arXiv 1902.025922019-02math.GTmath.AT

曲面和同调柱面上的广义Dehn扭曲

Generalized Dehn twists on surfaces and homology cylinders

Yusuke Kuno, Gwenael Massuyeau

备注：: 45 pages; minor modifications with respect to the first version
发表信息：: Algebr. Geom. Topol. 21 (2021) 697-754
DOI：: 10.2140/agt.2021.21.697

摘要：

设$Σ$是一个紧致定向曲面。沿着每一条简单闭曲线$γ\subset Σ$的Dehn扭曲导致$Σ$的基本群$π$的自同构。如果允许曲线$γ$有自相交，有两种可能的方法来推广这样的自同构。一种方法是考虑沿$γ$的“广义Dehn扭曲”：$π$的Malcev完备的自同构，其定义涉及交集运算，并且仅依赖于$γ$的同伦类$[γ]\in π$。另一种方法是在通常的圆柱体$U:=Σ\times [-1,+1]$中选择一个投影到$γ$上的结$L$，沿着$L$进行一个外科手术，以得到一个同调圆柱体$U_L$，并让$U_L$作用于$π$的每个幂零商$π/Γ_{j} π$（其中$Γ_jπ$表示由长度为$j$的商生成的$π$的子群）。在本文中，假设$[γ]$是在$Γ_k π$的一些$k\geq 2$，我们证明了（无论$L$的选择是）的自同构的$π/Γ_{2k+1} π$诱导$U_L$同意的广义Dehn扭曲沿$γ$和我们明确地计算这个自同构的$[γ]$模${Γ_{k+2}}π$。作为应用，我们获得新的公式，某些评价的约翰逊同态显示，特别是，如何实现任何元素的目标，由一些明确的同源缸和/或广义德恩扭曲。

英文摘要：

Let $Σ$ be a compact oriented surface. The Dehn twist along every simple closed curve $γ\subset Σ$ induces an automorphism of the fundamental group $π$ of $Σ$. There are two possible ways to generalize such automorphisms if the curve $γ$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $γ$: an automorphism of the Malcev completion of $π$ whose definition involves intersection operations and only depends on the homotopy class $[γ]\in π$ of $γ$. Another way is to choose in the usual cylinder $U:=Σ\times [-1,+1]$ a knot $L$ projecting onto $γ$, to perform a surgery along $L$ so as to get a homology cylinder $U_L$, and let $U_L$ act on every nilpotent quotient $π/Γ_{j} π$ of $π$ (where $Γ_jπ$ denotes the subgroup of $π$ generated by commutators of length $j$). In this paper, assuming that $[γ]$ is in $Γ_k π$ for some $k\geq 2$, we prove that (whatever the choice of $L$ is) the automorphism of $π/Γ_{2k+1} π$ induced by $U_L$ agrees with the generalized Dehn twist along $γ$ and we explicitly compute this automorphism in terms of $[γ]$ modulo ${Γ_{k+2}}π$. As applications, we obtain new formulas for certain evaluations of the Johnson homomorphisms showing, in particular, how to realize any element of their targets by some explicit homology cylinders and/or generalized Dehn twists.

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