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
Famous quotes containing the words formal and/or spaces:
“Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more formal one said, “Do you think the Holy Dove could fly down with only one wing?””
—Horace Walpole (1717–1797)
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)