# Cobordism maps on periodic Floer homology induced by elementary Lefschetz fibrations

## TL;DR

This paper defines and computes cobordism maps on periodic Floer homology induced by elementary Lefschetz fibrations, providing a direct holomorphic curve approach to these invariants.

## Contribution

It introduces a new method to define PFH cobordism maps via holomorphic curves for elementary Lefschetz fibrations, moving beyond indirect Seiberg-Witten theory definitions.

## Key findings

- Defined PFH cobordism maps using holomorphic curves.
- Computed these maps for specific elementary Lefschetz fibrations.
- Established a direct geometric approach to PFH cobordism maps.

## 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.