedded homology groups of hypergraphs, which provides a theoretical basis for further study of cohomology theory of hypergraphs. In fact, the Künneth formula here can be extended to the Künneth formula of embedded homology of graded abelian groups of chain complexes, which can be used to extend the Künneth formula for digraphs with coefficients in a field.},
 year = {2021}
}

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TY  - JOUR
T1  - A Künneth Formula for the Embedded Homology
AU  - Chong Wang
Y1  - 2021/04/10
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210901.15
DO  - 10.11648/j.ajam.20210901.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 31
EP  - 37
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210901.15
AB  - Hypergraph is an important model for complex networks. A hypergraph can be regarded as a virtual simplicial complex with some faces missing and it is the key hub to connect the simplicial complex in topology and graph in combinatorics. The embedded homology groups of hypergraphs are new developments in mathematics in recent years, and the embedded homology groups of hypergraphs can reflect the topological and geometric characteristics of complex network which can not be reflected by the associated simplicial complex of hypergraphs. Künneth formulas describe the homology or cohomology of a product space in terms of the homology or cohomology of the factors. In this paper, we prove that the infimum chain complex of tensor products of free R-modules generated by hypergraphs is isomorphic to the tensor product of their respective infimum chain complexes, and give an analogues of Künneth formula for hypergraphs by classical algebraic Künneth formula based on the embedded homology groups of hypergraphs, which provides a theoretical basis for further study of cohomology theory of hypergraphs. In fact, the Künneth formula here can be extended to the Künneth formula of embedded homology of graded abelian groups of chain complexes, which can be used to extend the Künneth formula for digraphs with coefficients in a field.
VL  - 9
IS  - 1
ER  -

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Author Information

Chong Wang

School of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China

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APA Style

Chong Wang. (2021). A Künneth Formula for the Embedded Homology. American Journal of Applied Mathematics, 9(1), 31-37. https://doi.org/10.11648/j.ajam.20210901.15

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ACS Style

Chong Wang. A Künneth Formula for the Embedded Homology. Am. J. Appl. Math. 2021, 9(1), 31-37. doi: 10.11648/j.ajam.20210901.15

Copy | Download

AMA Style

Chong Wang. A Künneth Formula for the Embedded Homology. Am J Appl Math. 2021;9(1):31-37. doi: 10.11648/j.ajam.20210901.15

Copy | Download

@article{10.11648/j.ajam.20210901.15,
  author = {Chong Wang},
  title = {A Künneth Formula for the Embedded Homology},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {1},
  pages = {31-37},
  doi = {10.11648/j.ajam.20210901.15},
  url = {https://doi.org/10.11648/j.ajam.20210901.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210901.15},
  abstract = {Hypergraph is an important model for complex networks. A hypergraph can be regarded as a virtual simplicial complex with some faces missing and it is the key hub to connect the simplicial complex in topology and graph in combinatorics. The embedded homology groups of hypergraphs are new developments in mathematics in recent years, and the embedded homology groups of hypergraphs can reflect the topological and geometric characteristics of complex network which can not be reflected by the associated simplicial complex of hypergraphs. Künneth formulas describe the homology or cohomology of a product space in terms of the homology or cohomology of the factors. In this paper, we prove that the infimum chain complex of tensor products of free R-modules generated by hypergraphs is isomorphic to the tensor product of their respective infimum chain complexes, and give an analogues of Künneth formula for hypergraphs by classical algebraic Künneth formula based on the embedded homology groups of hypergraphs, which provides a theoretical basis for further study of cohomology theory of hypergraphs. In fact, the Künneth formula here can be extended to the Künneth formula of embedded homology of graded abelian groups of chain complexes, which can be used to extend the Künneth formula for digraphs with coefficients in a field.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - A Künneth Formula for the Embedded Homology
AU  - Chong Wang
Y1  - 2021/04/10
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210901.15
DO  - 10.11648/j.ajam.20210901.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 31
EP  - 37
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210901.15
AB  - Hypergraph is an important model for complex networks. A hypergraph can be regarded as a virtual simplicial complex with some faces missing and it is the key hub to connect the simplicial complex in topology and graph in combinatorics. The embedded homology groups of hypergraphs are new developments in mathematics in recent years, and the embedded homology groups of hypergraphs can reflect the topological and geometric characteristics of complex network which can not be reflected by the associated simplicial complex of hypergraphs. Künneth formulas describe the homology or cohomology of a product space in terms of the homology or cohomology of the factors. In this paper, we prove that the infimum chain complex of tensor products of free R-modules generated by hypergraphs is isomorphic to the tensor product of their respective infimum chain complexes, and give an analogues of Künneth formula for hypergraphs by classical algebraic Künneth formula based on the embedded homology groups of hypergraphs, which provides a theoretical basis for further study of cohomology theory of hypergraphs. In fact, the Künneth formula here can be extended to the Künneth formula of embedded homology of graded abelian groups of chain complexes, which can be used to extend the Künneth formula for digraphs with coefficients in a field.
VL  - 9
IS  - 1
ER  -

Copy | Download