Perelman's theorem states
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
It is stated in this video at the 2 minute mark that proving this in dimensions other than 3 was much easier. Dimensions $1$ and $2$ are relatively easy, dimension $6$ was proven in 1962, and dimensions $5$ and $\geq7$ were proven in 1961. Michael Freedman won the Fields medal in 1982 for proving the dimension $4$ case. This left dimension $3$ which as we know was solved by Grigori Perelman in 2003. Can anyone give an explanation that I, a third year university student who has taken and understood a course in differential/Riemannian geometry and topology on manifolds, of why dimensions $3$ and $4$ were so much harder to prove than all the others?