Product and Coproduct on Fixed Point Floer Homology of Positive Dehn Twists

TL;DR

The paper determines the product and coproduct structures on fixed-point Floer homology for iterations of a single positive Dehn twist on a surface, expressing the algebraic structures in terms of Morse homology on the twist complement Σ_0 and sectors from the symplectic homology of T^*S^1. The authors implement a two-region (twist and non-twist) decomposition, establish a local energy inequality, and apply Morse–Bott techniques to count relevant holomorphic sections, ultimately obtaining explicit formulas for both product and coproduct and showing how these structures reflect the underlying geometry of Σ_0 and the Dehn twist region. They also discuss extensions to multiple, disjoint twists and relate the coproduct to dualities with negative twists. The results provide a precise algebraic description of the ⊕_{n>0}HF_*(φ^n) structure and offer a framework for PFH extensions in similar settings, with potential applications to more general symplectomorphisms and cobordism maps.

Abstract

We compute the product and coproduct structures on the fixed point Floer homology of iterations on the single Dehn twist, subject to some mild topological restrictions. We show that the resulting product and coproduct structures are determined by the product and coproduct on Morse homology of the complement of the twist region, together with certain sectors of product and coproduct structures on the symplectic homology of $T^*S^1$. The computation is done via a direct enumeration of $J-$holomorphic sections: we use a local energy inequality to show that some of the putative holomorphic sections do not exist, and we use a gluing construction plus some Morse-Bott theory to construct the sections we could not rule out.

Theorems & Definitions (67)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Definition 2.1
• Definition 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['local energy lemma']}