nLab Barlow surface

Idea

The Barlow surface is a simply connected complex surface of general type that is homeomorphic (Barlow 1984) but not diffeomorphic (Kotschick 1989) to $\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$ (the eight-fold blow-up of the complex projective plane), and thus the first known example of an exotic smooth structure on this topological manifold.

While it is now established that the underlying topological manifold $\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$ admits countably infinitely many non-diffeomorphic smooth structures, these infinite families are generally constructed via later techniques (such as rational blowdowns or knot surgery) and are not themselves “Barlow surfaces”.

(For context, the topological manifold obtained by one further blow-up is $\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$, which is the space famously underlying the Dolgachev surfaces.)

Description

Freedman's classification provides:

Proposition

Every simply connected oriented closed smooth 4-manifold with intersection form $[-1]\oplus 8[+1]$ is homeomorphic to $\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$.

(Instead of smoothness, a vanishing Kirby-Siebenmann invariant is also sufficient.)

For example with the E₈ form $E_8$, the intersection form of the E₈ manifold, one has $-E_8\oplus[+1]\cong [+1]\oplus 8[-1]$ from Serre’s classification theorem, with Freedman's classification translating it into a homeomorphism:

$$\overline{M_{E_8}}\#*\mathbb{C}P^2 \cong\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}.$$

($*\mathbb{C}P^2$ is the fake second complex projective space, which is required instead of the second complex projective space $\mathbb{C}P^2$ to make the connected sum on the left side have vanishing Kirby-Siebenmann invariant like the right side.)

As another example with the hyperbolic form $H=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, the intersection form of the complex surface $S^2\times S^2\cong\mathbb{C}P^1\times\mathbb{C}P^1$, one has $H\oplus[-1]\cong[+1]\oplus 2[-1]$ from Serre’s classification, with Freedman's classification translating it into a homeomorphism:

$$(S^2\times S^2)\#\overline{\mathbb{C}P^2} \cong\mathbb{C}P^2\# 2\overline{\mathbb{C}P^2}.$$

It can be shown that it’s even a diffeomorphism, hence further gives a diffeomorphism:

$$(S^2\times S^2)\# 7\overline{\mathbb{C}P^2} \cong\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}.$$

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

• Rebecca Barlow, Some new surfaces with $p_g=0$, Duke Mathematical Journal 51 4 (1984), pp. 889–904 [doi:10.1215/S0012-7094-84-05139-1 ISSN 0012-7094 MR 0771386]

• Dieter Kotschick, On manifolds homeomorphic to $\mathbb{C}P^2\#8\overline{\mathbb{C}P^2}$, Invent. Math. 95 3, pp. 591-600 (1989) [doi:10.1007/BF01393892]

See also:

• Wikipedia, Barlow surface