Hirzebruch signature theorem

Sketch of proof of the signature theorem

René Thom had earlier proved that the signature was given by some linear combination of Pontryagin numbers, and Hirzebruch found the exact formula for this linear combination by introducing the notion of the genus of a multiplicative sequence.

Since the rational oriented cobordism ring ${\displaystyle {\mathrm{\Omega }}_{\ast }^{\text{SO}}\otimes \mathbb{Q}}$ is equal to

${\displaystyle {\mathrm{\Omega }}_{\ast }^{\text{SO}}\otimes \mathbb{Q}=\mathbb{Q}[{\mathbb{P}}^2(\mathbb{C}),{\mathbb{P}}^4(\mathbb{C}),\dots ],}$

the polynomial algebra generated by the oriented cobordism classes ${\displaystyle [{\mathbb{P}}^{2i}(\mathbb{C})]}$ of the even dimensional complex projective spaces, it is enough to verify that

${\displaystyle \sigma ({\mathbb{P}}^{2i})=1=⟨L_i(p_1({\mathbb{P}}^{2i}),\dots ,p_n({\mathbb{P}}^{2i})),[{\mathbb{P}}^{2i}]⟩}$

for all i.

Generalizations

The signature theorem is a special case of the Atiyah–Singer index theorem for the signature operator. The analytic index of the signature operator equals the signature of the manifold, and its topological index is the L-genus of the manifold. By the Atiyah–Singer index theorem these are equal.