The Seidel Morphism of Cartesian Products

We prove that the Seidel morphism of (M × M , ω ⊕ ω) is naturally related to the Seidel morphisms of (M,ω) and (M , ω), when these manifolds are monotone. We deduce a condition for loops of Hamiltonian diffeomorphisms of the product to be homotopically non trivial. This result was inspired by and extends results obtained by Pedroza [P]. All the symplectic manifolds we consider in this note are closed. A symplectic manifold (M,ω) is strongly semi-positive if (at least) one of the following conditions holds (c1 denotes the first Chern class c1(TM,ω)): (a) there exists λ ≥ 0, such that for all A ∈ π2(M), ω(A) = λc1(A), (b) c1 vanishes on π2(M), (c) the minimal Chern number N (c1(π2(M)) = NZ) satisfies N ≥ n− 1. Under this assumption, Seidel introduced [S] a group morphism: qM : π̃1(Ham(M,ω)) −→ QH∗(M,ω) , where QH∗(M,ω) × denotes the group of invertible elements of QH∗(M,ω), the quantum homology of (M,ω). We recall that the identity of the groupQH∗(M,ω) × is the fundamental class of M , which is denoted [M ]. As usual, Ham(M,ω) denotes the group of Hamiltonian diffeomorphisms of (M,ω) and π̃1(Ham(M,ω)) is a covering of π1(Ham(M,ω)) which will be defined below. The inclusions of Ham(M,ω) and Ham(M , ω) in Ham(M × M , ω ⊕ ω) induce a map between the respective fundamental groups: ([g], [g]) 7→ [g, g], where [g, g] stands for [(g, g)], the homotopy class of the loop (g, g). The extension of this map to the coverings π̃1 is straightforward. We denote it by i : π̃1(Ham(M,ω))× π̃1(Ham(M , ω)) −→ π̃1(Ham(M ×M , ω ⊕ ω)). We also denote by κQ : QH∗(M,ω)⊗QH∗(M , ω) −→ QH∗(M ×M , ω ⊕ ω) the inclusion given by Künneth formula and the compatibility of the Novikov rings with the cartesian product (see §1 for definitions). Let (M,ω) and (M , ω) be strongly semi-positive symplectic manifolds and let φ ∈ π̃1(Ham(M,ω)) and φ ∈ π̃1(Ham(M , ω)). When (M ×M , ω⊕ω) is strongly semi-positive (this is not necessarily the case, see discussion in Remark 8), one can, on one hand, compute the images of φ and φ via the respective Seidel’s morphisms and then see the result as an element in QH∗(M ×M , ω ⊕ ω) via κQ. On the other hand, one can compute the image of i(φ, φ), via the Seidel morphism of the product. 2000 Mathematics Subject Classification. Primary 57R17; Secondary 57R58 57S05.