Product and coproduct on fixed point Floer homology of positive Dehn twists

Yao, Yuan; Zhao, Ziwen

doi:10.2140/agt.2025.25.3875

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ISSN (electronic): 1472-2739

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Product and coproduct on fixed point Floer homology of positive Dehn twists

Yuan Yao and Ziwen Zhao

Algebraic & Geometric Topology 25 (2025) 3875–3919

DOI: 10.2140/agt.2025.25.3875

Abstract

We compute the product and coproduct structures on the fixed point Floer homology of iterations on a single positive 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.

Keywords

Dehn twist, fixed point Floer homology, product, coproduct

Mathematical Subject Classification

Primary: 53D40, 57R58

References

Publication

Received: 14 October 2022

Revised: 22 May 2023

Accepted: 27 August 2024

Published: 29 October 2025

Authors

Yuan Yao
Nantes Université Nantes France

Ziwen Zhao
University of California, Los Angeles Los Angeles, CA United States

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