On the closed geodesics problem

TL;DR

This paper proves that for simply connected finite CW complexes with cohomology algebra generated by at least two elements, the Betti numbers of the free loop space grow unbounded, solving the closed geodesics problem.

Contribution

It provides a complete solution to the closed geodesics problem by linking cohomology algebra structure to Betti number growth.

Findings

Betti numbers of free loop spaces grow unbounded under given conditions

Cohomology algebra generated by multiple elements implies infinite closed geodesics

Complete resolution of the closed geodesics problem for certain spaces

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.