nLab 11/8 conjecture

Idea

The 11/8 conjecture arises from the still not fully answered question, which symmetric unimodular bilinear forms are the intersection forms of oriented closed smooth 4-manifolds.

According to Donaldson's theorem, a definite form is the intersection form of an oriented closed smooth 4-manifolds if and only if it’s either $\oplus n[-1]$ or $\oplus n[+1]$, with both realized by the even simply connected examples $\#n\mathbb{C}P^2$ or $\#n\overline{\mathbb{C}P^2}$. According to Serre’s classification theorem, an indefinite odd form is isomorphic to $\oplus m[-1]\oplus n[+1]$, which are all realized by the even simply connected examples $\#m\mathbb{C}P^2\#m\overline{\mathbb{C}P^2}$. (According to Freedman's theorem, all these simply connected examples are even unique up to homeomorphism.)

Left over is, which indefinite even forms are realized. It comes down to the forms:

$$\pm 2mE_8\oplus nH$$

according to Serre’s classification theorem with Rokhlin's theorem requiring an even number of definite $\pm E_8$ and indefiniteness requiring $n\geq 1$. Simon Donaldson furthermore showed, that if $m\neq 0$, then $n\geq 3$.

Currently, the strongest result approaching the 11/8 conjecture is Furuta's theorem, which is also called 10/8 theorem.

Statement

The open 11/8 conjecture assumes: Every oriented closed smooth 4-manifold $M$ with indefinite even intersection form fulfills:

$$b_2(M) \geq\frac{11}{8}|\sigma(M)|.$$

Hence, if $\sigma(M)\geq 0$, then $19b_2^-(M)\geq 3b_2^+(M)$. If $\sigma(M)\leq 0$, then $19b_2^+(M)\geq 3b_2^-(M)$. Without loss of generality, the 11/8 conjecture can be restricted to one of these cases by reversing orientation with $b_2(\overline{M})=b_2 (M)$ and $\sigma(\overline{M})=-\sigma(M)$.

Alternatively, for the above structure of the intersection form, the 3/2 conjecture is equivalent to:

$$n\geq 3|m|.$$

This follows from:

$$b_2(M) =16|m|+2n;$$

$$|\sigma(M)| =16|m|.$$

(Scorpan 05, p. 247)

Both reformulations of the 11/8 conjecture show, that the difference to Furuta's theorem or 10/8 theorem is bigger than the small difference between the fractions makes it seem.

If the 11/8 conjecture holds for two oriented closed smooth 4-manifolds $M$ and $N$ with indefinite even intersection form, then it automatically holds for their connected sum when additionally using the triangle inequality:

$$\frac{11}{8}|\sigma(M\# N)| =\frac{11}{8}|\sigma(M) +\frac{11}{8}\sigma(N)| \leq b_2(M) +b_2(N) =b_2(M\# N).$$

But that doesn’t mean, that it’s sufficient to prove the 11/8 conjecture just for irreducible (not splitting in a non-trivial connected sum) oriented closed smooth 4-manifolds with indefinite even intersection form. While all other properties reduce back to the splittings, indefiniteness doesn’t. For example, a connected sum of manifolds with a positive and negative definite intersection form has indefinite intersection form. Definite forms are extremely numeruous and therefore lack a simple classification.

Example

Further improvement of the 11/8 conjecture isn’t possible as shown by the simply connected oriented closed K3 surface $K3\coloneqq V(z_0^3+z_1^3+z_2^3+z_3^3)\subset\mathbb{C}P^3$ with intersection form:

$$Q_{K3} =-2E_8 \oplus 3H.$$

It therefore has $b_2(K3)=22$ and $\sigma(K3)=16$, meaning the 11/8 conjecture is an equality for it. Taking connected sums $\#m K3$ including reversing orientation (corresponding to flipping the sign of $m$) furthermore produces examples for equalities for every $m\in\mathbb{Z}$.

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

  • Dolgachev surface$\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$
  • E₈ manifold$M_{E_8}$
  • exotic R^4
  • 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

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