Rational homotopy theory

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

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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.