Geometric and algebraic aspects of 1-formality

Details Geometric and algebraic aspects of 1-formality Geometric and algebraic aspects of 1-formality

2009-03-13

arxiv.org/abs/0903.2307v2

Bull. Math. Soc. Sci. Math. Roumanie 52 (2009), no. 3, 355-375

Mathematics

Algebraic Topology

22 pages

Scientific paper

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 M\times W has the rational homotopy type of a K\"ahler manifold, yet M\times W admits no K\"ahler metric.

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