Non-trivial polynomial isolated singularities

AbstractMilnor considered real polynomial maps f:Rm → Rk (m ⩾ k ⩾ 2) with 0 an isolated critical point, found an associated fiber bundle, and called the singularity trivial if the fiber F is diffeomorphic to a disk. He asked: for what (m, k) are there non-trivial examples? Using a result of Looijenga the authors answer this question for m−k ≠ 3, and show that the answer depends precisely on the Poincaré Conjecture if m−k = 3. For m−k ≠ 4, 5 they also show that triviality is equivalent to: f at 0 is locally topologically equivalent to the projection map Rm → Rk