Cutting and pasting in the Torelli group

Abstract

We introduce machinery to allow “cut-and-paste”-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives different groups depending on how the subsurfaces are embedded. We define a category $TSur$ whose objects are surfaces together with a decoration restricting how they can be embedded into larger surfaces and whose morphisms are embeddings which respect the decoration. There is a natural “Torelli functor” on this category which extends the usual definition of the Torelli group on a closed surface. Additionally, we prove an analogue of the Birman exact sequence for the Torelli groups of surfaces with boundary and use the action of the Torelli group on the complex of curves to find generators for the Torelli group. For genus $g\ge 1$ only twists about (certain) separating curves and bounding pairs are needed, while for genus $g=0$ a new type of generator (a “commutator of a simply intersecting pair”) is needed. As a special case, our methods provide a new, more conceptual proof of the classical result of Birman and Powell which says that the Torelli group on a closed surface is generated by twists about separating curves and bounding pairs.