Floer homology and Novikov rings. (English) Zbl 0842.58029

Hofer, H.; Salamon, D. A.

Hofer, Helmut (ed.) et al., The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 483-524 (1995).

Considering a period one periodic Hamiltonian system on a compact symplectic manifold \((M,\omega)\), the Arnol’d conjecture states that if the 1-periodic solutions are all nondegenerate, their number is bounded below by the sum of the Betti numbers of \(M\). Symplectic manifolds \((\dim = 2n)\) with the property that for every \(A \in \pi_2(M)\), \(3 - n \leq c_1(A) < 0\) implies \(\omega(A) \leq 0\), are called weakly monotone. It is shown that either such manifolds are monotone, or the first Chern class \(c_1(A)\) vanishes for each \(A\), or the minimal Chern number \(N\) defined by \(c_1(\pi_2(M)) - N\mathbb{Z}\) is greater than or equal to \(n - 2\). Floer proved the Arnol’d conjecture in the first case. The present authors extend this important result to the second and third case. One of the difficulties to overcome is the presence of nonconstant \(J\)-holomorphic spheres with \(c_1 \leq 0\). It is dealt with by proving some genericity results for almost complex structures \(J\) satisfying a regularity assumption. A key result is that for a large class of weakly monotone symplectic manifolds (assuming \(N \geq n\) in the third case) the Floer cohomology groups agree with the cohomology of \(M\) with coefficients in the Novikov ring of generalized Laurent series.
For the entire collection see [Zbl 0824.00019].

Reviewer: W.Sarlet (Gent)

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99	Local and nonlocal bifurcation theory for dynamical systems
34C25	Periodic solutions to ordinary differential equations
53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57R20	Characteristic classes and numbers in differential topology

Keywords:

periodic Hamiltonian systems; Arnol’d’s conjecture; Floer homology; weakly monotone symplectic manifolds