Rational Ljusternik-Schnirelmann category and the Arnol'd conjecture for nilmanifolds
Let \((M,w)\) be a compact symplectic manifold whose universal cover is the Euclidean space; so \(\pi_ i(M) = 0\) for \(i \geq 2\). The authors prove that a smooth 1-periodic Hamiltonian system on \(M\) has at least \(e_ 0(M) + 1\) contractible 1-periodic solutions. Here \(e_ 0(M)\) is an invariant from rational homotopy theory which is a lower bound for the Lyusternik-Shnirel'man category. If \(M\) is a nilmanifold then \(e_ 0(M) = \dim(M)\). Thus for these manifolds the Arnol'd conjecture is true since \(1 + \dim(M)\) is the minimum number of critical points of a smooth function \(M \to \mathbb{R}\). For the proof the authors use variational methods. The finite-dimensional reduction as in \textit{Conley-Zehnder} [Invent. Math. 73, 33-49 (1983; Zbl 0516.58017)] who dealt with the case \(M = T^{2N}\) a torus, carries over almost verbatim. Then they apply methods from rational homotopy theory related to the rational Lyusternik- Shnirel'man category.
rational Lyusternik-Shnirel'man category
scientific article; zbMATH DE number 550340