Formal Spaces
A commutative differential graded algebra A, again with A0 = Q, is called formal if A has a model with vanishing differential. This is equivalent to requiring that the cohomology algebra of A (viewed as a differential algebra with trivial differential) is a model for A. Thus two formal commutative differential graded algebras with the isomorphic cohomology algebras have the same Sullivan minimal model. A space is called formal if its minimal Sullivan model is formal, so the minimal Sullivan model of a simply connected formal topological space is determined by the rational cohomology ring. This means that the rational homotopy of a formal space is particularly easy to work out.
Examples of formal spaces include spheres, H-spaces, symmetric spaces, and compact Kähler manifolds (Deligne et al. 1989). Formality is preserved under wedge sums and direct products; it is also preserved under connected sums for manifolds.
On the other hand, nilmanifolds are almost never formal: if Mn is a compact formal nilmanifold, then Mn=Tn, the n-dimensional torus (Hasegawa 1975). The simplest example of a non-formal compact nilmanifold is the Heisenberg manifold, the quotient of the Heisenberg group of 3×3 upper triangular matrices with 1's on the diagonal by its subgroup of matrices with integral coefficients. Symplectic manifolds need not be formal: the simplest example is the Kodaira-Thurston manifold (the product of the Heisenberg manifold with a circle). Examples of non-formal, simply connected symplectic manifolds were given in Babenko & Taimanov (2000).
Non-formality may often be detected by Massey products. Indeed, if a differential graded algebra A is formal, then all (higher order) Massey products must vanish. The converse is not true: formality means, roughly speaking, the "uniform" vanishing of all Massey products. The complement of the Borromean rings is a non-formal space: it supports a non-trivial triple Massey product.
Halperin & Stasheff (1979) gave an algorithm for deciding whether or not a commutative differential graded algebra is formal.
Read more about this topic: Rational Homotopy Theory
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