Geometric Composition in Quilted Floer Theory

Given two symplectic manifolds (M1, ω1), (M2, ω2) a Lagrangian correspondence is a Lagrangian submanifold L ⊂ (M1 ×M2,−ω1 ⊕ ω2). These are the central objects of the theory of holomorphic quilts as developed by Wehrheim and Woodward in [21]. Consider two Lagrangian correspondences Li ⊂ (Mi−1 ×Mi,−ωi−1 ⊕ ωi) for i = 1, 2. Let ∆ = {(x, y, z, t) ∈ M0 ×M1 ×M1 ×M2 | y = z} If L1 × L2 is transverse to ∆, we may form the fibre product L1 ×M1 L2 ⊂ M0 ×M1 ×M1 ×M2 by intersecting ∆ with L1 × L2 . If the projection L1 ×M1 L2 → M0 ×M2 is an embedding, we say that L1 and L2 are composable and L1 ◦ L2 is naturally a Lagrangian submanifold of M0 ×M2 and is called the (geometric) composition of L1 and L2 . As a point set one has