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==Four dimensions== {{Main|4-manifold}} A '''4-manifold''' is a 4-dimensional [[topological manifold]]. A '''smooth 4-manifold''' is a 4-manifold with a [[smooth structure]]. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds that admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds that are [[homeomorphic]] but not [[diffeomorphic]]). 4-manifolds are of importance in physics because, in [[General Relativity]], [[spacetime]] is modeled as a [[pseudo-Riemannian]] 4-manifold. ===Exotic R<sup>4</sup>=== {{main|Exotic R4|l1=Exotic '''R'''<sup>4</sup>}} An '''exotic''' '''R'''<sup>4</sup> is a [[differentiable manifold]] that is [[homeomorphic]] but not [[diffeomorphism|diffeomorphic]] to the [[Euclidean space]] '''R'''<sup>4</sup>. The first examples were found in the early 1980s by [[Michael Freedman]], by using the contrast between Freedman's theorems about topological 4-manifolds, and [[Simon Donaldson]]'s theorems about smooth 4-manifolds.<ref>{{citation | last = Gompf | first = Robert E. | authorlink = Robert Gompf | issue = 2 | journal = [[Journal of Differential Geometry]] | mr = 710057 | pages = 317–328 | title = Three exotic '''R'''<sup>4</sup>'s and other anomalies | url = http://projecteuclid.org/euclid.jdg/1214437666 | volume = 18 | year = 1983}}.</ref> There is a [[cardinality of the continuum|continuum]] of non-diffeomorphic [[differentiable structure]]s of '''R'''<sup>4</sup>, as was shown first by [[Clifford Taubes]].<ref>Theorem 1.1 of {{citation | last = Taubes | first = Clifford Henry | authorlink = Clifford Taubes | issue = 3 | journal = [[Journal of Differential Geometry]] | mr = 882829 | pages = 363–430 | title = Gauge theory on asymptotically periodic 4-manifolds | url = http://projecteuclid.org/euclid.jdg/1214440981 | volume = 25 | year = 1987}}</ref> Prior to this construction, non-diffeomorphic [[smooth structure]]s on spheres—[[exotic sphere]]s—were already known to exist, although the question of the existence of such structures for the particular case of the [[4-sphere]] remained open (and still remains open to this day). For any positive integer ''n'' other than 4, there are no exotic smooth structures on '''R'''<sup>''n''</sup>; in other words, if ''n'' ≠ 4 then any smooth manifold homeomorphic to '''R'''<sup>''n''</sup> is diffeomorphic to '''R'''<sup>''n''</sup>.<ref>Corollary 5.2 of {{citation | last = Stallings | first = John | authorlink = John R. Stallings | doi = 10.1017/S0305004100036756 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]] | mr = 0149457 | pages = 481–488 | title = The piecewise-linear structure of Euclidean space | volume = 58 | issue = 3 | year = 1962}}.</ref> ===Other special phenomena in four dimensions=== There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in four dimensions. Here are some examples: * In dimensions other than 4, the [[Kirby–Siebenmann invariant]] provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H<sup>4</sup>(''M'','''Z'''/2'''Z''') vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure. * In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countable infinite number of non-diffeomorphic smooth structures. * Four is the only dimension ''n'' for which '''R'''<sup>''n''</sup> can have an exotic smooth structure. '''R'''<sup>4</sup> has an uncountable number of exotic smooth structures; see [[exotic R4|exotic '''R'''<sup>4</sup>]]. * The solution to the smooth [[Poincaré conjecture]] is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see [[exotic sphere]]). The Poincaré conjecture for [[PL manifold]]s has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions). * The smooth [[h-cobordism theorem]] holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson). If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds. * A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable. * There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem. In 2013, Ciprian Manolescu posted a preprint on ArXiv showing that there are manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.
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