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.

81 pages; prepared for a volume dedicated to Dennis Sullivan's 80th birthday

References in corpus (6)

• Algebraic invariants for right-angled Artin groups
• The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces
• Toric complexes and Artin kernels
• Algebraic invariants for Bestvina-Brady groups
• Geometric and algebraic aspects of 1-formality
• Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Cited by in corpus (2)

• Milnor fibrations of arrangements with trivial algebraic monodromy
• Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures