#### Volume 21, issue 2 (2021)

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Generalized Dehn twists on surfaces and homology cylinders

### Yusuke Kuno and Gwénaël Massuyeau

Algebraic & Geometric Topology 21 (2021) 697–754
##### Abstract

Let $\Sigma$ be a compact oriented surface. The Dehn twist along every simple closed curve $\gamma \subset \Sigma$ induces an automorphism of the fundamental group $\pi$ of $\Sigma$. There are two possible ways to generalize such automorphisms if the curve $\gamma$ is allowed to have self-intersections. One way is to consider the “generalized Dehn twist” along $\gamma$: an automorphism of the Maltsev completion of $\pi$ whose definition involves intersection operations and only depends on the homotopy class $\left[\gamma \right]\in \pi$ of $\gamma$. Another way is to choose in the usual cylinder $U:=\Sigma ×\left[-1,+1\right]$ a knot $L$ projecting onto $\gamma$, to perform a surgery along $L$ so as to get a homology cylinder ${U}_{L}$, and let ${U}_{L}$ act on every nilpotent quotient $\pi ∕{\Gamma }_{\phantom{\rule{-0.17em}{0ex}}j}\pi$ of $\pi$ (where ${\Gamma }_{\phantom{\rule{-0.17em}{0ex}}j}\pi$ denotes the subgroup of $\pi$ generated by commutators of length $j$). In this paper, assuming that $\left[\gamma \right]$ is in ${\Gamma }_{\phantom{\rule{-0.17em}{0ex}}k}\pi$ for some $k\ge 2$, we prove that (whatever the choice of $L$ is) the automorphism of $\pi ∕{\Gamma }_{\phantom{\rule{-0.17em}{0ex}}2k+1}\pi$ induced by ${U}_{L}$ agrees with the generalized Dehn twist along $\gamma$ and we explicitly compute this automorphism in terms of $\left[\gamma \right]$ modulo ${\Gamma }_{\phantom{\rule{-0.17em}{0ex}}k+2}\pi$. 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.

##### Keywords
Dehn twist, homology cylinder, Dehn–Nielsen representation, Johnson homomorphism
##### Mathematical Subject Classification 2010
Primary: 20F34, 57M27, 57N10
Secondary: 20F14, 20F28, 20F38