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The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology (2401.01248v1)

Published 2 Jan 2024 in math.AT, math.CO, math.CT, and math.RT

Abstract: Building upon previous works of Proudfoot and Ramos, and using the categorical framework of Sam and Snowden, we extend the weak categorical minor theorem from undirected graphs to quivers. As case of study, we investigate the consequences on the homology of multipath complexes; eg. on its torsion. Further, we prove a comparison result: we show that, when restricted to directed graphs without oriented cycles, multipath complexes and matching complexes yield functors which commute up to a blow-up operation on directed graphs. We use this fact to compute the homotopy type of matching complexes for a certain class of bipartite graphs also known as half-graphs or ladders. We complement the work with a study of the (representation) category of cones, and with analysing related consequences on magnitude cohomology of quivers.

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References (51)
  1. S. Ault and Z. Fiedorowicz. Symmetric homology of algebras, 2007. ArXiv:0708.1575.
  2. Girth, magnitude homology and phase transition of diagonality. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, page 1–27, 2023.
  3. Y. Asao. Magnitude homology of geodesic metric spaces with an upper curvature bound. Algebr. Geom. Topol., 21(2):647–664, 2021.
  4. Y. Asao. Magnitude homology and path homology. Bull. Lond. Math. Soc., 55(1):375–398, 2023.
  5. S. Ault. Symmetric homology of algebras. Algebr. Geom. Topol., 10(4):2343–2408, 2010.
  6. D. Barter. Noetherianity and rooted trees, 2015.
  7. A. Björner and V. Welker. Complexes of directed graphs. SIAM J. Discrete Math., 12:413–424, 10 1999.
  8. L. Caputi and C. Collari. On finite generation in magnitude (co)homology, and its torsion, 2023. ArXiv:2302.06525.
  9. Monotone cohomologies and oriented matchings. Homology, Homotopy & Applications. In press. ArXiv:2203.03476.
  10. Multipath cohomology of directed graphs. Algebraic & Geometric Topology. In press. ArXiv:2108.02690.
  11. Combinatorial and topological aspects of path posets, and multipath cohomology. J. Algebr. Comb., 57(2):617–658, 2023.
  12. On the homotopy type of multipath complexes. Mathematika, 70(1):e12235, 2024.
  13. FIFI\mathrm{FI}roman_FI-modules over Noetherian rings. Geometry & Topology, 18(5):2951 – 2984, 2014.
  14. L. Caputi and H. Riihimäki. On reachability categories and commuting algebras of quivers, 2023.
  15. A. Engström. Complexes of directed trees and independence complexes. Discrete Math., 309(10):3299–3309, 2009.
  16. Path complexes and their homologies. Journal of Mathematical Sciences, 248:564–599, 2020.
  17. Path homology theory of multigraphs and quivers. Forum Mathematicum, 30(5):1319–1337, 2018.
  18. Graphs associated with simplicial complexes. Homology, Homotopy and Applications, 16(1):295 – 311, 2014.
  19. On a cohomology of digraphs and Hochschild cohomology. Journal of Homotopy and Related Structures, 11:209–230, 2016.
  20. K. Gomi. Magnitude homology of geodesic space, 2019.
  21. R. Hepworth. Magnitude cohomology. Math. Z., 301(4):3617–3640, 2022.
  22. R. Hepworth and E. Roff. The reachability homology of a directed graph, 2023.
  23. R. Hepworth and S. Willerton. Categorifying the magnitude of a graph. Homology Homotopy Appl., 19(2):31–60, 2017.
  24. S. O Ivanov and F. Pavutnitskiy. Simplicial approach to path homology of quivers, subsets of groups and submodules of algebras. Journal of the LMS, 2023.
  25. Generalized chessboard complexes and discrete Morse theory. Chebyshevskiĭ Sb., 21(2):207–227, 2020.
  26. S. O. Ivanov. Nested homotopy models of finite metric spaces and their spectral homology, 2023.
  27. J. Jonsson. Exact sequences for the homology of the matching complex. Journal of Combinatorial Theory, Series A, 115(8):1504–1526, 2008.
  28. J. Jonsson. Simplicial complexes of graphs, volume 1928 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008.
  29. J. Jonsson. More torsion in the homology of the matching complex. Experiment. Math., 19(3):363–383, 2010.
  30. D. N. Kozlov. Combinatorial algebraic topology, volume 21 of Algorithms and computation in mathematics. Springer, Berlin
  31. R. Kaneta and M. Yoshinaga. Magnitude homology of metric spaces and order complexes. Bull. Lond. Math. Soc., 53(3):893–905, 2021.
  32. T. Leinster. Entropy and diversity. The axiomatic approach. Cambridge: Cambridge University Press, 2021.
  33. T. Matsushita. Matching complexes of polygonal line tilings. Hokkaido Mathematical Journal, 51(3):339 – 359, 2022.
  34. Matching complexes of trees and applications of the matching tree algorithm. Ann. Comb., 26(4):1041–1075, 2022.
  35. Discrete morse theoretic computations in the matching complex of k7subscript𝑘7k_{7}italic_k start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT, 2023.
  36. D. Miyata and E. Ramos. The graph minor theorem in topological combinatorics. Advances in Mathematics, 430:109203, 2023.
  37. M. Marietti and D. Testa. A uniform approach to complexes arising from forests. Electron. J. Combin., 15(1):Research Paper 101, 18, 2008.
  38. Classes of graphs with low complexity: The case of classes with bounded linear rankwidth. European Journal of Combinatorics, 91:103223, 2021. Colorings and structural graph theory in context (a tribute to Xuding Zhu).
  39. N. Proudfoot and E. Ramos. Functorial invariants of trees and their cones. Selecta Mathematica, 25:1–28, 2019.
  40. N. Proudfoot and E. Ramos. The contraction category of graphs. Representation Theory of the American Mathematical Society, 26(23):673–697, 2022.
  41. E. Ramos. Hilbert series in the category of trees with contractions. Math. Z., 298(3-4):1831–1852, 2021.
  42. E. Roff. Iterated magnitude homology, 2023.
  43. N. Robertson and P. D. Seymour. Graph minors. XX: Wagner’s conjecture. J. Comb. Theory, Ser. B, 92(2):325–357, 2004.
  44. A. Singh. Higher matching complexes of complete graphs and complete bipartite graphs. Discrete Mathematics, 345(4):112761, 2022.
  45. S. Sam and A. Snowden. Gröbner methods for representations of combinatorial categories. Journal of the American Mathematical Society, 30(1):159–203, 2017.
  46. R. Sazdanovic and V. Summers. Torsion in the magnitude homology of graphs. J. Homotopy Relat. Struct., 16(2):275–296, 2021.
  47. J. Shareshian and M. L. Wachs. Torsion in the matching complex and chessboard complex. Advances in Mathematics, 212:525–570, 2004.
  48. P. Turner and E. Wagner. The homology of digraphs as a generalisation of Hochschild homology. Journal of Algebra and Its Applications, 11(02):1250031, 2012.
  49. Y. Tajima and M. Yoshinaga. Causal order complex and magnitude homotopy type of metric spaces, 2023.
  50. Cycle-free chessboard complexes and symmetric homology of algebras. European J. Combin., 30(2):542–554, 2009.
  51. M. L. Wachs. Topology of matching, chessboard, and general bounded degree graph complexes. volume 49, pages 345–385. 2003. Dedicated to the memory of Gian-Carlo Rota.
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