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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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##### References:
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