Hirzebruch signature theorem

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, 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 $p_1[M] = 3 \sigma(M)$.

Going deeper

For a student

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. 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.