Geometric and algebraic aspects of 1-formality
arXiv:0903.2307
Abstract
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.
22 pages
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Cited by in corpus (15)
- Fundamental groups, Alexander invariants, and cohomology jumping loci
- Chen ranks and resonance
- Around the tangent cone theorem
- Resonance varieties and Dwyer-Fried invariants
- Homotopical Height
- Cup products, lower central series, and holonomy Lie algebras
- Chen ranks and resonance varieties of the upper McCool groups
- Geometric and homological finiteness in free abelian covers
- Finite Galois covers, cohomology jump loci, formality properties, and multinets
- Algebraic models, Alexander-type invariants, and Green-Lazarsfeld sets
- Homology, lower central series, and hyperplane arrangements
- Formality and finiteness in rational homotopy theory
- Taylor expansions of groups and filtered-formality
- Remarks on the product of harmonic forms
- Alexander invariants and cohomology jump loci in group extensions