A contribution to the closed geodesic problem
It is conjectured that if \(X\) is a simply connected finite complex such that its cohomology algebra over a field \(\mathbb F\) is generated by at least two generators, then \(\{\dim H^i(LX,\mathbb F )\}_{i\geq 0}\) grows unbounded. Here \(LX\) is the space of free loops on \(X\). This conjecture is known to hold for \(\mathbb F=\mathbb Q\) by work of \textit{D. Sullivan} and \textit{M. Vigué-Poirrier} [J. Differ. Geom. 11, 633--644 (1976; Zbl 0361.53058)]. The authors in this paper give a sufficient algebraic condition for when this conjecture holds. With \(X\) as in the hypothesis of the conjecture, they first exhibit a commutative differential graded algebra \(\Gamma\) such that \(H^0(\Gamma)=0=H^1(\Gamma)\) and \(H(\Gamma )\) is generated by two generators, with a surjective homomorphism of DGA \(p : T\to\Gamma\), where \(T\) is a free DGA quasi-isomorphic to the singular cochains \(C^*(X,\mathbb F )\). The existence of \(T\), with certain additional properties, can be deduced from work of Halperin-Lemaire, Lambrechts and McCleary. Then they show that if \(p\) admits an \(A_{\infty}\)-map \({\mathbf\sigma}:=\{\sigma_i\}_{i\geq 1}: \Gamma\to T\) with \(p\circ\sigma_1=id_\Gamma\), then the conjecture holds for \(X\). An \(A_{\infty}\)-map between two DGAs is what is needed to construct a homomorphism of graded coalgebras between their bar constructions. The authors go on to identify a finite set of obstructions to the existence of such an \(A_{\infty}\)-section \(\mathbf\sigma\), and then use this to deduce that if \(M\) is a simply connected closed smooth manifold of dimension \(n\) such that for some field coefficients, the graded algebra \(H^*(M)\) is not monogenic and vanishes in degrees \(2\leq i\leq r-1\) with \(n<3(2r-1)\), then \(M\) has infinitely many geometrically distinct closed geodesics for any choice of Riemannian metric.
scientific article; zbMATH DE number 5673214
scientific article