Editing Mapping class group (section)

== Torelli group ==
Notice that there is an induced action of the mapping class group on the [[homology (mathematics)|homology]] (and [[cohomology]]) of the space ''X''.  This is because (co)homology is functorial and Homeo<sub>0</sub> acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy).  The kernel of this action is the ''Torelli group'', named after the [[Torelli theorem]].

In the case of orientable surfaces, this is the action on first cohomology ''H''<sup>1</sup>(Σ) ≅ '''Z'''<sup>2''g''</sup>. Orientation-preserving maps are precisely those that act trivially on top cohomology ''H''<sup>2</sup>(Σ) ≅ '''Z'''. ''H''<sup>1</sup>(Σ) has a [[Symplectic geometry|symplectic]] structure, coming from the [[cup product]]; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are realized, yielding the [[short exact sequence]]:
:<math>1 \to \operatorname{Tor}(\Sigma) \to \operatorname{MCG}(\Sigma) \to \operatorname{Sp}(H^1(\Sigma)) \cong \operatorname{Sp}_{2g}(\mathbf{Z}) \to 1</math>

One can extend this to
:<math>1 \to \operatorname{Tor}(\Sigma) \to \operatorname{MCG}^*(\Sigma) \to \operatorname{Sp}^{\pm}(H^1(\Sigma)) \cong \operatorname{Sp}^{\pm}_{2g}(\mathbf{Z}) \to 1</math>

The [[symplectic group]] is well understood.  Hence understanding the algebraic structure of the mapping class group often reduces to questions about the Torelli group.

Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the Torelli group vanishes.