nLab Wu formula

Idea

Wu’s formula is a relation between the cup product and the second Stiefel-Whitney class, which connects spin topological 4-manifold and even intersection forms.

Statement

Proposition

For a oriented closed topological 4-manifold $M$, one has:

$$x \smile x = w_2(T M)\smile x$$

for all classes $x\in H^2(M,\mathbb{Z}_2)$.

(Scorpan 05, p. 163)

Application

An important application of Wu’s formula is the connection with the intersection form $Q_M\colon H^2(M,\mathbb{Z})\times H^2(M,\mathbb{Z})\rightarrow\mathbb{Z},(x,y)\mapsto\langle x\smile y,[M]\rangle$. Considering the reduction of its diagonal, one has:

$$Q_M(x,x)mod 2 =Q_M(x mod 2,x mod 2) =Q_M(w_2(T M),x mod 2)$$

for all classes $x\in H^2(M,\mathbb{Z})$. It leads to the following results:

Proposition

The intersection form of a closed spin topological 4-manifold is even.

(Scorpan 05, p. 163)

Proposition

A simply connected oriented closed topological 4-manifold with even intersection form is spin.

In particular, a simply connected oriented closed topological 4-manifold is spin if and only if its intersection form is even.

(Scorpan 05, p. 164)

Related concepts

• Wu class

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

Named after

• Wen-Tsun Wu: On Pontrjagin classes: II, Sientia Sinica 4 (1955) 455–90

See also:

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