Torelli Group

Description

There is a natural action of the Mapping class group on the Homology of a surface. This gives us a natural Representation of the group. The kernel of this representation is called the Torelli Group. If you understand the MCG as a decomposition into a linear and a non-linear part, the Torelli group is the complicated part.

Definition

Let $S$ be a surface and $MCG(S)$ its Mapping class group. Homeomorphisms of $S$ act on curves and therefore on $1$-chains of the surface. This gives us a well-defined action of the Mapping class group on the Singular homology of $S$: $MCG(S)\to Aut(H_1(\Sigma ,\mathbb{Z}))$ The kernel of this map is a group called the Torelli group $\mathcal{I}(\Sigma )$

Properties

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Examples

Example