Π1 of Symplectic Automorphism Groups and Invertibles in Quantum Homology Rings

The aim of this paper is to establish a connection between the topology of the automorphism group of a symplectic manifold (M,ω) and the quantum product on its homology. More precisely, we assume that M is closed and connected, and consider the group Ham(M,ω) of Hamiltonian automorphisms with the C-topology. Ham(M,ω) is a path-connected subgroup of the symplectic automorphism group Aut(M,ω); if H1(M,R) = 0, it is the connected component of the identity in Aut(M,ω). We introduce a homomorphism q from a certain extension of the fundamental group π1(Ham(M,ω)) to the group of invertibles in the quantum homology ring of M . This invariant can be used to detect nontrivial elements in π1(Ham(M,ω)). For example, consider M = S2 × S2 with the family of product structures ωλ = λ(ωS2 × 1) + 1 × ωS2, λ > 0. This example has been studied by Gromov [6], McDuff [8] and Abreu [1]. McDuff showed that for λ 6= 1, π1(Ham(M,ωλ)) contains an element of infinite order. This result can be recovered by our methods. In a less direct way, the existence of q imposes topological restrictions on all elements of π1(Ham(M,ω)). An example of this kind of reasoning can be found in section 10; a more important one will appear in forthcoming work by Lalonde, McDuff and Polterovich.