Differential structure
A differential structure describes important properties of a manifold which lying between topology and geometry. A manifold is described by charts <math>\varphi_{i}</math>: homeomorphic maps from the manifold <math>M</math> in the linear space <math>\mathbb{R}^{n}</math>
<math>\varphi_{i}:W_{i}\subset M\rightarrow U_{i}\subset\mathbb{R}^{n}.</math>
This charts describes the local properties of the manifold captured
by linear spaces. But the really interesting property is the structure
between this charts. Suppose two charts <math>\varphi_{i}:W_{i}\rightarrow U_{i}</math>
and <math>\varphi_{j}:W_{j}\rightarrow U_{j}</math>. The overlapping origin
<math>W_{ij}=W_{i}\cap W_{j}</math> will be maped in two (probably different)
images <math>U_{ij}=\varphi_{i}\left(W_{ij}\right)</math> and <math>U_{ji}=\varphi_{j}\left(W_{ij}\right)</math>.
A coordinate transformation between two charts is a map between
subsets of linear spaces:
<math>
\varphi_{ij}:U_{ij}\rightarrow U_{ji},\,\,\varphi_{ij}(x)=\varphi_{j}\left(\varphi_{i}^{-1}\left(x\right)\right).
</math>
Two charts <math>\varphi_{i},\,\varphi_{j}</math> are compatible if
<math>U_{ij},\, U_{ji}</math> are open (maybe empty), and the coordinate transformations
<math>\varphi_{ij},\,\varphi_{ji}</math> (with <math>W_{i}\cap W_{j}\neq\emptyset</math>)
are diffeomorphisms. A family of compatible charts covering
the whole manifold is an atlas and two atlases are equivalent,
if all their charts are compatible. The equivalence classes
of atlases are the diffential structures of the manifold.
There is only one differential structure of any manifold of dimension smaller than four.
For all manifolds larger than four dimensions there is only a finite
number of possible differential structures. The following
table listed the numbers of differential structures up to dimension 11.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 1 | 1 | 1 | <math>\infty</math> | 1 | 1 | 28 | 2 | 8 | 6 | 992 |
In dimension four there is a countable number of differential
structures on most compact four-manifolds and an uncountable number
for most non-compact four-manifolds.