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

TL;DR

This paper computes the algebraic structures of product and coproduct on fixed point Floer homology for positive Dehn twists, linking them to Morse and symplectic homology, using holomorphic section enumeration.

Contribution

It provides explicit calculations of Floer homology structures for Dehn twists, connecting them to Morse and symplectic homology, with novel enumeration and gluing techniques.

Findings

Product and coproduct structures are determined by Morse homology of the twist complement.

Holomorphic sections are enumerated using energy inequalities and gluing methods.

Certain holomorphic sections are shown not to exist, refining the structure calculations.

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.