Homology theory of graphs (Q2454137)

The authors define ``graphical'' homology groups for reflexive nonoriented graphs. If \(G\) is such a graph, these homology groups result from a chain complex based on \textit{singular \(n\)-simplices} defined as graph homomorphisms \(\overline{2}^n \to G\) where \(\overline{2}\) is the looped path with three vertices and \(\overline{2}^n\) is the product of \(n\) copies of \(\overline{2}\) given by the categorical product (which is the product used for defining the \(\times\)-homotopy theory introduced by \textit{A. Dochtermann} [Eur. J. Comb. 30, No. 2, 490--509 (2009; Zbl 1167.05017)]). These groups induce a relative exact sequence \( \ldots \to H_{n+1}(G,A) \to H_n(A) \to H_n(G) \to H_{n}(G,A) \to \ldots\) for every pair \((G,A)\), verify the homotopy invariance (two homotopic graph homomorphisms induce identical homomorphisms between homology groups), additivity : \(H_n(\coprod G_i)=\bigoplus H_n(G_i)\) for all \(n \in \mathbb N\) and suspension isomorphism \(H_n(\mathbb S G) \approx H_{n-1}(G)\) for all \(n \geq 2\). Finally, \(H_0(G)=\mathbb Z\) when \(G\) is connected (and \(\mathbb Z\) is the coefficient ring) and the ``graphical'' spheres (obtained by successive suspensions of the \(0\)-sphere, the graph with two non adjacent vertices) have the same homology as their topological counterparts. The authors proves also that, as in the topological case, \(H_1(G,\mathbb Z)\) is the abelianization of the first homotopy group \(\Pi_1(G,x_0)\). As noted by the authors, the first homotopy group that they consider is different from that obtained from \(A\)-homotopy by \textit{H. Barcelo} et al. [Adv. Appl. Math. 26, No. 2, 97--128 (2001; Zbl 0984.57014)] by using the cartesian product of graphs. Concerning higher homotopy groups, the authors refer to their previous work proving the existence of a homotopy exact sequence associated to a pair of graphs [Acta Sci., Technol. 35, No. 4, 733--738 (2013)] but the product used in that paper was also the cartesian one, contrarily to the present paper where they have adopted the categorical product.