Exotic R4

Short description: Smooth 4-manifold homeomorphic yet not diffeomorphic to Euclidean space

In mathematics, an exotic ${\mathbb{ℝ}}^4$ is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space ${\mathbb{ℝ}}^4.$ 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 ${\mathbb{ℝ}}^4,$ as was shown first by Clifford Taubes.

Prior to this construction, non-diffeomorphic smooth structures on spheres – exotic spheres – were 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 other than 4, there are no exotic smooth structures ${\mathbb{ℝ}}^n;$ in other words, if n ≠ 4 then any smooth manifold homeomorphic to ${\mathbb{ℝ}}^n$ is diffeomorphic to ${\mathbb{ℝ}}^n.$

Construction

Construction from the failure of smooth surgery theory

Exotic ${\mathbb{ℝ}}^4$ arise from the failure of smooth surgery theory, in particular glueing a disc $D^4$ to its boundary sphere $S^3$ not resulting in a smooth 4-manifold. Exotic 7-spheres called Milnor spheres arise from a fully analogous failure of glueing a disc $D^8$ to its boundary sphere $S^7$ not resulting in a smooth 8-manifold. But the conclusions are different: In the latter case, $D^8\cong {\mathbb{ℝ}}^8$ has no exotic smooth structure, so $S^7$ has to in order to explain the failure. In the former case, $S^3$ has no exotic smooth structure due to Moise's theorem, so $D^4\cong {\mathbb{ℝ}}^4$ has to in order to explain the failure.

Construction from the K3 surface

A concrete construction is the K3 surface $\mathrm{K}\mathrm{3}$, described by the homogenous equation $x^4+y^4+z^4+w^4=0$ in twistor space $\mathbb{ℂ}{P}^3$. According to Freedman's classification, it is homeomorphic to the connected sum $2\overline{M_{E_8}}\mathrm{\#}3(S^2\times S^2)$ of twice the orientation-reversed E8 manifold $M_{E_8}$ and thrice the complex surface $S^2\times S^2\cong \mathbb{ℂ}{P}^1\times \mathbb{ℂ}{P}^1$ gained from the product of the Riemann sphere with itself. Although $\mathrm{K}\mathrm{3}$ is a smooth 4-manifold, $2\overline{M_{E_8}}\mathrm{\#}k(S^2\times S^2)$ isn't for $k=0,1,2$. For $k=0$, this follows from Donaldson's theorem, and for $k=1,2$, 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.) Hence a smooth surgery along the connecting sphere $S^3$ to remove one, two or three $S^2\times S^2$ from the smooth K3 surface $\mathrm{K}\mathrm{3}$ must fail. (It is an open question whether this also holds for any connected sum of the K3 surface $\mathrm{K}\mathrm{3}$, which is known as 11/8 conjecture.) In the simplest surgery of removing three $S^2\times S^2$, the failure spawns an exotic ${\mathbb{ℝ}}^4$ in $\mathrm{\#}3(S^2\times S^2)$.

Construction from a nine-fold blow-up

The intersection form of the smooth complex surface $\mathbb{ℂ}{P}^2\mathrm{\#}9\overline{\mathbb{ℂ}{P}^9}$, the simplest example of an elliptic surface known as Dolgachev surface, is $[+1]\oplus 9[-1]\cong -E_8\oplus [+1]\oplus [-1]$ with the isomorphism coming from the Hesse–Minkowski theorem. It is therefore the same as for the topological 4-manifold $\overline{M_{E_8}}\mathrm{\#}\mathbb{ℂ}{P}^2\mathrm{\#}*\overline{\mathbb{ℂ}{P}^2}$ using the fake second complex projective space, so Freedman's classification claims both to be homeomorphic. In particular, that makes $\overline{M_{E_8}}\mathrm{\#}\mathbb{ℂ}{P}^2\mathrm{\#}*\overline{\mathbb{ℂ}{P}^2}$ smoothable. But a smooth surgery to remove $\mathbb{ℂ}{P}^2$ results in the non-smoothable manifold $\overline{M_{E_8}}\mathrm{\#}*\overline{\mathbb{ℂ}{P}^2}$, which follows from its intersection form $-E_8\oplus [-1]$ being negative definite and not diagonalizable, which Donaldson's theorem forbids if it was smoothable. The failure of this smooth surgery spawns an exotic ${\mathbb{ℝ}}^4$ in $\mathbb{ℂ}{P}^2$.

Construction from a ten-fold blow-up

The previous construction can be expanded: The intersection form of the smooth complex surface $\mathbb{ℂ}{P}^2\mathrm{\#}10\overline{\mathbb{ℂ}{P}^9}$ is $[+1]\oplus 10[-1]\cong -E_8\oplus H\oplus [-1]$, the same as for the topological 4-manifold $\overline{M_{E_8}}\mathrm{\#}(S^2\times S^2)\mathrm{\#}*\overline{\mathbb{ℂ}{P}^2}$, so Freedman's classification claims both to be homeomorphic and in particular the latter to be smoothable. But a smooth surgery to remove $S^2\times S^2$ again results in the non-smoothable manifold $\overline{M_{E_8}}\mathrm{\#}*\overline{\mathbb{ℂ}{P}^2}$. The failure of this smooth surgery spawns an exotic ${\mathbb{ℝ}}^4$ in $S^2\times S^2$.

Construction from the failure of the smooth h-cobordism theorem

According to Michael Freedman, the four-dimensional topological h-cobordism theorem holds, while according to Simon Donaldson, the four-dimensional smooth h-cobordism theorem doesn't hold. Let $W$ be a smooth h-cobordism between smooth 4-manifolds $M$ and $N$, for which there is no diffeomorphism $W\cong M\times [0,1]$, although there must be a homeomorphism $W\cong M\times [0,1]$. In this case there exists an open sub-cobordism $U\subset W$ with a homeomorphism $U\cong D^4\times [0,1]\cong {\mathbb{ℝ}}^4\times [0,1]$ and a compact contractible sub-cobordism $K\subset U$, so that $W$ and $U$ are indeed smoothly trivial cobordisms outside of $K$, hence there are diffeomorphisms:

$W\setminus K\cong (M\setminus K)\times [0,1];$

$U\setminus K\cong (U\cap M\setminus K)\times [0,1].$

Now $U$ connects the two open subsets $U\cap M\setminus K$ and $U\cap N\setminus K$, which are homeomorphic to ${\mathbb{ℝ}}^4$, but cannot diffeomorphic to ${\mathbb{ℝ}}^4$ as this would cause a diffeomorphism $W\cong M\times [0,1]$.

Small exotic R⁴s

An exotic ${\mathbb{ℝ}}^4$ is called small if it can be smoothly embedded as an open subset of the standard ${\mathbb{ℝ}}^4.$

Small exotic ${\mathbb{ℝ}}^4$ 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

An exotic ${\mathbb{ℝ}}^4$ is called large if it cannot be smoothly embedded as an open subset of the standard ${\mathbb{ℝ}}^4.$

Examples of large exotic ${\mathbb{ℝ}}^4$ 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).

Michael Hartley Freedman and Laurence R. Taylor (1986) showed that there is a maximal exotic ${\mathbb{ℝ}}^4,$ into which all other ${\mathbb{ℝ}}^4$ can be smoothly embedded as open subsets.

Related exotic structures

Casson handles are homeomorphic to ${\mathbb{𝔻}}^2\times {\mathbb{ℝ}}^2$ by Freedman's theorem (where ${\mathbb{𝔻}}^2$ is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to ${\mathbb{𝔻}}^2\times {\mathbb{ℝ}}^2.$ In other words, some Casson handles are exotic ${\mathbb{𝔻}}^2\times {\mathbb{ℝ}}^2.$

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

• Akbulut cork - tool used to construct exotic ${\mathbb{ℝ}}^4$'s from classes in $H^3(S^3,\mathbb{ℝ})$
• Atlas (topology)

References

• Uhlenbeck, Karen K.; Freed, Daniel S. (1984) (in en). Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications. 1. Springer. doi:10.1007/978-1-4613-9703-8. ISBN 978-1-4613-9703-8. 
• Freedman, Michael H.; Quinn, Frank (1990). Topology of 4-manifolds. Princeton Mathematical Series. 39. Princeton, NJ: Princeton University Press. ISBN 0-691-08577-3. https://archive.org/details/topologyof4manif0000free. 
• Freedman, Michael H.; Taylor, Laurence R. (1986). "A universal smoothing of four-space". Journal of Differential Geometry 24 (1): 69–78. doi:10.4310/jdg/1214440258. ISSN 0022-040X. https://projecteuclid.org/euclid.jdg/1214440258. 
• Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirby calculus. Graduate Studies in Mathematics. 20. Providence, RI: American Mathematical Society. ISBN 0-8218-0994-6. 
• Kirby, Robion C. (1989). The topology of 4-manifolds. Lecture Notes in Mathematics. 1374. Berlin: Springer-Verlag. ISBN 3-540-51148-2. 
• Scorpan, Alexandru (2005). The Wild World of 4-Manifolds. Mathematical Sciences Research Institute Publications. 1. American Mathematical Society. ISBN 978-1-4704-6861-3. 
• Stallings, John (1962). "The piecewise-linear structure of Euclidean space". Mathematical Proceedings of the Cambridge Philosophical Society 58 (3): 481–488. doi:10.1017/s0305004100036756. Bibcode: 1962PCPS...58..481S. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2053736&fulltextType=RA&fileId=S0305004100036756.  MR0149457
• Taubes, Clifford Henry (1987). "Gauge theory on asymptotically periodic 4-manifolds". Journal of Differential Geometry 25 (3): 363–430. doi:10.4310/jdg/1214440981. Template:Project Euclid. http://projecteuclid.org/euclid.jdg/1214440981.