Rational homotopy types of simply connected spaces can be identified with (isomorphism classes of) certain algebraic objects called Sullivan minimal models, which are commutative differential graded algebras over the rational numbers satisfying certain conditions.
A geometric application was the theorem of Sullivan and Micheline Vigué-Poirrier (1976): every simply connected closed Riemannian manifold X whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct closed geodesics.
[2] The proof used rational homotopy theory to show that the Betti numbers of the free loop space of X are unbounded.
[3] The rational homotopy category (of simply connected spaces) is defined to be the localization of the category of simply connected spaces with respect to rational homotopy equivalences.
This is a special case of Sullivan's construction of the localization of a space at a given set of prime numbers.
More generally, let X be any space whose rational cohomology ring is a free graded-commutative algebra (a tensor product of a polynomial ring on generators of even degree and an exterior algebra on generators of odd degree).
The hypothesis on the cohomology ring applies to any compact Lie group (or more generally, any loop space).
, and the homotopy groups form a graded Lie algebra via the Whitehead product.
of finite dimension, there is a simply connected space X whose rational cohomology ring is isomorphic to A.
(By contrast, there are many restrictions, not completely understood, on the integral or mod p cohomology rings of topological spaces, for prime numbers p.) In the same spirit, Sullivan showed that any graded-commutative
that satisfies Poincaré duality is the cohomology ring of some simply connected smooth closed manifold, except in dimension 4a; in that case, one also needs to assume that the intersection pairing on
For spaces whose rational homology in each degree has finite dimension, Sullivan classified all rational homotopy types in terms of simpler algebraic objects, Sullivan algebras.
In the context of differential graded algebras A, "commutative" is used to mean graded-commutative; that is, for a in
There are examples of non-isomorphic minimal Sullivan models with isomorphic cohomology algebras.)
For any topological space X, Sullivan defined a commutative differential graded algebra
An element of this algebra consists of (roughly) a polynomial form on each singular simplex of X, compatible with face and degeneracy maps.
More precisely, any differential graded algebra with the same Sullivan minimal model as
To any simply connected CW complex X with all rational homology groups of finite dimension, there is a minimal Sullivan model
[12] This gives an equivalence between rational homotopy types of such spaces and such algebras, with the properties: When X is a smooth manifold, the differential algebra of smooth differential forms on X (the de Rham complex) is almost a model for X; more precisely it is the tensor product of a model for X with the reals and therefore determines the real homotopy type.
One can go further and define the p-completed homotopy type of X for a prime number p. Sullivan's "arithmetic square" reduces many problems in homotopy theory to the combination of rational and p-completed homotopy theory, for all primes p.[13] The construction of Sullivan minimal models for simply connected spaces extends to nilpotent spaces.
Thus the rational homotopy type of a formal space is completely determined by its cohomology ring.
[15] The simplest example of a non-formal nilmanifold is the Heisenberg manifold, the quotient of the Heisenberg group of real 3×3 upper triangular matrices with 1's on the diagonal by its subgroup of matrices with integral coefficients.
There are also examples of non-formal, simply connected symplectic closed manifolds.
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 nontrivial triple Massey product.
Rational homotopy theory revealed an unexpected dichotomy among finite CW complexes: either the rational homotopy groups are zero in sufficiently high degrees, or they grow exponentially.
[21] Bott's conjecture predicts that every simply connected closed Riemannian manifold with nonnegative sectional curvature should be rationally elliptic.
[22] Halperin's conjecture asserts that the rational Serre spectral sequence of a fiber sequence of simply-connected spaces with rationally elliptic fiber of non-zero Euler characteristic vanishes at the second page.
grows at most polynomially, for every prime number p. All known Riemannian manifolds with nonnegative sectional curvature are in fact integrally elliptic.