In the early 20th Century, Henri Poincare
conjectured that the three-dimensional sphere
is the only compact
which is simply connected
This is now called the Poincare Conjecture.
This conjecture is the dimension three case of the Generalized Poincare Conjecture, which makes a similar assertion for manifolds in an arbitrary dimension. Amazingly, the dimension three case (which the original case conjectured by Poincare) is the only one which has not yet been solved.
This conjecture is one of the seven Millenium Prize Problems proposed by the Clay Mathematics Institute.