Rational Ljusternik-schnirelmann Category and the Arnold Conjecture for Nilmanifolds

A nilmanifold is a homogenous space of a nilpotent Lie group. If M is a compact symplectic nilmanifold, then any 1-periodic Hamiltonian system on M has at least dim(M) + 1 contractible periodic orbits with period 1. This provides an aarmative answer to the Arnold conjecture for such manifolds. The proof uses the techniques of rational homotopy theory, and in particular an extension of the rational L.-S. category invariant e 0 to maps. 1. The Arnold Conjecture The Arnold conjecture 2] is a well-known problem in Hamiltonian dynamics, which has been studied in various forms. This paper deals with the following formulation of the conjecture: let (M; !) be a compact symplectic manifold, H : M R ! R a smooth function which is 1-periodic in time. Then the Hamiltonian ow associated with H has at least MC(M) 1-periodic orbits, where MC(M) is the minimum number of critical points of a real-valued function on M. In this form, the rst results were obtained by Conley and Zehnder 5] for the case of M a torus with the standard symplectic form. Several other authors have made contributions, introducing new methods and considering diierent classes of manifolds 10, 11, 12, 13, 24]. In the form stated above, the most general results are those of Floer, who shows 11, 12] for all compact symplectic manifolds (M; !) with 2 (M) = 0 that the Hamiltonian ow associated with H has at least 1 + CL 2 (M) 1-periodic orbits, where CL 2 (M) is the mod 2 cup-length of M { the longest nontrivial product in ~ H (M; Z 2). The cuplength is a lower bound for MC(M), but in general the two are not equal, so these results do not provide a complete answer to the conjecture. In this paper, we show that this cuplength estimate can be improved for some manifolds. We consider compact symplectic manifolds which have Euclidean space as their universal cover. Examples include tori and surfaces with non-positive Euler characteristic. These manifolds form a large subclass of the the class of aspherical manifolds (manifolds with i (M) = 0 for i 6 = 1), though not all aspherical manifolds have a Euclidean cover 6]. To study these manifolds, we make use of an invariant from rational homotopy theory, e 0 (M), which is often (but not universally) a better estimate of MC(M) than the mod 2 cuplength. …