Formality and finiteness in rational homotopy theory

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.