Generalized Dehn twists on surfaces and homology cylinders

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 $\pi$ $\Sigma$. There are two possible ways to generalize such automorphisms if $\gamma$ is allowed have self-intersections. One way consider `generalized twist' $\gamma$: Malcev completion whose definition involves intersection operations and only depends on homotopy class $[\gamma]\in \pi$ $\gamma$. Another choose in usual cylinder $U:=\Sigma \times [-1,+1]$ knot $L$ projecting onto $\gamma$, perform surgery so as get homology $U_L$, let $U_L$ act nilpotent quotient $\pi/\Gamma_{j} (where $\Gamma_j\pi$ denotes subgroup generated by commutators length $j$). In this paper, assuming that $[\gamma]$ $\Gamma_k for some $k\geq 2$, we prove (whatever choice is) $\pi/\Gamma_{2k+1} induced agrees with generalized explicitly compute terms modulo ${\Gamma_{k+2}}\pi$. As applications, obtain new formulas certain evaluations Johnson homomorphisms showing, particular, how realize any element their targets explicit cylinders and/or twists.