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 Dennis Sullivan (1977) and Daniel Quillen (1969). This simplification of homotopy theory makes certain 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 f : X → Y {\displaystyle f\colon X\to Y} of simply connected topological spaces is called a rational homotopy equivalence if it induces an isomorphism on homotopy groups tensored with the rational numbers Q {\displaystyle \mathbb {Q} } . 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 (i.e. to determine the information that can be recovered from rational homotopy equivalences). 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 X {\displaystyle X} , there is a rational space X Q {\displaystyle X_{\mathbb {Q} }} , unique up to homotopy equivalence, with a map X → X Q {\displaystyle X\to X_{\mathbb {Q} }} that induces an isomorphism on homotopy groups tensored with the rational numbers. The space X Q {\displaystyle X_{\mathbb {Q} }} is called the rationalization of X {\displaystyle X} . This is a special case of Sullivan's construction of the localization of a space at a given set of prime numbers. One obtains equivalent definitions using homology rather than homotopy groups. Namely, a simply connected CW complex X {\displaystyle X} is a rational space if and only if its homology groups H i ( X , Z ) {\displaystyle H_{i}(X,\mathbb {Z} )} are rational vector spaces for all i > 0 {\displaystyle i>0} . The rationalization of a simply connected CW complex X {\displaystyle X} is the unique rational space X → X Q {\displaystyle X\to X_{\mathbb {Q} }} (up to homotopy equivalence) with a map X → X Q {\displaystyle X\to X_{\mathbb {Q} }} that induces an isomorphism on rational homology. Thus, one has
π i ( X Q ) ≅ π i ( X ) ⊗ Q {\displaystyle \pi _{i}(X_{\mathbb {Q} })\cong \pi _{i}(X)\otimes {\mathbb {Q} }}
and
H i ( X Q , Z ) ≅ H i ( X , Z ) ⊗ Q ≅ H i ( X , Q ) {\displaystyle H_{i}(X_{\mathbb {Q} },{\mathbb {Z} })\cong H_{i}(X,{\mathbb {Z} })\otimes {\mathbb {Q} }\cong H_{i}(X,{\mathbb {Q} })}
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