Here $T_g$ denotes the g-fold torus, and $T_i\times T_j$ denotes the cartesian product. My major trouble is that I can't use Kunneth's Theorem. Thanks for helping!
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Maybe try induction and the long exact sequence for a pair. (There are many pairs you could choose; some slicker than others.) – Dylan Wilson Nov 29 '12 at 15:58