Geometric and algebraic aspects of 1-formality

TL;DR

This paper explores the concept of 1-formality in topology, examining its geometric and algebraic implications, and providing examples of manifolds with specific rational homotopy properties related to Kähler structures.

Contribution

It surveys various aspects of 1-formality, highlighting its implications and relations to cohomology jump loci and the Bieri-Neumann-Strebel invariant, and constructs examples of manifolds with particular rational homotopy types.

Findings

1-formality relates to the reconstruction of the fundamental group's rational pro-unipotent completion.

Examples of 4-manifolds are provided where products with Kähler manifolds have specific homotopy types but no Kähler metric.

The survey links 1-formality to geometric properties and cohomology invariants.

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 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.