# Rational homotopy theory

## Mathematical theory of topological spaces / From Wikipedia, the free encyclopedia

#### Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about Rational homotopy theory?

Summarize this article for a 10 years old

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.[1] It was founded by Dennis Sullivan (1977) and Daniel Quillen (1969).[1] 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.[2] 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.