On cobordism maps on periodic Floer homology

TL;DR

This paper explores the definition and properties of cobordism maps on periodic Floer homology, establishing their relation to Seiberg Witten theory and holomorphic curves under specific conditions.

Contribution

It introduces two equivalent definitions of cobordism maps on PFH, connecting Seiberg Witten theory and holomorphic curve methods, under certain assumptions.

Findings

Cobordism maps on PFH are defined via Seiberg Witten theory.

An alternative holomorphic curve-based definition is provided.

The two definitions are shown to be equivalent under monotonicity conditions.

Abstract

In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the isomorphism between PFH and Seiberg Witten cohomology. Furthermore, we show that the maps satisfy the holomorphic curve axiom. In the second part of the paper, we give an alternative definition of these maps by using holomorphic curve method, provided that the symplectic cobordisms are Lefschetz fibration satisfying certain nice conditions. Under additionally certain monotonicity assumptions, we show that these two definitions are equivalent.