On the homology theory of the closed geodesic problem

Abstract

Let $\Lambda X$ be the free loop space on a simply connected finite $CW$-complex $X$ and $\beta_{i}(\Lambda X;\Bbbk)$ be the cardinality of a minimal generating set of $H^{i}(\Lambda X;\Bbbk)$ for $\Bbbk$ to be a commutative ring with unit. The sequence $ \beta_{i}(\Lambda X;\Bbbk) $ grows unbounded if and only if $\tilde {H}^{\ast}(X;\Bbbk)$ requires at least two algebra generators. This in particular answers to a long standing problem whether a simply connected closed smooth manifold has infinitely many geometrically distinct closed geodesics in any Riemannian metric.