Generalized Dehn twists on surfaces and homology cylinders

Abstract

Let $\mathrm{\Sigma }$ be a compact oriented surface. The Dehn twist along every simple closed curve $\gamma \subset \mathrm{\Sigma }$ induces an automorphism of the fundamental group $\pi$ of $\mathrm{\Sigma }$. 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 $U:=\mathrm{\Sigma }\times \left[-1,+1\right]$ 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 $\pi ∕{\mathrm{\Gamma }}_j\pi$ of $\pi$ (where ${\mathrm{\Gamma }}_j\pi$ denotes the subgroup of $\pi$ generated by commutators of length $j$). In this paper, assuming that $\left[\gamma \right]$ is in ${\mathrm{\Gamma }}_k\pi$ for some $k\ge 2$, we prove that (whatever the choice of $L$ is) the automorphism of $\pi ∕{\mathrm{\Gamma }}_{2k+1}\pi$ induced by $U_L$ agrees with the generalized Dehn twist along $\gamma$ and we explicitly compute this automorphism in terms of $\left[\gamma \right]$ modulo ${\mathrm{\Gamma }}_{k+2}\pi$. 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.