Editing Differential structure (section)

==Differential structures on topological manifolds==
As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by [[Tibor Radó]] for dimension 1 and 2, and by [[Edwin E. Moise]] in dimension 3.<ref>{{cite journal |last=Moise |first=Edwin E.|authorlink=Edwin E. Moise |title=Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung |journal=[[Annals of Mathematics]] |series=Second Series |volume=56 |issue=1 |pages=96–114 |year=1952 |jstor=1969769 |doi=10.2307/1969769|mr=0048805}}</ref> By using [[obstruction theory]], [[Robion Kirby]] and [[Laurent C. Siebenmann]] were able to show that the number of [[PL structure]]s for compact topological manifolds of dimension greater than 4 is finite.<ref>{{cite book |last=Kirby |first=Robion C. |authorlink1=Robion Kirby|last2=Siebenmann |first2=Laurence C. |authorlink2=Laurent C. Siebenmann|title=Foundational Essays on Topological Manifolds. Smoothings, and Triangulations |url=https://archive.org/details/foundationalessa0000kirb |url-access=registration |location=Princeton, New Jersey |publisher=Princeton University Press |year=1977 |isbn=0-691-08190-5 }}</ref> [[John Milnor]], [[Michel Kervaire]], and [[Morris Hirsch]] proved that the number of smooth structures on a compact PL manifold is finite and agrees with the number of differential structures on the sphere for the same dimension (see the book Asselmeyer-Maluga, Brans chapter 7) <!-- This needs checking: this number agrees with the number of differential structures on the sphere of the same dimension. Thus the table above lists also the number of differential structures for any (metrizable) topological manifold of dimension <math>n</math>.-->. By combining these results, the number of smooth structures on a compact topological manifold of dimension not equal to 4 is finite. 

[[4-manifold|Dimension 4]] is  more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second [[Betti number]]&nbsp;''b''<sub>2</sub>. For large Betti numbers ''b''<sub>2</sub>&nbsp;>&nbsp;18 in a [[simply connected]] 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many differential structures. But even for simple spaces such as <math>S^4, {\mathbb C}P^2,...</math> one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like <math>{\mathbb R}^4,S^3\times {\mathbb R},M^4\smallsetminus\{*\},...</math> having uncountably many differential structures.