Formality and finiteness in rational homotopy theory
arXiv:2210.08310 · doi:10.4171/EMSS/74
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.
arXiv comment81 pages; prepared for a volume dedicated to Dennis Sullivan's 80th birthday
References in corpus (17)
- Geometry of cohomology support loci for local systems I
- Algebraic invariants for right-angled Artin groups
- Chen Lie algebras
- 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
- Massey products in symplectic manifolds
- Geometric and algebraic aspects of 1-formality
- Formality properties of finitely generated groups and Lie algebras
- Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces
- A finite Q-bad space
- Around the tangent cone theorem
- Cup products, lower central series, and holonomy Lie algebras
- Non-abelian resonance: product and coproduct formulas
- Quadratic-linear duality and rational homotopy theory of chordal arrangements
- On Thom spaces, Massey products and non-formal symplectic manifolds
- Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products