\(\pi_1\) of symplectic automorphism groups and invertibles in quantum homology rings. (English) Zbl 0928.53042

Seidel, P.

Geom. Funct. Anal. 7, No. 6, 1046-1095 (1997).

Let \((M,\omega)\) be a closed connected symplectic manifold. Further, let \(\text{Ham} (M,\omega)\) denote the group of Hamiltonian automorphisms of \((M,\omega)\) equipped with the \(C^{\infty}\)-topology. Defining a homomorphism from a certain extension of the fundamental group \(\pi_1(\text{Ham} (M,\omega))\) to the group of invertibles in the quantum homology ring of \((M,\omega)\), the author studies relations between the topology of the automorphism group of \((M,\omega)\) and the quantum product on its homology. Methods used to define this homomorphism are Hamiltonian fibre bundles, Floer homology, compatible almost complex structures, pseudoholomorphic curves as well as a gluing argument. Since the author allows time-dependent almost complex structures, the manifold has to satisfy a technical condition that replaces weak monotonicity. Finally, some examples and applications are given. For instance, a known result of D. McDuff for the Hamiltonian automorphism group of \(S^2\times S^2\) is recovered.

Reviewer: Katharina Habermann (Leipzig)

MSC:

53D40	Symplectic aspects of Floer homology and cohomology
32Q60	Almost complex manifolds
32Q65	Pseudoholomorphic curves

Keywords:

fundamental group of the group of Hamiltonian automorphisms; Hamiltonian fiber bundles; Floer homology; quantum homology