Rational homotopy theory

In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homotopy theory makes calculations much easier.
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. The proof used rational homotopy theory to show that the Betti numbers of the free loop space of "X" are unbounded. The theorem then follows from a 1969 result of Detlef Gromoll and Wolfgang Meyer. 
Rational spaces.
A continuous map formula_1 of simply connected topological spaces is called a rational homotopy equivalence if it induces an isomorphism on homotopy groups tensored with the rational numbers formula_2. Equivalently: "f" is a rational homotopy equivalence if and only if it induces an isomorphism on singular homology groups with rational coefficients. 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. The goal of rational homotopy theory is to understand this category. That is, if one declares all rational homotopy equivalences to be isomorphisms, how much information is left?
One basic result is that the rational homotopy category is equivalent to a full subcategory of the homotopy category of topological spaces, the subcategory of rational spaces. By definition, a rational space is a simply connected CW complex all of whose homotopy groups are vector spaces over the rational numbers. For any simply connected CW complex formula_3, there is a rational space formula_4, unique up to homotopy equivalence, with a map formula_5 that induces an isomorphism on homotopy groups tensored with the rational numbers. The space formula_4 is called the rationalization of formula_3. This is a special case of Sullivan's construction of the localization of a space at a given set of prime numbers.
One gets equivalent definitions using homology rather than homotopy groups. Namely, a simply connected CW complex formula_3 is a rational space if and only if its homology groups formula_9 are rational vector spaces for all formula_10. The rationalization of a simply connected CW complex formula_3 is the unique rational space formula_5 (up to homotopy equivalence) with a map formula_5 that induces an isomorphism on rational homology. Thus one has
and
for all formula_16.
These results for simply connected spaces extend with little change to nilpotent spaces (spaces whose fundamental group is nilpotent and acts nilpotently on the higher homotopy groups).
Computing the homotopy groups of spheres is a central open problem in homotopy theory. However, the "rational" homotopy groups of spheres were computed by Jean-Pierre Serre in 1951:
and
This suggests the possibility of describing the whole rational homotopy category in a practically computable way. Rational homotopy theory has realized much of that goal.
In homotopy theory, spheres and Eilenberg–MacLane spaces are two very different types of basic spaces from which all spaces can be built. In rational homotopy theory, these two types of spaces become much closer. In particular, Serre's calculation implies that formula_19 is the Eilenberg–MacLane space formula_20. 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). Then the rationalization formula_4 is a product of Eilenberg–MacLane spaces. The hypothesis on the cohomology ring applies to any compact Lie group (or more generally, any loop space). For example, for the unitary group SU("n"),
Cohomology ring and homotopy Lie algebra.
There are two basic invariants of a space "X" in the rational homotopy category: the rational cohomology ring formula_23 and the homotopy Lie algebra formula_24. The rational cohomology is a graded-commutative algebra over formula_2, and the homotopy groups form a graded Lie algebra via the Whitehead product. (More precisely, writing formula_26 for the loop space of "X", we have that formula_27 is a graded Lie algebra over formula_2. In view of the isomorphism formula_29, this just amounts to a shift of the grading by 1.) For example, Serre's theorem above says that formula_30 is the free graded Lie algebra on one generator of degree formula_31.
Another way to think of the homotopy Lie algebra is that the homology of the loop space of "X" is the universal enveloping algebra of the homotopy Lie algebra:
Conversely, one can reconstruct the rational homotopy Lie algebra from the homology of the loop space as the subspace of primitive elements in the Hopf algebra formula_33.
A central result of the theory is that the rational homotopy category can be described in a purely algebraic way; in fact, in two different algebraic ways. First, Quillen showed that the rational homotopy category is equivalent to the homotopy category of connected differential graded Lie algebras. (The associated graded Lie algebra formula_34 is the homotopy Lie algebra.) Second, Quillen showed that the rational homotopy category is equivalent to the homotopy category of 1-connected differential graded cocommutative coalgebras. (The associated coalgebra is the rational homology of "X" as a coalgebra; the dual vector space is the rational cohomology ring.) These equivalences were among the first applications of Quillen's theory of model categories.
In particular, the second description implies that for any graded-commutative formula_2-algebra "A" of the form
with each vector space formula_37 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 formula_2-algebra with formula_39 that satisfies Poincaré duality is the cohomology ring of some simply connected smooth closed manifold, except in dimension 4"a"; in that case, one also needs to assume that the intersection pairing on formula_40 is of the form formula_41 over formula_2.
One may ask how to pass between the two algebraic descriptions of the rational homotopy category. In short, a Lie algebra determines a graded-commutative algebra by Lie algebra cohomology, and an augmented commutative algebra determines a graded Lie algebra by reduced André–Quillen cohomology. More generally, there are versions of these constructions for differential graded algebras. This duality between commutative algebras and Lie algebras is a version of Koszul duality.
Sullivan algebras.
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. By definition, a Sullivan algebra is a commutative differential graded algebra over the rationals formula_2, whose underlying algebra is the free commutative graded algebra formula_44 on a graded vector space
satisfying the following "nilpotence condition" on its differential "d": the space "V" is the union of an increasing series of graded subspaces, formula_46, where formula_47 on formula_48 and formula_49 is contained in formula_50. In the context of differential graded algebras "A", "commutative" is used to mean graded-commutative; that is,
for "a" in formula_37 and "b" in formula_53.
The Sullivan algebra is called minimal if the image of "d" is contained in formula_54, where formula_55 is the direct sum of the positive-degree subspaces of formula_56.
A Sullivan model for a commutative differential graded algebra "A" is a Sullivan algebra formula_56 with a homomorphism formula_58 which induces an isomorphism on cohomology. If formula_59, then "A" has a minimal Sullivan model which is unique up to isomorphism. (Warning: a minimal Sullivan algebra with the same cohomology algebra as "A" need not be a minimal Sullivan model for "A": it is also necessary that the isomorphism of cohomology be induced by a homomorphism of differential graded algebras. There are examples of non-isomorphic minimal Sullivan models with isomorphic cohomology algebras.)
The Sullivan minimal model of a topological space.
For any topological space "X", Sullivan defined a commutative differential graded algebra formula_60, called the algebra of polynomial differential forms on "X" with rational coefficients. An element of this algebra consists of (roughly) a polynomial form on each singular simplex of "X", compatible with face and degeneracy maps. This algebra is usually very large (uncountable dimension) but can be replaced by a much smaller algebra. More precisely, any differential graded algebra with the same Sullivan minimal model as formula_60 is called a model for the space "X". When "X" is simply connected, such a model determines the rational homotopy type of "X".
To any simply connected CW complex "X" with all rational homology groups of finite dimension, there is a minimal Sullivan model formula_62 for formula_60, which has the property that formula_64 and all the formula_65 have finite dimension. This is called the Sullivan minimal model of "X"; it is unique up to isomorphism. 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".
The construction of Sullivan minimal models for simply connected spaces extends to nilpotent spaces. For more general fundamental groups, things get more complicated; for example, the rational homotopy groups of a finite CW complex (such as the wedge formula_68) can be infinite-dimensional vector spaces.
Formal spaces.
A commutative differential graded algebra "A", again with formula_59, 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" (though it does not have to be the "minimal" model). Thus the rational homotopy type of a formal space is completely determined by its cohomology ring.
Examples of formal spaces include spheres, H-spaces, symmetric spaces, and compact Kähler manifolds. Formality is preserved under products and wedge sums. For manifolds, formality is preserved by connected sums.
On the other hand, closed nilmanifolds are almost never formal: if "M" is a formal nilmanifold, then "M" must be the torus of some dimension. 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. Closed symplectic manifolds need not be formal: the simplest example is the Kodaira–Thurston manifold (the product of the Heisenberg manifold with a circle). There are also examples of non-formal, simply connected symplectic closed manifolds.
Non-formality can 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 nontrivial triple Massey product.
Elliptic and hyperbolic spaces.
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. Namely, let "X" be a simply connected space such that formula_97 is a finite-dimensional formula_2-vector space (for example, a finite CW complex has this property). Define "X" to be rationally elliptic if formula_99 is also a finite-dimensional formula_2-vector space, and otherwise rationally hyperbolic. Then Félix and Halperin showed: if "X" is rationally hyperbolic, then there is a real number formula_101 and an integer "N" such that
for all formula_103.
For example, spheres, complex projective spaces, and homogeneous spaces for compact Lie groups are elliptic. On the other hand, "most" finite complexes are hyperbolic. For example:
There are many other restrictions on the rational cohomology ring of an elliptic space.
Bott's conjecture predicts that every simply connected closed Riemannian manifold with nonnegative sectional curvature should be rationally elliptic. Very little is known about the conjecture, although it holds for all known examples of such manifolds.
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.
A simply connected finite complex "X" is rationally elliptic if and only if the rational homology of the loop space formula_26 grows at most polynomially. More generally, "X" is called integrally elliptic if the mod "p" homology of formula_26 grows at most polynomially, for every prime number "p". All known Riemannian manifolds with nonnegative sectional curvature are in fact integrally elliptic.