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Article Dans Une Revue Algebraic and Geometric Topology Année : 2021

Generalized Dehn twists on surfaces and homology cylinders

Résumé

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 Malcev completion of $\pi$ whose definition involves intersection operations and only depends on the homotopy class $[\gamma]\in \pi$ of $\gamma$. Another way is to choose in the usual cylinder $U:=\Sigma \times [-1,+1]$ 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 $\pi/\Gamma_{j} \pi$ of $\pi$ (where $\Gamma_j\pi$ denotes the subgroup of $\pi$ generated by commutators of length $j$). In this paper, assuming that $[\gamma]$ is in $\Gamma_k \pi$ for some $k\geq 2$, we prove that (whatever the choice of $L$ is) the automorphism of $\pi/\Gamma_{2k+1} \pi$ induced by $U_L$ agrees with the generalized Dehn twist along $\gamma$ and we explicitly compute this automorphism in terms of $[\gamma]$ modulo ${\Gamma_{k+2}}\pi$. 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.

Dates et versions

hal-02011582 , version 1 (08-02-2019)

Identifiants

Citer

Yusuke Kuno, Gwénaël Massuyeau. Generalized Dehn twists on surfaces and homology cylinders. Algebraic and Geometric Topology, 2021, 21 (2), pp.697-754. ⟨10.2140/agt.2021.21.697⟩. ⟨hal-02011582⟩
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