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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).

MSC:
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
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