The homology of a free loop space
Denote by \(X^{S^ 1}\) the space of all continuous maps from the circle into a simply connected finite CW complex X. Theorem: Let \({\mathbb{K}}\) be a field and suppose that either char \({\mathbb{K}}>\dim X\) or that X is \({\mathbb{K}}\)-formal. Then the Betti numbers \(b_ q=\dim H_ q(X^{S^ 1};{\mathbb{K}})\) are uniformly bounded above if and only if the \({\mathbb{K}}\)- algebra H*(X;\({\mathbb{K}})\) is generated by a single cohomology class. Corollary: If, in addition, X is a smooth closed manifold and \({\mathbb{K}}\) is as in the theorem, and if H*(X;\({\mathbb{K}})\) is not generated by a single class then X has infinitely many distinct closed geodesics in any Riemannian metric.
infinitely many distinct closed geodesics
scientific article; zbMATH DE number 4090367
scientific article