# “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.
Thanks in Advance.

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Yes, but how do we define this action? Is this just the induced map on homology, or is the action defined on the intersection form itself?. Thanks. – gary May 9 '11 at 2:03
It's the induced map on homology. – Ryan Budney May 9 '11 at 2:41
@gary: please do not use answers to make comments. – Qiaochu Yuan May 9 '11 at 3:02
Qiaochu: Sorry, I tried to log-on, but could not, and the comments were disabled. I am also having trouble with assigning points, which also seems to be disabled, unless I am missing something obvious. – gary May 9 '11 at 23:27

## 1 Answer

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 $Sp(2g,\mathbb{Z})$.

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