Differentiable manifold
Classification

Every one-dimensional connected smooth manifold is diffeomorphic to either R {\displaystyle \mathbb {R} } or S 1, {\displaystyle S^{1},} each with their standard smooth structures.

For a classification of smooth 2-manifolds, see surface. A particular result is that every two-dimensional connected compact smooth manifold is diffeomorphic to one of the following: S 2, {\displaystyle S^{2},} or (S 1 × S 1) ♯ ⋯ ♯ (S 1 × S 1), {\displaystyle (S^{1}\times S^{1})\sharp \cdots \sharp (S^{1}\times S^{1}),} or R P 2 ♯ ⋯ ♯ R P 2. {\displaystyle \mathbb {RP} ^{2}\sharp \cdots \sharp \mathbb {RP} ^{2}.} The situation is more nontrivial if one considers complex-differentiable structure instead of smooth structure.

The situation in three dimensions is quite a bit more complicated, and known results are more indirect. A remarkable result, proved in 2002 by methods of partial differential equations, is the geometrization conjecture, stating loosely that any compact smooth 3-manifold can be split up into different parts, each of which admits Riemannian metrics which possess many symmetries. There are also various "recognition results" for geometrizable 3-manifolds, such as Mostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups.

The classification of n -manifolds for n greater than three is known to be impossible, even up to homotopy equivalence. Given any finitely presented group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to decide the isomorphism problem for finitely presented groups, there is no algorithm to decide whether two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is undecidable. In addition, since even recognizing the trivial group is undecidable, it is not even possible in general to decide whether a manifold has trivial fundamental group, i.e. is simply connected.

Simply connected 4-manifolds have been classified up to homeomorphism by Freedman using the intersection form and Kirby–Siebenmann invariant. Smooth 4-manifold theory is known to be much more complicated, as the exotic smooth structures on R demonstrate.

However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where the h-cobordism theorem can be used to reduce the classification to a classification up to homotopy equivalence, and surgery theory can be applied. This has been carried out to provide an explicit classification of simply connected 5-manifolds by Dennis Barden.
keywords:forbidden_words, length_constraints:number_paragraphs[5]