On the homology theory of the closed geodesic problem

2011/10/24 by Samson Saneblidze, Saneblidze, Samson
Mathematics · #53C22 (Primary) 55U20 (Secondary) #55P35 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:53C22 #msc:55P35 #msc:55U20

paper · pdf · doi:10.48550/arxiv.1110.5233

18 pages, the reference is added, typos corrected

openalex publication_date 2011/10/24 · arxiv created 2011/11/01 · arxiv updated 2011/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ΛX be the free loop space on a simply connected finite CW-complex X and β_i(ΛX;\Bbbk) be the cardinality of a minimal generating set of Hⁱ(ΛX;\Bbbk) for \Bbbk to be a commutative ring with unit. The sequence β_i(ΛX;\Bbbk) grows unbounded if and only if H^∗(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.

Citations

• On the Betti numbers of a loop space

Related

• Morse theory, closed geodesics, and the homology of free loop spaces
• On "small geodesics" and free loop spaces
• Closed geodesics on orbifolds
• The Existence of Infinitely Many Geometrically Distinct Non-Constant Prime Closed Geodesics on Riemannian Manifolds
• Counting closed geodesics on rank one manifolds without focal points