Geometric approach to graph magnitude homology (Q2214755)

The authors study the geometric approach to graph magnitude homology. Given a graph \(G\) and a non-negative integer \(l\), the magnitude chain complex of length \(l\) on \(G\) is given by the free modules \(MC_{k,l}(G)\) generated by all \((k+1)\)-sequences \[ (x_0,x_1,\dots,x_k) \in V(G)^{k+1} \] satisfying \[ \sum_{i=1}^k d(x_{k-1},x_k)=l. \] The boundary map is defined by \[ \partial=\sum_{i=1}^{k-1} (-1)^i \partial_i \] where \( \partial_i(x_0,x_1,\dots,x_k)=(x_0,\dots,x_{i-1},x_{i+1},\dots,x_k) \) if \( d(x_{i-1},x_{i+1})=d(x_{i-1},x_i)+d(x_i,x_{i+1}) \) and \( \partial_i(x_0,x_1,\dots,x_k)=0 \) otherwise. It is proved in the reference [\textit{R. Hepworth} and \textit{S. Willerton}, Homology Homotopy Appl. 19, No. 2, 31--60 (2017; Zbl 1377.05088)] that the magnitude chain complex of length \(l\) is a chain complex. The homology of this chain complex is called the magnitude homology. In Section 3, the authors study the magnitude homology for trees. They use simplicial homology to calculate the magnitude homology of a tree. They prove a direct sum decomposition of the magnitude chain complex in Proposition 3.2. Then they prove a chain isomorphism in Proposition 3.4. Finally, they calculate the magnitude homology of a tree in Theorem 3.6 by the number of the edges. In Section 4, the authors give some attempts to calculate the magnitude homology of a general graph. They prove some chain map and some isomorphism for the magnitude chain complex in Theorem 4.3.