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How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an epimorphism? My idea was to work with generators, but I was not able to prove it this way.

I would love to get detailed answers in order to understand this better.

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This is well-known material, so insted of a "detailed answer" let me give you a standard reference. See

B. Farb and D. Margalit: A primer on mapping class groups, Theorem 6.4.

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I agree with Francesco that for a detailed answer you should read Farb/Margalit, but for an executive summary:

The mapping class group is generated by Dehn twists, the symplectic group by transvections (elementary matrices). Dehn twists get mapped to transvections by the Torelli map (your $\phi.$)

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