Symplectic fixed points and holomorphic spheres

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LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.