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.
Categories: math.AT