// Workers AI · dad joke modeWhat did Exotic R4 say to its date? You're a 4-getful companion.

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 $\mathrm {K3}$, 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 $2{\overline {M_{E_{8}}}}\#3(S^{2}\times S^{2})$ 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 $\mathrm {K3}$ is a smooth 4-manifold, $2{\overline {M_{E_{8}}}}\#k(S^{2}\times S^{2})$ 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 $\mathrm {K3}$ must fail. (It is an open question whether this also holds for any connected sum of the K3 surface $\mathrm {K3}$, 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 $\#3(S^{2}\times S^{2})$.

Construction from a nine-fold blow-up

The intersection form of the smooth complex surface $\mathbb {C} P^{2}\#9{\overline {\mathbb {C} P^{9}}}$, 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 ${\overline {M_{E_{8}}}}\#\mathbb {C} P^{2}\#*{\overline {\mathbb {C} P^{2}}}$ using the fake second complex projective space, so Freedman's classification claims both to be homeomorphic. In particular, that makes ${\overline {M_{E_{8}}}}\#\mathbb {C} P^{2}\#*{\overline {\mathbb {C} P^{2}}}$ smoothable. But a smooth surgery to remove $\mathbb {C} P^{2}$ results in the non-smoothable manifold ${\overline {M_{E_{8}}}}\#*{\overline {\mathbb {C} P^{2}}}$, 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 $\mathbb {C} P^{2}\#10{\overline {\mathbb {C} P^{9}}}$ is $[+1]\oplus 10[-1]\cong -E_{8}\oplus H\oplus [-1]$, the same as for the topological 4-manifold ${\overline {M_{E_{8}}}}\#(S^{2}\times S^{2})\#*{\overline {\mathbb {C} P^{2}}}$, 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 ${\overline {M_{E_{8}}}}\#*{\overline {\mathbb {C} P^{2}}}$. 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). Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications. Vol. 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. Vol. 39. Princeton, NJ: Princeton University Press. ISBN 0-691-08577-3.
• 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. MR 0857376.
• Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirby calculus. Graduate Studies in Mathematics. Vol. 20. Providence, RI: American Mathematical Society. ISBN 0-8218-0994-6.
• Kirby, Robion C. (1989). The topology of 4-manifolds. Lecture Notes in Mathematics. Vol. 1374. Berlin: Springer-Verlag. ISBN 3-540-51148-2.
• Scorpan, Alexandru (2005). The Wild World of 4-Manifolds. Mathematical Sciences Research Institute Publications. Vol. 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. Bibcode:1962PCPS...58..481S. doi:10.1017/s0305004100036756. S2CID 120418488. MR 0149457
• Taubes, Clifford Henry (1987). "Gauge theory on asymptotically periodic 4-manifolds". Journal of Differential Geometry. 25 (3): 363–430. doi:10.4310/jdg/1214440981. MR 0882829. PE 1214440981.