Geometric approach to graph magnitude homology

In this paper, we introduce a new method to compute the magnitude homology of general graphs. To each direct sum component of the magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we state our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.

Keywords

magnitude homology, simplicial homology

2010 Mathematics Subject Classification

55U05

Full Text (PDF format)

Received 18 March 2020

Received revised 1 July 2020

Accepted 7 July 2020

Published 4 November 2020