Cobordism maps on periodic Floer homology induced by elementary Lefschetz fibrations

Abstract: Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps induced by a class of symplectic cobordisms constructed by P. Seidel, called elementary Lefschetz brations. We define the PFH cobordism maps induced by elementary Lefschetz fibrations in terms of holomorphic curves. Moreover, we compute these maps for some cases.