Generalized Dehn twists on surfaces and homology cylinders

Kuno, Yusuke; Massuyeau, Gwénaël

doi:10.2140/agt.2021.21.697

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Generalized Dehn twists on surfaces and homology cylinders

Yusuke Kuno and Gwénaël Massuyeau

Algebraic & Geometric Topology 21 (2021) 697–754

DOI: 10.2140/agt.2021.21.697

Abstract

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 Maltsev 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 $Γ_{j} π$ 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 $π ∕ Γ_{2 k + 1} π$ 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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Keywords

Dehn twist, homology cylinder, Dehn–Nielsen representation, Johnson homomorphism

Mathematical Subject Classification 2010

Primary: 20F34, 57M27, 57N10

Secondary: 20F14, 20F28, 20F38

References

Publication

Received: 15 July 2019

Revised: 10 May 2020

Accepted: 15 June 2020

Published: 25 April 2021

Authors

Yusuke Kuno
Department of Mathematics Tsuda University Tokyo Japan

Gwénaël Massuyeau
IMB Université Bourgogne Franche-Comté & CNRS Dijon France

Volume 21, issue 2 (2021)