nLab Hirzebruch signature theorem

Idea

Hirzebruch’s signature theorem relates the signature of oriented smooth $4n$-manifolds with its Pontrjagin numbers.

Statement

Proposition

(Hirzebruch signature theorem for 4-manifolds) For an oriented smooth 4-manifold $M$ with fundamental class $[M]\in H^4(M,\mathbb{Z})\cong\mathbb{Z}$, its signature is given by Pontrjagin numbers as:

$$\sigma(M) =\frac{1}{3}\langle p_1(M),[M]\rangle \in\mathbb{Z}.$$

Proposition

(Hirzebruch signature theorem for 8-manifolds) For an oriented smooth 8-manifold $M$ with fundamental class $[M]\in H^8(M,\mathbb{Z})\cong\mathbb{Z}$, its signature is given by Pontrjagin numbers as:

$$\sigma(M) =\frac{1}{45}\langle(7p_2-2p_1^2)(M),[M]\rangle \in\mathbb{Z}.$$

Proposition

(Hirzebruch signature theorem for 12-manifolds) For an oriented smooth 12-manifold $M$ with fundamental class $[M]\in H^12(M,\mathbb{Z})\cong\mathbb{Z}$, its signature is given by Pontrjagin numbers as:

$$\sigma(M) =\frac{1}{945}\langle(62p_3-13p_1p_2+2p_1^3)(M),[M]\rangle \in\mathbb{Z}.$$

Proposition

(Hirzebruch signature theorem for 16-manifolds) For an oriented smooth 16-manifold $M$ with fundamental class $[M]\in H^16(M,\mathbb{Z})\cong\mathbb{Z}$, its signature is given by Pontrjagin numbers as:

$$\sigma(M) =\frac{1}{14175}\langle(381 p_4-71p_1p_3-19p_2^2+22p_1^2p_2-3p_1^4)(M),[M]\rangle \in\mathbb{Z}.$$

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:

  • 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

• Wikipedia, Hirzebruch signature theorem