Generalized Dehn twists in low-dimensional topology

Y. Kuno, G. Massuyeau, Shunsuke Tsuji
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引用次数: 1

Abstract

The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description in terms of decorated trees and the Hopf-algebraic framework underlying their construction. Going to the dimension three, we also overview the relation between generalized Dehn twists and $3$-dimensional homology cobordisms, and we survey the variants of generalized Dehn twists for skein algebras of the surface.
低维拓扑中的广义Dehn扭转
有向曲面上沿封闭曲线的广义Dehn扭转是一个涉及曲面上环的交点的代数构造。它被定义为曲面基群Malcev补全的自同构。顾名思义,对于曲线没有自交的情况,它是由通常沿曲线的Dehn扭转引起的。在这篇解释性的文章中,在解释了广义Dehn扭曲的定义之后,我们回顾了一些关于广义Dehn扭曲的结果,如它们作为表面的微分同态的可实现性、它们用装饰树表示的图解描述以及它们构造背后的hopf -代数框架。在三维空间中,我们还概述了广义Dehn扭转与$3$维同调配合的关系,并研究了曲面上串代数的广义Dehn扭转的变体。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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