On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture

Abstract

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, \omega)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $\pi_2(M)=0$. The Arnold conjecture asserts that the number of fixed points of a Hamiltonian symplectomorphism of $M$ is greater than or equal to the number of critical points of some function on $M$. A modified form of the conjecture, replacing the latter quantity (via Lusternik-Schnirelmann theory) by $cup(M) + 1$, has been proved recently by various authors using techniques of Floer. The first author has also recently shown that the original form of the conjecture holds when $cat(M) =dim(M)$. Thus, this paper completes the proof of the original Arnold conjecture for closed symplectic manifolds with, for example, $\pi_2(M)=0$.