7 dimensions

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Maybe I'll post something to explain what is going on:

A homeomorphism between two spaces is a bijection such that it is continuous and it's inverse is continuous to. For example, ]0,1[ and ]0,2[ are homeomorphic because there is a bijection ([tex]f:]0,1[\rightarrow ]0,2[:x\rightarrow 2x[/tex]) which is evidently continuous, and also it's inverse [tex]]0,2[\rightarrow ]0,1[[/tex] is continuous.

Now, a somewhat stronger property is that of a diffeomorphism. While a homeomorphism can be a very rough function, diffeomorphisms are very smooth and nice. Formally, a diffeomorphism is a bijection which is smooth (i.e. all derivatives exist) and whose inverse is smooth too.

Now, what are exotic spheres? Well, an exotic sphere is a space that is homeomorphic to the sphere, but not diffeomorphic. So there exists a bijection which is continuous in both ways, but there is no bijection that is smooth in both ways.
The existence of exotic spheres came as a real surprise to many differential geometers, because it shows that there are spaces which look alike the sphere, but which are no spheres.

The fun (and annoying) thing is that there is no way for us to visualize the exotic spheres. Indeed, there are no exotic spheres in three dimensions (thus any three-dimensional space which is homeomorphic to the sphere is also diffeomorphic). You'll need to go to higher dimensions to visualize this. This makes it quite clear that exotic spheres are very difficult things to handle. And this is indeed true: exotic spheres are (to me) a quite difficult matter...
 

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