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

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