  Let G be a compact Lie group or a complex reductive affine algebraic group.
We explore induced mappings between G-character varieties of surface groups by
mappings between corresponding surfaces. It is shown that these mappings are
generally Poisson. We also given an effective algorithm to compute the Poisson
bi-vectors when G=SL(2,C). We demonstrate this algorithm by explicitly
calculating the Poisson bi-vector for the 5-holed sphere, the first example for
an Euler characteristic -3 surface.
