Generalized Dehn twists on surfaces and homology cylinders

Abstract

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 $\pi$ 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 Malcev completion of $\pi$ whose definition involves intersection operations 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 $L$ 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 $k\ge 2$, 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 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.

45 pages; minor modifications with respect to the first version

References in corpus (3)

• Cohomological Aspects of Magnus Expansions
• A functorial LMO invariant for Lagrangian cobordisms
• Intersections of curves on surfaces and their applications to mapping class groups

Cited by in corpus (1)

• The total Johnson homomorphism on the homology cylinder and the bracket-quantization HOMFLY-PT skein algebra