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Register a new userSquared Dehn twists and deformed symplectic invariants
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Authors
Kyler Siegel
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We establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed two-form or bulk deforming it by a half-dimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in cite{murphysiegel}.
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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 Seidels exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodwards 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-Thomass 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.
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We give detailed descriptions of gluing pseudoholomorphic maps in symplectic geometry, especially in the presence of an obstruction bundle. The main motivation is to try to compare the symplectic and enumerative invariants of algebraic manifolds. These descriptions can also be used to enumerate rational curves with high-order degeneracies of local nature in projective spaces.
A textit{multicurve} $C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{C}$ about $C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.
Let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g geq 1$. In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order $4g$ and $4g+2$, for $ggeq 2$, whose product is pseudo-Anosov.
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