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Torsion in the magnitude homology of graphs

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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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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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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, USA

    Radmila Sazdanovic

  2. Division of Mathematics and Computer Science, University of South Carolina Upstate, Spartanburg, USA

    Victor Summers

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  1. Radmila Sazdanovic
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  2. Victor Summers
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Correspondence to Radmila Sazdanovic.

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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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