---
tag: TEG1ZX1AGJ
kind: statement
role: theorem
name: Hirzebruch signature theorem
msc: [57R20, 55R40]
uses: [F45VQ47GPV, 913WEQERBE, DN11FKWN63, WK8GKRKAVD, 07EZA80NEH, T8BVGPPSZP]
about: [DN11FKWN63, WK8GKRKAVD]
sources:
  - url: https://en.wikipedia.org/wiki/Hirzebruch_signature_theorem
same_as:
  wikidata: Q7512855
  wikipedia: Hirzebruch signature theorem
---

## Statement

Let $M$ be a closed oriented [[smooth manifold|F45VQ47GPV]] of dimension $4k$ with fundamental class $[M]$.
Let $\sigma(M)$ be the [[signature|07EZA80NEH]] of its [[intersection form|WK8GKRKAVD]] on $H^{2k}(M; \R)$, the symmetric bilinear form $(a, b) \mapsto \langle a \smile b, [M] \rangle$.
Let $p_i = p_i(TM) \in H^{4i}(M; \Q)$ be the [[Pontryagin classes|DN11FKWN63]] of the tangent bundle.
Let $L_k(p_1, \ldots, p_k)$ be the degree $4k$ part of the multiplicative sequence of Hirzebruch with characteristic power series $\sqrt{z} / \tanh \sqrt{z}$.
Then
$$\sigma(M) = \langle L_k(p_1, \ldots, p_k), [M] \rangle.$$

## Remarks

The first two polynomials are $L_1 = \tfrac{1}{3} p_1$ and $L_2 = \tfrac{1}{45}(7 p_2 - p_1^2)$.
Idea of the proof: both sides are invariants of oriented [[cobordism|T8BVGPPSZP]], additive under disjoint union and multiplicative under products.
So both sides are ring homomorphisms from the rational oriented cobordism ring to $\Q$.
Thom showed that this ring is a polynomial ring on the classes of the complex projective spaces $\C P^{2i}$.
The power series is chosen so that the right side equals $1 = \sigma(\C P^{2i})$ on each generator, and so the two homomorphisms agree.

Example: for $M = \C P^2$ with generator $h \in H^2$, the total Pontryagin class is $(1 + h^2)^3$, so $p_1 = 3h^2$ and $\langle L_1, [M] \rangle = 1$.
The intersection form is $(1)$, so $\sigma(\C P^2) = 1$, as the theorem says.
For a closed oriented smooth $4$-manifold the theorem gives $p_1[M] = 3 \sigma(M)$.

## Explanations

### L1 human

The signature is a whole number read off from how middle-dimensional cycles in a manifold intersect.
Pontryagin classes measure how the tangent bundle is twisted.
The theorem says that one fixed combination of the Pontryagin numbers always equals the signature.
So a number defined by pure topology is computed by curvature-type data.

### L2 human

The theorem shows that a certain rational combination of Pontryagin numbers is always an integer, which constrains which manifolds can exist.
Milnor used the case $k = 2$ to show that some $7$-manifolds homeomorphic to $S^7$ are not diffeomorphic to it, the first [[exotic spheres|6WYEMXSHD9]].
The theorem is a special case of the Atiyah–Singer index theorem applied to the signature operator.
It is also the model for the Hirzebruch–Riemann–Roch theorem.
In surgery theory, the signature formula is what makes the simply connected surgery obstruction in dimensions $4k$ computable.