Cutting and pasting in the Torelli group - [pdf]
Geom. Topol. 11 (2007) 829-865.

Abstract :
We introduce machinery to allow ``cut-and-paste''-style inductive arguments in the Torelli subgroups of the mapping class groups. In the past these arguments have been problematic because the restriction of the Torelli groups 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 groups on the complex of curves to find generators for the Torelli groups. For genus $g \geq 1$ 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-Powell which says that the Torelli groups on closed surfaces are generated by twists about separating curves and bounding pairs.