zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Atkin, R., An algebra for patterns on a complex, I, Internat. J. Man-Machine Stud., 6, 285-307, (1974)
[2] Atkin, R., An algebra for patterns on a complex, II, Internat. J. Man-Machine Stud., 8, 483-448, (1976)
[3] Barcelo, Héléne; Laubenbacher, Reinhard, Perspectives on A-homotopy theory and its applications, Discr. Math., 298, 39-61, (2002) · Zbl 1082.37050
[4] Barcelo, Héléne; Kramer, Xenia; Laubenbacher, Reinhard; Weaver, Christopher, Foundations of a connectivity theory for simplicial complexes, Adv. Appl. Math., 26, 97-128, (2001) · Zbl 0984.57014
[5] A. Björner, private communication.
[6] Björner, A.; Welker, V., The homology of “\(k\)-Equal” manifolds and related partition lattices, Adv. in Math., 110, 277-313, (1995) · Zbl 0845.57020
[7] X. Kramer and R. Laubenbacher, “Combinatorial homotopy of simplicial complexes and complex information networks,” in “Applications of computational algebraic geometry,” D. Cox and B. Sturmfels (eds.), Proc. Sympos. inAppl. Math., vol. 53, Amer. Math. Soc., Providence, 1998. · Zbl 0886.05122
[8] P. May, Simplicial Objects in Algebraic Topology, The University of Chicago Press, Chicago, 1967.
[9] V. Voevodsky, A1-homotopy theory, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. I, 579-604 (1998). · Zbl 0907.19002
[10] D. West, Introduction to Graph Theory, second edition, Prentice-Hall, Upper Saddle River, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.