Editing Exotic sphere (section)

==Introduction==
The unit ''n''-sphere, <math>S^n</math>, is the set of all [[tuple|(''n''+1)-tuples]] <math>(x_1, x_2, \ldots , x_{n+1})</math> of real numbers, such that the sum <math>x_1^2 + x_2^2 + \cdots + x_{n+1}^2 = 1</math>. For instance, <math>S^1</math> is a circle, while <math>S^2</math> is the surface of an ordinary ball of radius one in 3 dimensions. Topologists consider a space ''X'' to be an ''n''-sphere if there is a [[homeomorphism]] between them, i.e. every point in ''X'' may be assigned to exactly one point in the unit ''n''-sphere by a continuous bijection with continuous inverse. For example, a point ''x'' on an ''n''-sphere of radius ''r'' can be matched homeomorphically with a point on the unit ''n''-sphere by multiplying its distance from the origin by <math>1/r</math>. Similarly, an ''n''-cube of any radius is homeomorphic to an ''n''-sphere.

In [[differential topology]], two smooth manifolds are considered smoothly equivalent if there exists a [[diffeomorphism]] from one to the other, which is a homeomorphism between them, with the additional condition that it be [[smooth function|smooth]] — that is, it should have [[derivative]]s of all orders at all its points — and its inverse homeomorphism must also be smooth. To calculate derivatives, one needs to have local coordinate systems defined consistently in ''X''. Mathematicians (including Milnor himself) were surprised in 1956 when Milnor showed that consistent local coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6-, 12-, 56- or 61-sphere.<ref>{{Cite journal|last1=Behrens|first1=M.|last2=Hill|first2=M.|last3=Hopkins|first3=M. J.|last4=Mahowald|first4=M.|date=2020|title=Detecting exotic spheres in low dimensions using coker J|url=https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12301|journal=Journal of the London Mathematical Society|language=en|volume=101|issue=3|pages=1173–1218|arxiv=1708.06854|doi=10.1112/jlms.12301|s2cid=119170255 |issn=1469-7750}}</ref> Some higher-dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an [[List of unsolved problems in mathematics|unsolved problem]].