Discrete Differential Calculus on Simplicial Complexes and Constrained Homology

Abstract

Let V be a finite set. Let ${\cal K}$ be a simplicial complex with its vertices in V. In this paper, the author discusses some differential calculus on V. He constructs some constrained homology groups of ${\cal K}$ by using the differential calculus on V. Moreover, he defines an independence hypergraph to be the complement of a simplicial complex in the complete hypergraph on V. Let ${\cal L}$ be an independence hypergraph with its vertices in V. He constructs some constrained cohomology groups of ${\cal L}$ by using the differential calculus on V.

Keywords

Simplicial complexes / Hypergraphs / Chain complexes / Homology / Differential calculus

References

[1] Berge C. Graphs and Hypergraphs, 1973, Amsterdam: North-Holland Mathematical Library

[2] Bott R, Tu L W. Differential Forms in Algebraic Topology, 1982, Berlin and Heidelberg: Springer-Verlag

[3] Bressan S, Li J Y, Ren S Q, Wu J. The embedded homology of hypergraphs and applications. Asian Journal of Mathematics, 2019, 23(3): 479-500

[4] Chern S S, Chen W H, Lam K S. Lectures on Differential Geometry, 2000, River Edge, NJ: World Scientific

[5] Cohen F R, Wu J. On braid groups and homotopy groups. Geometry and Topology Monographs, 2008, 13: 169-193

[6] Dawson R J MacG. Homology of weighted simplicial complexes. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 1990, 31(3): 229-243

[7] Dimakis A, Müller-Hoissen F. Differential calculus and gauge theory on finite sets. Journal of Physics A: Mathematical and General, 1994, 27(9): 3159-3178

[8] Dimakis A, Müller-Hoissen F. Discrete differential calculus: Graphs, topologies, and gauge theory. Journal of Mathematical Physics, 1994, 35(12): 6703-6735

[9] Dimakis A, Müller-Hoissen F. Discrete Riemannian geometry. Journal of Mathematical Physics, 1999, 40(3): 1518-1548

[10] Curtis E B. Simplicial homotopy theory. Advance in mathematics, 1971, 6: 107-209

[11] Eilenberg S, Zilber J A. Semi-simplicial complexes and singular homology. Annals of Mathematics, 1950, 51: 499-513

[12] Goerss P G, Jardine J. Simplicial Homotopy Theory, 1999, Basel: Birkhäuser

[13] Grbić J, Wu J, Xia K L, Wei G-W. Aspects of topological approaches for data science. Foundations of Data Science, 2022, 4: 165-216

[14] Grigor’yan, A., Lin, Y. and Yau, S.-T., Torsion of digraphs and path complexes, 2020, arXiv: 2012.07302v1.

[15] Grigor’yan, A., Lin, Y., Muranov, Y. and Yau, S.-T., Homologies of path complexes and digraphs, 2013, arXiv: 1207.2834.

[16] Grigor’yan A, Lin Y, Muranov Y, Yau S-T. Homotopy theory for digraphs. Pure and Applied Mathematics Quarterly, 2014, 10(4): 619-674

[17] Grigor’yan A, Lin Y, Muranov Y, Yau S-T. Cohomology of digraphs and (undirected) graphs. Asian Journal of Mathematics, 2015, 15(5): 887-932

[18] Grigor’yan A, Lin Y, Muranov Y, Yau S-T. Path complexes and their homologies. Journal of Mathematical Sciences, 2020, 248(5): 564-599

[19] Grigor’yan A, Muranov Y, Yau S-T. Homologies of digraphs and Künneth formulas. Communications in Analysis and Geometry, 2017, 25: 969-1018

[20] Hatcher A. Algebraic Topology, 2002, Cambridge: Cambridge University Press

[21] Lei F C, Li F L, Wu J. On simplicial resolutions of framed links. Transactions of The American Mathematical Society, 2014, 366(6): 3075-3093

[22] Madsen I H, Tornehave J. From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, 1997, Cambridge: Cambridge University Press

[23] Milnor J. The geometric realization of a semi-simplicial complex. Annals of Mathematics, 2nd Ser., 1957, 65(2): 357-362

[24] Munkres J R. Elements of Algebraic Topology, 1984, California: Addison-Wesley Publishing Company

[25] Parks, A. D. and Lipscomb, S. L., Homology and Hypergraph Acyclicity: A Combinatorial Invariant for Hypergraphs, Naval Surface Warfare Center, 1991, http://www.dtic.mil/docs/citations/ADA241584.

[26] Pavutnitskiy F, Wu J. A simplicial James-Hopf map and decompositions of the unstable Adams spectral sequence for suspensions. Algebraic and Geometric Topology, 2019, 19(1): 77-108

[27] Ren, S. Q., Simplicial-like identities for the paths and the regular paths on discrete sets (unpublished manuscript), 2021, arXiv 2107.09868.

[28] Ren S Q, Wu C Y, Wu J. Weighted persistent homology. Rocky Mountain Journal of Mathematics, 2018, 48(8): 2661-2687

[29] Ren S Q, Wu C Y, Wu J. Computational tools in weighted persistent homology. Chinese Annals of Mathematics, Ser. B, 2021, 42(2): 237-258

[30] Wu, C. Y., Ren, S. Q., Wu, J. and Xia, K. L., Discrete Morse theory for weighted simplicial complexes, Topology and its Applications, 270, 2020, Article 107038.

[31] Wu J. Simplicial objects and homotopy groups, Braids, 31–181, 2010, Hackensack, NJ: World Scientific Publishing