THE GENERALIZED DEHN TWIST ALONG A FIGURE EIGHT

For any unoriented loop on a compact connected oriented surface with one boundary component, we introduce a generalized Dehn twist along the loop as a certain automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this corresponds to the right-handed Dehn twist, and in particular is realized as a diffeomorphism of the surface. We investigate the case where the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a diffeomorphism.