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Cutting and pasting in the Torelli group
Andrew Putman
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Geometry & Topology 11 (2007)
829–865
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Abstract
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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=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.
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Keywords
Torelli group, mapping class group,
Birman exact sequence, curve complex
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Mathematical Subject Classification
Primary: 57S05
Secondary: 20F05, 57M07, 57N05
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Publication
Received: 25 August 2006
Accepted: 10 April 2007
Published: 10 May 2007
Proposed: Joan Birman
Seconded: Walter Neumann, Martin Bridson
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