Smale proved the cases for n greater or equal to seven in 1961, and Michael Freedman won the 1986 Fields Medal for proving the case for n=4.
Volume 2 of Whitehead's Mathematical Works contains an incorrect proof and his namesake link to this problem.  Rubinstein's theorem attempts to explain it by assuming the shape of the original problem is algorithmically decidable.  Smale proved the cases for n greater or equal to seven in 1961, and Michael Freedman won the 1986 Fields Medal for proving the case for n=4. FTP, what is this conjecture that every simply connected closed n-manifold is homeomorphic to an n-sphere?
Volume 2 of Whitehead's Mathematical Works contains an incorrect proof and his namesake link to this problem.  
Poincare Conjecture