  For any complex affine reductive group G and a fixed choice of maximal
compact subgroup K, we show that the G-character variety of a free group
strongly deformation retracts to the corresponding K-character space, which is
a real semi-algebraic set. Combining this with constructive invariant theory
and classical topological methods, we show that the SL(3,C)-character variety
of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character
variety of a rank 3 free group is homotopic to a 6 sphere.
