Editing Talk:Exotic sphere (section)

== Truly bad writing ==

In the section "Introduction" the second paragraph begins as follows:

"''In differential topology, the relevant notion of sameness is witnessed by a diffeomorphism, which is a homeomorphism with the additional condition that it is smooth, that is, it should have derivatives of all orders everywhere.''"

But nobody has the slightest idea what the words "''relevant''" or "''sameness''" or "''witnessed''" mean here.

This is as incomprehensible to most readers as any writing I have ever seen in Wikipedia.

I hope someone knowledgeable about this subject and also capable of clear writing can fix this.

'''ALSO''': The claim that a diffeomorphism "is a homeomorphism with the additional condition that it is smooth, that is, it should have derivatives of all orders everywhere" '''is simply wrong'''.

In order to qualify as a smooth diffeomorphism, a homeomorphism between two smooth manifolds must not only be smooth but its inverse function must '''also''' be smooth.

'''Why this is wrong''': The real line ℝ is a smooth manifold. Consider the map f : ℝ  →  ℝ defined via f(x) = x<sup>3</sup>. 

The function f is a homeomorphism with the additional condition that it has derivatives of all orders.  

But '''it is not a diffeomorphism''', because its inverse function is defined by f<sup>-1</sup>(x)  =  x<sup>1/3</sup>, which is not differentiable at x = 0.)