Rational homotopy theory

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In mathematics, rational homotopy theory is the study of the rational homotopy type of a space, which means roughly that one ignores all torsion in the homotopy groups. It was started by Dennis Sullivan (1977) and Daniel Quillen (1969).

Rational homotopy types of simply connected spaces can be identified with certain algebraic objects called minimal Sullivan algebras, which are commutative differential graded algebras over the rational numbers satisfying certain conditions.

The standard textbook on rational homotopy theory is (Félix, Halperin & Thomas 2001).

Contents

[edit] Rational spaces

A rational space is a simply connected space all of whose homotopy groups are vector spaces over the rational numbers. If X is any simply connected CW complex, then there is a rational space Y, unique up to homotopy equivalence, and a map from X to Y inducing an isomorphism on homotopy groups tensored with the rational numbers. The space Y is called the rationalization of X, and is the localization of X at the rationals, and is the rational homotopy type of X. Informally, it is obtained from X by killing all torsion in the homotopy groups of X.

[edit] Sullivan algebras

A Sullivan algebra is a commutative differential graded algebra over the rationals Q, whose underlying algebra is the free supercommutative algebra Λ(V) on a graded vector space

V=\oplus_{n>0}V^n,\,

satisfying the following "nilpotence condition on d": V is the union of an increasing series of graded subspaces V(0)⊆V(1)⊆ where d = 0 on V(0) and d(V(k)) is contained in Λ(V(k − 1)). ("Commutative" means commutative in the graded sense, so ab = (−1)deg(a)deg(b)ba, and so on.)

The Sullivan algebra is called minimal if the image of d is contained in Λ+(V)2, where Λ+(V) is the direct sum of the positive degree subspaces of Λ(V).

A Sullivan model for a commutative differential graded algebra A is an algebra homomorphism from a Sullivan algebra Λ(V) that is an isomorphism on cohomology. If A0 = Q then A has a minimal Sullivan model which is unique up to isomorphism. (Warning: a minimal Sullivan algebra with the same cohomology as A need not be a minimal Sullivan model for A: it is also necessary that the ismorphism of cohomology be induced by an algebra homomorphism. There are examples of non-isomorphic minimal Sullivan models with the same cohomology algebra.)

[edit] The Sullivan minimal model of a topological space

For any topological space X Sullivan defined a commutative differential graded algebra APL(X), 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 mimimal model as APL(X) is called a model for the space X, and determines the rational homotopy type of X when X is simply connected.

To any simply connected CW complex X with all rational homology groups of finite dimension one can assign a minimal Sullivan algebra ΛV of APL(X), which has the property that V1 = 0 and all the Vk of finite dimension. This is called the Sullivan minimal model of X, and is unique up to isomorphism. This gives an equivalence between rational homotopy types of such spaces and such algebras, such that:

  • The rational cohomology of the space is the cohomology of its Sullivan minimal model.
  • The space V is the dual of the rational homotopy groups of the space X.
  • The Whitehead product on rational homotopy is the dual of the "quadratic part" of the differential d.
  • Two spaces have the same rational homotopy type if and only if their minimal Sullivan algebras are isomorphic.
  • There is a simply connected space X corresponding to each possible Sullivan algebra with V1 = 0 and all the Vk of finite dimension.

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.

The results above for simply connected spaces can easily be extended to nilpotent spaces (whose fundamental group is nilpotent and acts nilpotently on the higher homotopy groups). For more general fundamental groups things get more complicated; for example, the homotopy groups need not be finitely generated even if there are only a finite number of cells of the CW complex in each dimension.

[edit] Formal spaces

A commutative differential graded algebra A is called formal if A0 = Q and it has the same minimal Sullivan algebra as a commutative differential graded algebra with vanishing differential. So any two formal commutative differential graded algebras with the same cohomology algebra have the same Sullivan minimal model. A space is called formal if its minimal Sullivan model is formal, so the minimal Sullivan models of simply connected formal topological spaces are 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.

[edit] Examples

  • If X is a sphere of odd dimension 2n + 1 > 1, its minimal Sullivan model has 1 generator a of degree 2n + 1 with da = 0, and a basis of elements 1, a.
  • If X is a sphere of even dimension 2n > 0, its minimal Sullivan model has 2 generators a and b of degrees 2n and 4n − 1, with db = a2, da = 0, and a basis of elements 1, a, b→ a2, ab→a3, a2b→a4, ... where the arrow indicated the action of d.
  • Suppose that V has 3 elements a, b, c of degrees 3, 3, and 5, with differentials da = 0, db = 0, dc = ab. Then this algebra is a minimal Sullivan algebra that is not formal. If A is the commutative differential graded algebra with zero differential given by the cohomology of ΛV, then A and ΛV are examples of two non-isomorphic minimal Sullivan models with isomorphic cohomology. The corresponding topological spaces are two spaces with the same rational cohomology ring but different rational homotopy types.

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