Abstract
Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular, that torsion of a given prime order can appear in the magnitude homology of a graph and that there are infinitely many such graphs. Finally, we provide complete computations of magnitude homology of a class of outerplanar graphs and focus on the ranks of the groups along the main diagonal of magnitude homology.
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References
Björner, A., Lutz, F.H.: Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the poincaré homology 3-sphere. Exp. Math. 9(2), 275–289 (2000)
Gu, Y.: Graph magnitude homology via algebraic Morse theory (2018). arXiv:1809.07240
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hepworth, R.: Magnitude cohomology (2018). arXiv:1807.06832
Hepworth, R., Willerton, S.: Categorifying the magnitude of a graph. Homol. Homotopy Appl. 19(2), 31–60 (2017)
Kaneta, R., Yoshinaga, M.: Magnitude homology of metric spaces and order complexes (2018). arXiv:1803.04247
Leinster, T.: The Euler characteristic of a category. Doc. Math. 13, 21–49 (2008)
Leinster, T.: The magnitude of metric spaces. Doc. Math. 18, 857–905 (2013)
Leinster, T.: The magnitude of a graph. Math. Proc. Camb. Philos. Soc. 166(2), 247–264 (2019). https://doi.org/10.1017/S0305004117000810
Leinster, T., Willerton, S.: On the asymptotic magnitude of subsets of Euclidean space. Geom. Dedic. 164, 287–310 (2013). https://doi.org/10.1007/s10711-012-9773-6
Lickorish, W.B.R.: Simplicial moves on complexes and manifolds. Geom. Topol. Monogr. 2(299–320), 314 (1999)
Pachner, U.: P.L. homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Comb. 12(2), 129–145 (1991)
Rubinstein, J.H., Tillmann, S.: Generalized trisections in all dimensions. Proc. Natl. Acad. Sci. 115(43), 10908–10913 (2018)
Wachs, M.L.: Poset topology: tools and applications. Geometric Combinatorics, IAS/Park City Mathematics Series, vol. 13. American Mathematical Society (2007)
Willerton, S.: Torsion: Graph magnitude homology meets combinatorial topology (2018). https://golem.ph.utexas.edu/category/2018/04/torsion_graph_magnitude_homolo.html. Accessed 18 Aug 2020
Acknowledgements
We are grateful to Simon Willerton for bringing the topic to our attention and to Tye Lidman for the valuable insights. The authors would like to thank Simon Willerton and the referee for a thorough reading and insightful suggestions to improve the paper.
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Communicated by Simon Willerton.
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RS was partially supported by the Simons Collaboration Grant 318086 and NSF DMS 1854705.
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Sazdanovic, R., Summers, V. Torsion in the magnitude homology of graphs. J. Homotopy Relat. Struct. 16, 275–296 (2021). https://doi.org/10.1007/s40062-021-00281-9
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DOI: https://doi.org/10.1007/s40062-021-00281-9