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 1 only twists about (certain) separating curves and bounding pairs are needed, while for genus g D 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. 57S05; 20F05, 57M07, 57N05

Crosscap slides and the level 2 mapping class group of a nonorientable surface

“…The group I( ) defined above is the same as I( , {{δ 1 , δ 2 }}) in the notation of [16] and it agrees with the definition of the Torelli group of a surface with boundary given by Johnson [6]. Note that I( ) is not the kernel of the action of M( ) on H 1 ( ; Z).…”

“…Consider an embedding i : → S such that S\i( ) is an annulus with core β g . Such embedding is called capping in [16]. Since every homeomorphism of i( ) equal to the identity on i(∂ ) can be extended by the identity on S\i( ) to a homeomorphism of S, we have an induced homomorphism i * : M( ) → M(S).…”

“…For example, T δ i for i = 1, 2 act trivially on the homology group, but they are not elements of I( ). Nevertheless, I( ) can also be defined intrinsically, without reference to S, by using relative homology, see [6,16]. Note that γ is a bounding curve in if and only if i(γ ) is separating in S, and (γ , γ ) is a bounding pair in if and only if (i(γ ), i(γ )) is a bounding pair in S.…”

An Infinite Presentation of the Torelli Group

“…Like in the closed surface case, the group I g,1 is defined to be the subgroup of Mod g,1 consisting of mapping classes that act trivially on H 1 (Σ g,1 ). For surfaces with more than 1 boundary component, there is more than one useful definition for the Torelli group (see [Pu1] for a discussion). We discuss one special definition in section 3.2.1.…”

“…• The fact that I g is generated by separating twists and bounding pair maps follows from work of Birman and Powell ([Bi2,Po]; see also [Pu1] for a different proof, as well as generalizations) • Warning: Traditionally, the curves in a bounding pair are required to be nonseparating; however, to simplify our statements we allow them to be separating. • Commutators of simply intersecting pairs are not needed to generate I g , but their presence greatly simplifies our relations.…”