nLab Furuta theorem

Idea

Furuta’s theorem (also called 10/8 theorem) describes which indefinite even intersection forms arise from oriented closed smooth 4-manifolds. (According to Freedman's classification, all can from simply connected oriented closed topological 4-manifolds, which makes smoothness the crucial property.) 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 $n\geq 3$.

It’s assumed that Furuta’s theorem can be improved, which is known as the 11/8 conjecture. For this reason, Furuta’s theorem is also called 10/8 theorem and not 5/4 theorem.

Statement

Proposition

Every oriented closed smooth 4-manifold $M$ with indefinite even intersection form fulfills:

$$b_2(M) \geq\frac{10}{8}|\sigma(M)|+2.$$

Hence, if $\sigma(M)\geq 0$, then $9b_2^-(M)\geq b_2^+(M)$. If $\sigma(M)\leq 0$, then $9b_2^+(M)\geq b_2^-(M)$. Without loss of generality, Furuta’s theorem 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:

$$n\geq 2|m|+1.$$

This follows from:

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

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

(Scorpan 05, p. 248)

For $|m|=1$ with $n\geq 3$, Furata’s theorem is exactly Donaldson’s prior work. For $m=|2|$ with $n\geq 5$, it was later concluded that Furuta’s theorem doesn’t give the best possible lower bound, as is generally assumed by the 11/8 conjecture:

Proposition

The indefinite even intersection form $\pm 4E_8\oplus 5H$ isn’t the intersection form of a oriented closed smooth 4-manifold.

(Scorpan 05, p. 248)

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

• Mikio Furuta: Monopole Equation and the 11/8-Conjecture, Mathematical Research Letters 8 (2001) 279-291 [doi:10.4310/MRL.2001.v8.n3.a5]

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