On the closed geodesics problem

Abstract

Let $\bk $ be a field of characteristic $p\geq 0$ and $X$ a simply connected finite CW complex. In this text, we prove that: {\sl if the cohomology algebra $H^*(X;\bk)$ is generated, as an algebra, by at least two linearly independent elements, then the sequence of Betti numbers $ \left( \dim H^n(LX;\bk)\right)_{n\geq 1 }$ grows unbounded.} This provides a complete solution of the closed geodesics problem.