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Perspectives on \(A\)-homotopy theory and its applications. (English) Zbl 1082.37050
This article is a survey of a combinatorial homotopy theory, the \(A\)-theory, which concerns simplicial complexes and graphs. In the first section, the authors recall the definition of this homotopy theory in the two frameworks [see H. Barcelo, X. Kramer, R. Laubenbacher and C. Weaver, Adv. Appl. Math. 26, 97–128 (2001; Zbl 0984.57014)]. The theory has some similarities with the classical homotopy theory of a pointed topological space. For instance, the Seifert-van Kampen theorem for the fundamental group is valid and the higher-dimensional groups are abelian too. It is however different, as a contractible complex may have nontrivial \(A\)-groups, and there is no invariance under triangulation. The authors give also an algorithm for computing the abelianization of the \(A_1\)-groups. Finally, the last sections contain some links and applications of the \(A\)-theory, like, e.g., homotopy theory of matroid complexes (S. B. Maurer), graph theory (L. Lovász) or subspace arrangement (E. Babson, H. Barcelo, R. Laudenbacher).

37F20 Combinatorics and topology in relation with holomorphic dynamical systems
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