In 2006, Labourie defined a map from a bundle over Teichmuller space to
the Hitchin component of the representation variety $Rep(\pi_1(S),PSL(n,R))$, and conjectured that it is a homeomorphism for every $n$ (it
was known for $n =2,3$). I will describe some of the background to the
Labourie conjecture, and then show that it does not hold for any $n >3$.
Having shown that Labourie’s map is more interesting than a mere
homeomorphism, I will describe some new questions and conjectures about
how it might look.