{tag=Classification (mathematics)}

There are two cases:
* (topological) manifolds
* differential manifolds

Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
* 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:

  || Dimension
  || Smooth types

  | 1
  | 1

  | 2
  | 1

  | 3
  | 1

  | 4
  | ?

  | 5
  | 1

  | 6
  | 1

  | 7
  | 28

  | 8
  | 2

  | 9
  | 8

  | 10
  | 6

  | 11
  | 992

  | 12
  | 1

  | 13
  | 3

  | 14
  | 2

  | 15
  | 16256

  | 16
  | 2

  | 17
  | 16

  | 18
  | 16

  | 19
  | 523264

  | 20
  | 24

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


Generalized Poincaré conjecture

{wiki=Generalized_Poincaré_conjecture}

The Generalized Poincaré Conjecture extends the classical Poincaré Conjecture, which is a statement about the topology of 3-dimensional manifolds. The original Poincaré Conjecture, proposed by Henri Poincaré in 1904, asserts that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere.


 Generalized Poincaré conjecture

ID: generalized-poincare-conjecture