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

Abstract

In [Arn, Appendix 9] Arnold proposed a beautiful conjecture concerning the relation between the number of fixed points of certain (i.e., exact or Hamiltonian) selfdiffeomorphisms of a closed symplectic manifold (M,ω) and the minimum number of critical points of any smooth (= C∞) function on M . The first author succeeded in proving this form of the Arnold conjecture [R2] under the hypothesis that ω and c1 vanish on all spherical homology classes and that there is equality between the Lusternik–Schnirelmann category of M and the dimension of M . In this paper, we use a fundamental property of category weight to show that, for any closed symplectic manifold whose symplectic form vanishes on the image of the Hurewicz map, the required equality holds. Thus, we show that the original form of the Arnold Conjecture holds for all symplectic manifolds having ω|π2(M) = 0 = c1|π2(M).