String homology, and closed geodesics on manifolds which are elliptic spaces
Let \(M\) be a simply connected closed smooth manifold of dimension \(n\geq 2\). It is conjectured that if, for some field \({\Bbbk}\), the cohomology algebra \(H^*(M;{\Bbbk})\) is generated by at least two generators then for any Riemannian metric on \(M\) there exist infinitely many geometrically distinct closed geodesics. This conjecture, also called the closed geodesic problem, has been solved by many authors under additional hypotheses. Here the authors prove this conjecture for the prime fields \(\mathbb{F}_p\), \(p\geq 2\) and manifolds \(M\) which are \(\mathbb{F}_p\)-elliptic spaces. NEWLINENEWLINEEllipticity is a very severe restriction on topological spaces while many of the spaces which play a role in geometry are elliptic. For example, simply connected compact homogeneous spaces, simply connected positively curved closed manifolds, fibrations built out of elliptic spaces, et cetera. The proof uses string topology as well as older results on elliptic connected graded Hopf algebras.
scientific article; zbMATH DE number 6653754
scientific article