Dehn twists exact sequences through Lagrangian cobordism

TL;DR

This paper develops new Lagrangian cobordism techniques to derive exact sequences related to Dehn twists, connecting symplectic topology, Fukaya categories, and mirror symmetry.

Contribution

It introduces novel constructions in Lagrangian surgery and cobordism theory, extending existing frameworks to immersed categories and product manifolds.

Findings

Manifestation of Seidel's exact sequences as consequences of surgery and cobordism

Expression of Dehn twist autoequivalences matching mirror symmetry predictions

Proof of split generation of symplectomorphisms by Dehn twists in $ADE$-type Milnor fibers

Abstract

In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism theory to the immersed category. As a result, we manifest Seidel's exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodward's family Dehn twist sequence (including the codimension-1 case missing in the literature) as consequences of our surgery/cobordism constructions. Moreover, we obtain an expression of the autoequivalence of Fukaya category induced by Dehn twists along Lagrangian $\mathbb{RP}^n$, $\mathbb{CP}^n$ and $\mathbb{HP}^n$, which matches Huybrechts-Thomas's mirror prediction of the $\mathbb{CP}^n$ case modulo connecting maps. We also prove the split generation of any symplectomorphism by Dehn twists in $ADE$-type Milnor fibers.