an:02207449
Zbl 1082.37050
Barcelo, H??l??ne; Laubenbacher, Reinhard
Perspectives on \(A\)-homotopy theory and its applications
EN
Discrete Math. 298, No. 1-3, 39-61 (2005).
00119166
2005
j
37F20
\(A\)-theory; combinatorial homotopy; simplicial complexes; graphs
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 \textit{H. Barcelo, X. Kramer, R. Laubenbacher} and \textit{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. Lov??sz) or subspace arrangement (E. Babson, H. Barcelo, R. Laudenbacher).
Christophe Dupont (Orsay)
Zbl 0984.57014