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There is a standard map $T: Mod(S) \rightarrow Sp(2g, \mathbb{Z})$ from the mapping class group of a surface to the symplectic group (where a mapping class goes to its action on the first homology group of $S.$) The question is: is it reasonable to refer to this map $T$ as The Torelli Map?

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I've never seen it referred to like this. As evidence against it being reasonable, it would seem weird to name a map after its kernel. As evidence for it being reasonable, the associated map from the moduli space of curves to the moduli space of ppav's is injective by Torelli's theorem. However, in the end I think it's not a very good name. I always just call it the "classical symplectic representation" or the "homology representation". –  Andy Putman Aug 23 '12 at 20:26
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Igor, please, do not call this "Torelli map", maybe "Torelli homomorphism", since the actual Torelli map $M_g\to A_g$ induces the above homomorphism. –  Misha Aug 23 '12 at 20:36
    
@Andy: my belief is that the "Torelli group" is actually named after the "Torelli map", which comes from the "Torelli theorem", but I could be totally wrong. –  Igor Rivin Aug 23 '12 at 21:18
    
@Andy: but as Misha says, Torelli homomorphism may be better (and @Misha's comment largely supports my theory...) –  Igor Rivin Aug 23 '12 at 21:44
    
What I hear folks say when referring to this map is the "action on homology", it being understood that the action preserves intersection product. –  Lee Mosher Aug 23 '12 at 22:52
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