Generalized Poincaré conjecture

for topological manifolds: this is a generalization of the Poincaré conjecture.

Original problem posed, $n=3$ for topological manifolds.

Millennium Prize Problems.

Last to be proven, only the 4-differential manifold case missing as of 2013.

Even the truth for all $n>4$ was proven in the 60's!

Why is low dimension harder than high dimension?? Surprise!

AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.

For dimension two, we know there are infinitely many: classification of closed surfaces

for differential manifolds:

Not true in general. First counter example is $n=7$. Surprise: what is special about the number 7!?

Counter examples are called exotic spheres.

Totally unpredictable count table:

$n=4$ is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??