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Homotopy theory of graphs. (English) Zbl 1108.05030
Recently, in [H. Barcelo, X. Kramer, R. Laubenbacher and C. Weaver, Adv. Appl. Math. 26, No. 2, 97–128 (2001; Zbl 0984.57014)] and [H. Barcelo and R. Laubenbacher, Discrete Math. 298, No. 1–3, 39–61 (2005; Zbl 1082.37050)] a new combinatorial homotopy theory, called \(A\)-theory, was defined. It is sensitive to the combinatorics encoded in the complex, in particular to the level of connectivity of simplices. If \(\Delta\) is a simplicial complex of dimension \(d\) and \(\sigma_0\in\Delta\) is a simplex of dimension \(q\), where \(0\leq q\leq d\), then one obtains a family of groups \(A_n^q(\Delta,\sigma_0)\), \(n\geq 1\), the \(A\)-groups of \(\Delta\), based at \(\sigma_0\). The computation of these groups proceeds via the construction of a graph \(\Gamma^q(\Delta)\) whose vertices represent simplices in \(\Delta\). This suggests a natural definition of the \(A\)-theory of graphs. One of the results of the above cited papers shows that \(A_1\) of the complex can be obtained as the fundamental group of the space obtained by attaching 2-cells into all 3- and 4-cycles of \(\Gamma^q(\Delta)\).
The goal of the present paper is to generalize this result. For a simple undirected graph \(\Gamma\) with distinguished base vertex \(v_0\), the authors construct an infinite cell complex \(X_{\Gamma}\) together with a homomorphism \(A_n(\Gamma,v_0)\to \pi_n(X_{\Gamma},v_0)\) which is an isomorphism if a plausible cubical analogue of the simplicial approximation theorem holds. The authors give several reasons for this generalization. Besides the desire for a homotopy theory associated to the \(A\)-theory of a graph, there is a connection to the homotopy of the complements of certain subspace arrangements. Finally, the authors introduce the loop graph of a graph and establish that the \((n+1)\)st \(A\)-group of the graph is isomorphic to the \(n\)th \(A\)-group of the loop graph, in analogy to a standard result about classical homotopy.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05E99 Algebraic combinatorics
55P99 Homotopy theory
55Q05 Homotopy groups, general; sets of homotopy classes
57M15 Relations of low-dimensional topology with graph theory
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
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