  Suppose $C_1,\ldots , C_k$ are generic conjugacy classes in $SL_2({\mathbb
C})$. Consider the character variety of local systems on ${\mathbb P}^1-\{
y_1,\ldots , y_k\}$ whose monodromy transformations around the punctures $y_i$
are in the respective conjugacy classes $C_i$. We show that the dual boundary
complex of this character variety is homotopy equivalent to a sphere of
dimension $2(k-3)-1$.
