Classification of compact smooth manifolds of dimension 3. I know classification of compact dimension 2 smooth manifolds. They are diffeomorphic to a sphere with n "ears" (connected sum of n tori) or a sphere with m mobius strips (connected sum of m real projective planes).
I know only that geometric hypothesis proved by Perelman says something about 3 manifolds, but I can't find similar as above precise classification for compact smooth manifolds of dimension 2. Is there a similarsimple classification? IF Yes, could You leave a link to it or write it in comment?
 A: A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two summands is homeomorphic to a sphere.
In dimension two, the closed prime manifolds are $S^2$, $\mathbb{RP}^2$, and $S^1\times S^1$. By the classification of surfaces, every closed two-dimensional manifold is homeomorphic to a connected sum of prime manifolds. In the orientable case, the connected summands are unique up to $S^2$ summands (you can always connected sum with $S^2$ without changing anything). In the non-orientable case, we no longer have uniqueness as $(S^1\times S^1)\#\mathbb{RP}^2$ is homeomorphic to $\mathbb{RP}^2\#\mathbb{RP}^2\#\mathbb{RP}^2$. However, one can recover uniqueness (up to spherical summands) if one prohibits the use of $S^1\times S^1$ summands.
There is a similar story for closed three-manifolds. The prime decomposition theorem for three-manifolds states that every closed three-manifold is homeomorphic to a connected sum of prime manifolds. If the orientable case, the connected summands are unique up to $S^3$ summands. If $M$ is non-orientable, then uniqueness no longer holds, however one can recover uniqueness by prohibiting the use of $S^2\times S^1$ as one of the connected summands.
The key difference between dimensions two and three is that there are infinitely many prime three-manifolds.  In the orientable case, they fit into three categories:

*

*those manifolds covered by $S^3$,

*the manifold $S^2\times S^1$, and

*orientable aspherical manifolds.

These categories can also be characterised via the fundamental group: namely finite, infinite cyclic, and infinite non-cyclic respectively.
In the non-orientable case however, there are too many prime manifolds to admit a classification; see the answer to this question of mine.
In dimension four, we no longer have uniqueness, even in the orientable case. For example, $(S^2\times S^2)\#\overline{\mathbb{CP}^2}$ is homeomorphic to $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}\#\overline{\mathbb{CP}^2}$. Note the similarity to the fact that $(S^1\times S^1)\#\mathbb{RP}^2$ is homeomorphic to $\mathbb{RP}^2\#\mathbb{RP}^2\#\mathbb{RP}^2$.
