Volume 21, issue 1 (2021)

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On cobordism maps on periodic Floer homology

Guanheng Chen

Algebraic & Geometric Topology 21 (2021) 1–103
Abstract

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 curves, provided that the symplectic cobordisms are Lefschetz fibrations satisfying certain nice conditions. Under certain additional monotonicity assumptions, we show that these two definitions are equivalent.

Keywords
periodic Floer homology, Seiberg–Witten Floer homology, cobordism maps
Primary: 57R58
Secondary: 53D40