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
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
only
twists about (certain) separating curves and bounding pairs are needed, while for
genus
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.
Keywords
Torelli group, mapping class group, Birman exact sequence,
curve complex
