On the homology theory of the closed geodesic problem

TL;DR

This paper establishes a homological criterion linking the growth of minimal generators in the free loop space to the algebraic structure of the base space, providing a solution to a long-standing geodesic problem.

Contribution

It proves that the unbounded growth of minimal generators in the free loop space characterizes the algebraic complexity of the base space, resolving a major open question about closed geodesics.

Findings

Growth of minimal generators is unbounded iff the cohomology requires at least two algebra generators.

Provides a homological criterion for the existence of infinitely many closed geodesics.

Answers a long-standing problem in Riemannian geometry regarding closed geodesics on manifolds.

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.