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Generalized Dehn twists on surfaces and homology cylinders

Yusuke Kuno and Gwénaël Massuyeau

Algebraic & Geometric Topology 21 (2021) 697–754

Let Σ be a compact oriented surface. The Dehn twist along every simple closed curve γ 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] π of γ. Another way is to choose in the usual cylinder U := Σ × [1,+1] a knot L projecting onto γ, to perform a surgery along L so as to get a homology cylinder UL, and let UL act on every nilpotent quotient πΓjπ of π (where Γjπ denotes the subgroup of π generated by commutators of length j). In this paper, assuming that [γ] is in Γkπ for some k 2, we prove that (whatever the choice of L is) the automorphism of πΓ2k+1π induced by UL agrees with the generalized Dehn twist along γ and we explicitly compute this automorphism in terms of [γ] modulo Γk+2π. 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.

Dehn twist, homology cylinder, Dehn–Nielsen representation, Johnson homomorphism
Mathematical Subject Classification 2010
Primary: 20F34, 57M27, 57N10
Secondary: 20F14, 20F28, 20F38
Received: 15 July 2019
Revised: 10 May 2020
Accepted: 15 June 2020
Published: 25 April 2021
Yusuke Kuno
Department of Mathematics
Tsuda University
Gwénaël Massuyeau
Université Bourgogne Franche-Comté & CNRS