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Let $\Sigma$ be a compact oriented surface. The Dehn twist along every simpleclosed curve $\gamma \subset \Sigma$ induces an automorphism of the fundamentalgroup $\pi$ of $\Sigma$. There are two possible ways to generalize suchautomorphisms if the curve $\gamma$ is allowed to have self-intersections. Oneway is to consider the `generalized Dehn twist' along $\gamma$: an automorphismof the Malcev completion of $\pi$ whose definition involves intersectionoperations 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 $L$ soas to get a homology cylinder $U_L$, and let $U_L$ act on every nilpotentquotient $\pi/\Gamma_{j} \pi$ of $\pi$ (where $\Gamma_j\pi$ denotes thesubgroup 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 weexplicitly compute this automorphism in terms of $[\gamma]$ modulo${\Gamma_{k+2}}\pi$. As applications, we obtain new formulas for certainevaluations of the Johnson homomorphisms showing, in particular, how to realizeany element of their targets by some explicit homology cylinders and/orgeneralized Dehn twists.

Generalized Dehn twists on surfaces and homology cylinders