Smooth 4-dimensional Poincare conjecture
The original Poincare conjecture was the assertion that a simply-connected compact boundaryless -manifold is diffeomorphic (smooth Poincare conjecture) or homeomorphic (topological Poincare conjecture) to . Because of Poincare duality, this is equivalent to the assertion that a -manifold has the homotopy-type of then it is diffeomorphic/homeomorphic to . This gave birth to the generalized Poincare conjecture -- that an -manifold with the homotopy type of is diffeomorphic or homeomorphic to .
By the work of Smale and Stallings, the topological Poincare conjecture was shown to be true provided . But for Milnor and Kervaire showed that admits non-standard smooth structures so the smooth Poincare conjecture is false in general.
The generalized Poincare conjecture is an undergraduate-level point-set topology problem for .
The case was proven by Poincare.
The case was recently proven by Perelman.
The case is the only outstanding case. Mike Freedman has proven that a -manifold which is homotopy-equivalent to is homeomorphic to , so the smooth 4-dimensional Poincare conjecture is the only outstanding problem among the generalized Poincare conjectures. Moreover, it can be considered to be reduced to the question of if has an exotic smooth structure.
Technically, Poincare never asserted this conjecture. He only stated it was an interesting problem. So perhaps it should be called Poincare's Egregious Problem.
It is unknown whether or not admits an exotic smooth structure. If not, the smooth -dimensional Poincare conjecture would have an affirmative answer. Similarly, it's known that -dimensional euclidean space admits a continuum of pairwise non-diffeomorphic smooth structures. But it's unknown whether or not any of these exotic smooth structures extend to , thought of as a compactification of .
[FQ] Freedman, M. Quinn, F. Topology of 4-manifolds. Princeton University Press.
[S] Smale, S., "On the structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399
[MT] Morgan, John W.; Gang Tian. Ricci Flow and the Poincaré Conjecture. AMS/CMI (2009)
* indicates original appearance(s) of problem.