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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.
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Let $S_g$ be a closed orientable surface of genus $g geq 2$ and $C$ a simple closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist about $C$. A textit{fractional power} of $t_C$ of textit{exponent} $fraction{ell}{n}$ is an $h in Mod(S_g)$ such that $h^n = t_C^{ell}$. Unlike a root of a $t_C$, a fractional power $h$ can exchange the sides of $C$. We derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We show in the side-preserving case that if $gcd(ell,n) = 1$, then $h$ will be isotopic to the $ell^{th}$ power of an $n^{th}$ root of $t_C$ and that $n leq 2g+1$. In general, we show that $n leq 4g$, and that side-preserving fractional powers of exponents $fraction{2g}{2g+2}$ and $fraction{2g}{4g}$ always exist. For a side-exchanging fractional power of exponent $fraction{ell}{2n}$, we show that $2n geq 2g+2$, and that side-exchanging fractional powers of exponent $fraction{2g+2}{4g+2}$ and $fraction{4g+1}{4g+2}$ always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on $S_5$.
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.
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.
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}.
This article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.
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