nLab Dolgachev surface

Idea

Dolgachev surfaces provide countably infinitely many non-diffeomorphic smooth structures on the complex surface $\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$, which is the nine-fold blow-up of the second complex projective space. (One blow-up less yields the Barlow surfaces $\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$.)

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 $-E_8\oplus[+1]\oplus[-1]\cong [+1]\oplus 9[-1]$ from Serre’s classification theorem, with Freedman's classification translating it into a homeomorphism:

$$\overline{M_{E_8}}\#*\mathbb{C}P^2\#\overline{\mathbb{C}P^2} \cong\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}.$$

($*\mathbb{C}P^2$ 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:

$$(S^2\times S^2)\# 8\overline{\mathbb{C}P^2} \cong\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}.$$

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

• Selman Akbulut, The Dolgachev surface. Disproving the Harer–Kas–Kirby conjecture, Commentarii Mathematici Helvetici 87 1 (2012), pp. 187–241 [arXiv:0805.1524 doi:10.4171%2FCMH%2F252 Bibcode:2008arXiv0805.1524A MR 2874900]

• Selman Akbulut, 4-manifolds (2016) [ISBN 978-0198784869]

See also:

• Wikipedia, Dolgachev surface