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 of 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 Maltsev completion of pi whose definition involves intersection operations and only depends on the homotopy class [gamma] is an element of pi of gamma. Another way is to choose in the usual cylinder U := Sigma x [-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 >= 2, we prove that (whatever the choice of L is) the automorphism of pi/Gamma(k+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.