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, ${\pi }_2(M)=0$.

arXiv commentAMSTEX, 5 pages, submitted to Math. Zeitschrift

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