Torsion in the magnitude homology of graphs

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 ge...

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Bibliographic Details
Published inJournal of homotopy and related structures Vol. 16; no. 2; pp. 275 - 296
Main Authors Sazdanovic, Radmila, Summers, Victor
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021
Springer Nature B.V
Subjects
Algebra
Algebraic Topology
Functional Analysis
Graphs
Group theory
Homology
Mathematics
Mathematics and Statistics
Number Theory
Power series
Subgroups
Magnitude of a graph
Magnitude homology
Categorification
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Summary: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.
ISSN:2193-8407
1512-2891
DOI:10.1007/s40062-021-00281-9