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,ω)
equals the dimension
dim(M)
provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when
π
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,
π
2
(M)=0
.