Triviality of some representations of $MCG(S_g)$ in $GL(n,C),Diff(S^2)$ and $Homeo(T^2)$

We show the triviality of representations of the mapping class group of a genus $g$ surface in $GL(n,C),Diff(S^2)$ and $Homeo(T^2)$ when appropriate restrictions on the genus $g$ and the size of $n$ hold. For example, if $S_g$ is a surface of finite type and $\varphi :MCG(S_g)\to GL(n,C)$ is a homomorphism, then $\varphi$ is trivial provided the genus $g\ge 3$ and $n<2g$. We also show that if $S_g$ is a closed surface with genus $g\ge 7$, then every homomorphism $\varphi :MCG(S_g)\to Diff(S^2)$ is trivial and that if $g\ge 3$, then every homomorphism $\varphi :MCG(S_g)\to Homeo(T^2)$ is trivial.