nLab exotic R^4

Idea

There exists exotic smooth structure on the Euclidean space $\mathbb{R}^4$ of dimension 4.

(For all other dimensions there is a unique smooth structure, the standard one, see at smooth structure).

Construction

Construction from the failure of smooth surgery theory

Exotic$\phantom{,}\mathbb{R}^4$s, i.e. smooth manifolds homeomorphic but not diffeomorphic to $\mathbb{R}^4$, may be understood by comparison with exotic 7-spheres such as the Milnor spheres. In both cases there are closed topological $n$-manifolds $M$ ($n = 8$ resp. $4$) that admit no smooth structure, but become smoothable once a closed $n$-ball is removed. The failure lies in gluing the ball back in. The conclusions differ, though. For $n = 8$, a neighbourhood of the removed ball can be taken to be diffeomorphic to $\Sigma \times \mathbb{R}$, where $\Sigma$ is homeomorphic to $S^7$ but not diffeomorphic to it, since otherwise a smooth disc could be glued in along $\Sigma$. So $\Sigma$ is an exotic 7-sphere. For $n = 4$ this cannot happen, since $S^3$ has no exotic smooth structure (by Moise's theorem, (Sco. 05, p. 101)). Instead the obstruction lies in an exotic smooth structure on the 4-dimensional collar $S^3 \times \mathbb{R} \cong \mathbb{R}^4 \setminus \{0\}$ itself. This is a manifestation of the failure of smooth surgery theory in dimension 4, which also produces exotic $\mathbb{R}^4$s.

Construction from the K3 surface

A concrete construction is the K3 surface $K3$, described by the homogenous equation $x^4+y^4+z^4+w^4=0$ in twistor space $\mathbb{C}P^3$. (Fre. & Uhl. 84, p. 21, Sco. 05, p. 130) According to Freedman's classification, (Fre. & Uhl. 84, p. 22-24, Sco. 05, p. 240-243) 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 $K3$ 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, (Fre. & Uhl. 84, p. 22-24, Sco. 05, p. 243) and for $k=1,2$, it was also shown by Simon Donaldson, (Sco. 05, p. 248) 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 $K3$ must fail. (Fre. & Uhl. 84, p. 24-30) (It is an open question whether this also holds for any connected sum of the K3 surface $K3$, which is known as 11/8 conjecture.) In the simplest surgery of removing three $S^2\times S^2$, the failure spawns an exotic R^4 in $\#3(S^2\times S^2)$.

Construction from a nine-fold blow-up

The intersection form of the smooth complex surface $\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}$, the simplest example of an elliptic surface known as Dolgachev surface, is $[+1]\oplus 9[-1] \cong -E_8\oplus[+1]\oplus[-1]$ (Sco. 05, p. 250-259) 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 $\overline{M_{E_8}}\#*\overline{\mathbb{C}P^2}$, (Sco. 05, p. 240-243 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 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 $[+1]\oplus 10[-1] \cong -E_8\oplus H\oplus[-1]$, (Sco. 05, p. 250-259) 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 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 $W\cong M\times[0,1]$. (Sco. 05, p. 250-259)

Related entries

Articles on geometry and topology of 4-manifolds:

• Basic concepts:

  • Donaldson theory and related articles
  • Seiberg-Witten theory and related articles
  • intersection form
  • Kirby-Siebenmann invariant
  • signature

• Important examples:

  • Barlow surface$\;\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$
  • Dolgachev surface$\;\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$
  • E₈ manifold$\;M_{E_8}$
  • exotic $\mathbb{R}^n$
  • fake second complex projective space$\mathbb{C}P^2$

• Central results:

  • Donaldson theorem (Fields Medal 1986)
  • Freedman classification (Fields Medal 1986)
  • Furuta theorem (10/8 theorem)
  • Hirzebruch signature theorem
  • Hitchin-Thorpe inequality
  • Quinn theorem
  • Rokhlin theorem
  • van der Blij lemma
  • Wall theorems
  • Wu formula

• Open problems:

  • 3/2 conjecture
  • 11/8 conjecture
  • simple type conjecture
  • Poincaré conjecture

References

• Robert Gompf, Three exotic $\mathbb{R}^4$ and other anomalies, J. Differential Geom. 18 2 (1983), pp. 317-328 [doi:10.4310/jdg/1214437666.full]

• Robert Gompf, An infinite set of exotic $\mathbb{R}^4$‘s, J. Differential Geom. 21 2 (1985) [doi:10.4310/jdg/1214439566 pdf]

Reviews include:

• Daniel Freed, Karen Uhlenbeck, Instantons and Four-Manifolds (1984, revised 1991), Mathematical Sciences Research Institute Publications, Springer (doi:10.1007/978-1-4613-9703-8)

• Simon Donaldson, Peter Kronheimer: The Geometry of Four-Manifolds (1990, revised 1997), Oxford University Press and Claredon Press, [oup:52942, doi:10.1093/oso/9780198535539.001.0001, ISBN:978-0198502692, ISSN:0964-9174]

• Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society (2005) [ISBN 978-1470468613]

See also:

• Wikipedia, Exotic R4