Formality and finiteness in rational homotopy theory

TL;DR

This paper investigates the relationships between formality and finiteness properties in rational homotopy theory, linking algebraic models to geometric features of spaces, with applications in algebraic geometry, Lie group actions, and 3-manifolds.

Contribution

It reinterprets 1-formality in terms of Malcev Lie algebra properties and explores how algebraic model properties influence the geometry of cohomology jump loci.

Findings

1-formality relates to filtered and graded formality of Malcev Lie algebra

Finiteness properties of spaces mirror in algebraic models

Algebraic model properties impact the geometry of cohomology jump loci

Abstract

We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1-formality property of the space may be reinterpreted in terms of the filtered and graded formality properties of the Malcev Lie algebra of its fundamental group, while some of the finiteness properties of the space are mirrored in the finiteness properties of algebraic models associated with it. In turn, the formality and finiteness properties of algebraic models have strong implications on the geometry of the cohomology jump loci of the space. We illustrate the theory with examples drawn from complex algebraic geometry, actions of compact Lie groups, and 3-dimensional manifolds.