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 $ϕ: MCG(S_g) \to GL(n,C)$ is a homomorphism, then $ϕ$ 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 $ϕ: MCG(S_g) \to Diff(S^2)$ is trivial and that if $g \ge 3$, then every homomorphism $ϕ: MCG(S_g) \to Homeo(T^2)$ is trivial.