Wikipedia / Exotic R4
Wikipedia / Exotic R4
In mathematics, an exotic is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures as was shown first by Clifford Taubes.
Prior to this construction, non-diffeomorphic smooth structures on spheresexotic sphereswere already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remains open as of 2026. For any positive integer n in other words, if n
Construction
Construction from the failure of smooth surgery theory
arise from the failure of smooth surgery theory not resulting in a smooth 4-manifold. Exotic 7-spheres called Milnor spheres has no exotic smooth structure due to Moise's theorem, has to in order to explain the failure.
Construction from the K3 surface
A concrete construction is the K3 surface in twistor space. According to Freedman's classification, it is homeomorphic to the connected sum of twice the orientation-reversed E8 manifold and thrice the complex surface gained from the product of the Riemann sphere, this follows from Donaldson's theorem,, it was also shown by Simon Donaldson, although Donaldson's theorem itself no longer applies. (A simpler argument was later obtained with Seiberg–Witten invariants must fail., which is known as 11/8 conjecture.
Construction from a nine-fold blow-up
The intersection form of the smooth complex surface, the simplest example of an elliptic surface known as Dolgachev surface with the isomorphism coming from the Hesse–Minkowski theorem using the fake second complex projective space, so Freedman's classification,.
Construction from a ten-fold blow-up
,, so Freedman's classification.
Construction from the failure of the smooth h-cobordism theorem
According to Michael Freedman, the four-dimensional topological h-cobordism theorem, hence there are diffeomorphisms:
.
Small exotic R4s
is called small
can be constructed by starting with a non-trivial smooth 5-dimensional h-cobordism (which exists by Donaldson's proof that the h-cobordism theorem fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.
Large exotic R4s
is called large
can be constructed using the fact that compact 4-manifolds can often be split as a topological sum (by Freedman's work), but cannot be split as a smooth sum (by Donaldson's work).
can be smoothly embedded as open subsets.
Related exotic structures
It is not known (as of 2024) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.
See also
Notes
References