Stefan Papadima and Alexandru Suciu: Geometric and algebraic aspects of $1$-formality, p.355-375


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 Kähler manifold $M$, the product $M\times W$ has the rational homotopy type of a Kähler manifold, yet $M\times W$ admits no Kähler metric.

Key Words: Formality, fundamental group, cohomology jumping loci, holonomy Lie algebra, Bieri-Neumann-Strebel invariant, Malcev completion, lower central series, Kähler manifold, quasi-Kähler manifold, Milnor fiber, hyperplane arrangement, Artin group, Bestvina-Brady group, pencil, fibration, monodromy

2000 Mathematics Subject Classification: Primary: 55P62, 57M07,
Secondary: 14F35, 20J05, 55N25.

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