Geodesic
Path complexes and their homologies
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 5, pp. 79-128.

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We introduce the notions of a path complex and its homologies. Particular cases of path homologies are simplicial homologies and digraph homologies. We state and prove some properties of path homologies, in particular, the Künneth formulas for Cartesian product and join, which happen to be true at the level of chain complexes. 
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A. A. Grigor'yan; Yong Lin; Yu. V. Muranov; Shing-Tung Yau. Path complexes and their homologies. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 5, pp. 79-128. https://geodesic-test.mathdoc.fr/item/FPM_2016_21_5_a4/

[1] Babson E., Barcelo H., de Longueville M., Laubenbacher R., “Homotopy theory of graphs”, J. Algebraic Combin., 24 (2006), 31–44 | DOI | MR | Zbl

[2] Barcelo H., Kramer X., Laubenbacher R., Weaver Ch., “Foundations of a connectivity theory for simplicial complexes”, Adv. Appl. Math., 26 (2001), 97–128 | DOI | MR | Zbl

[3] Bourbaki N., Elements of Mathematics. Algebra I. Chapters 1–3, Springer, Berlin, 1989 | MR | Zbl

[4] Chen B., Yau S.-T., Yeh Y.-N., “Graph homotopy and Graham homotopy”, Discrete Math., 241 (2001), 153–170 | DOI | MR | Zbl

[5] Dimakis A., Müller-Hoissen F., “Differential calculus and gauge theory on finite sets”, J. Phys. A. Math. Gen., 27:9 (1994), 3159–3178 | DOI | MR | Zbl

[6] Dimakis A., Müller-Hoissen F., “Discrete differential calculus: graphs, topologies, and gauge theory”, J. Math. Phys., 35:12 (1994), 6703–6735 | DOI | MR | Zbl

[7] Gerstenhaber M., Schack S. D., “Simplicial cohomology is Hochschild cohomology”, J. Pure Appl. Algebra, 30 (1983), 143–156 | DOI | MR | Zbl

[8] Grigor'yan A., Lin Y., Muranov Yu., Yau S.-T., Homologies of path complexes and digraphs, 2013, arXiv: 1207.2834v4

[9] Grigor'yan A., Lin Y., Muranov Yu., Yau S.-T., “Homotopy theory for digraphs”, Pure Appl. Math. Q, 10:4 (2014), 619–674 | DOI | MR | Zbl

[10] Grigor'yan A., Muranov Yu., Yau S.-T., “Graphs associated with simplicial complexes”, Homology, Homotopy Appl., 16:1 (2014), 295–311 | DOI | MR | Zbl

[11] Grigor'yan A., Muranov Yu., Yau S.-T., “On a cohomology of digraphs and Hochschild cohomology”, J. Homotopy Relat. Struct., 11:2 (2016), 209–230 | DOI | MR | Zbl

[12] Grigor'yan A., Muranov Yu., Yau S.-T., “Cohomology of digraphs and (undirected) graphs”, Asian J. Math., 19 (2015), 887–932 | DOI | MR

[13] Grigor'yan A., Muranov Yu., Yau S.-T., “Homologies of digraphs and Künneth formulas”, Comm. Anal. Geom., 2016 | MR

[14] Happel D., “Hochschild cohomology of finite-dimensional algebras”, Sém. d'Algèbre Paul Dubreil et Marie-Paul Malliavin, Lect. Notes Math., 1404, Springer, Berlin, 1989, 108–126 | DOI | MR

[15] Ivashchenko A. V., “Contractible transformations do not change the homology groups of graphs”, Discrete Math., 126 (1994), 159–170 | DOI | MR | Zbl

[16] MacLane S., Homology, Grundlag. Math. Wissensch., 114, Springer, Berlin, 1963 | MR | Zbl

[17] Tahbaz-Salehi A., Jadbabaie A., “Distributed coverage verification in sensor networks without location information”, IEEE Trans. Automatic Control., 55 (2010), 1837–1849 | DOI | MR | Zbl

[18] Talbi M. E., Benayat D., “Homology theory of graphs”, Mediterranean J. Math., 11 (2014), 813–828 | DOI | MR | Zbl