On category weight and the Arnold conjecture
by
Yuli B. Rudyak
FB Mathematik, Universität Siegen, 57068 Siegen, Germany

Let (M²ⁿ, w) be a closed symplectic manifold. The well-known Arnold conjecture claims that, for every Hamiltonian symplectomorphism f: M ® M, the number of fixed points of f is at least the minimal number of critical points of a smooth function on M. Floer and Hofer proved that, under the condition p₂(M)=0, the number of fixed points of f is at least the cup-length of M. So, here we have a weak form of the Arnold conjecture.

I prove, also under the condition p₂(M)=0, the precise estimation from the Arnold conjecture, i.e., that the number of fixed points of f is at least the minimal number of critical points of a smooth function on M. More precisely, I prove that the number of fixed points of f is at least 2n+1 (provided p₂(M)=0). Since the minimal number of critical points of a function on every closed 2n-dimensional manifold is at most 2n+1 (actually, equal to 2n+1 if p₂(M)=0), this implies the conjecture.

The proof uses the analitical results of Floer and Hofer. I did not develop any new analitical things, but just added some topological arguments. Namely, making purely topological research, I have found a way to compute the Lusternik-Schnirelmann category of some maps and spaces (using the technique of so-called category weight). Using this, I was able to estimate the number of rest points of a gradient-like flow on a "bad" topological space. This enabled me, in turn, to improve the Floer-Hofer estimation.

Date received: June 17, 1999