“Natural Surjection” between Mapping Class Group for Sg (genus-g surface) and Symplectic 2gx2g-matrices?

Everyone: I am reading a paper that makes reference to this "natural surjection" between Mg ; the mapping-lass group of the genus-g surface, and Sp(2g,Z), the group of 2gx2g -symplectic matrices over Z. Does anyone know what this surjection is? . I know that the kernel is the Torelli group ( and, of course, that the kernel do chicken right!), and I have been trying to reconstruct the map this way (at least up to some equivalence with the original/intended map), using the first theorem on isomorphisms. ( I understand the Torelli group as being either the subgroup that preserves the intersection form, or the subgroup that stabilizes H_1(Sg,Z) ; H_1 is the first homology of Sg, the genus-g surface.). I would appreciate your suggestions, references, etc.
As you wrote, the mapping class group acts on $H_1(S_g,\mathbb{Z})\cong\mathbb{Z}^{2g}$. The action has to preserve the intersection pairing in $H_1$, i.e. the symplectic form, that's why we get a morphism to $Sp(2g,\mathbb{Z})$.