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Floer homology and Novikov rings

1995; Linguagem: Inglês

10.1007/978-3-0348-9217-9_20

Helmut Hofer, Dietmar Salamon,

Homotopy and Cohomology in Algebraic Topology

We prove the Arnold conjecture for compact symplectic manifolds under the assumption that either the first Chern class of the tangent bundle vanishes over π2(M) or the minimal Chern number is at least half the dimension of the manifold. This includes the important class of Calabi-Yau manifolds. The key observation is that the Floer homology groups of the loop space form a module over Novikov’s ring of generalized Laurent series. The main difficulties to overcome are the presence of holomorphic spheres and the fact that the action functional is only well defined on the universal cover of the loop space with a possibly dense set of critical levels.