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Autori: S. Papadima, A. Suciu
Editorial: Bulletin Math'ematique de la Soci'et'e des Sciences Math'ematiques de Roumanie, 52 (3), p.355-375, 2009.
Formality is a topological property, defined in terms of Sullivan’s model for a space. In the simply-connected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring. In the general setting, the weaker 1-formality property allows one to reconstruct the rational pro-unipotent completion of the fundamental group, solely from the cup products of degree 1 cohomology classes. In this note, we survey various facets of formality, with emphasis on the geometric and algebraic implications of 1-formality, and its relations to the cohomology jump loci and the Bieri-Neumann-Strebel invariant. We also produce examples of 4-manifolds W such that, for every compact K”ahler manifold M, the product Mtimes W has the rational homotopy type of a K”ahler manifold, yet M x W admits no K”ahler metric.
Cuvinte cheie: Formality, fundamental group, cohomology jumping loci, holonomy Lie algebra, Bieri-Neumann-Strebel invariant, Malcev completion, lower central series, K"ahler manifold, quasi-K"ahler manifold, Milnor fiber, hyperplane arrangement, Artin group, Bestvina-Brady group, pencil, fibration, monodromy