What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers?

Question

I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that the signature of a manifold had anything to do with pontrjagin numbers?

Answer 1

First one notes that the signature is a bordism invariant, additive, and multiplicative, hence defines $\sigma:\Omega \otimes \mathbb Q\to \mathbb Q$.

Next you note that every ring morphism from the oriented bordism ring $\Omega \to \mathbb Q$ factors through $\Omega \otimes \mathbb Q$. But we know that on the latter space the Pontryagin classes are a complete invariant. That means that we should able to write down any such morphism in terms of these classes.

Lastly, one computes necessary conditions on how a ring morphism looks like, and one obtains that these are always given by a multiplicative sequence (plugging in the Pontryagin classes), which are in correspondence to power series. This implies that the signature needs to come from a power series.

Now, verifying the power series for the signature is easier (and more mysterious!) than you think. This is just done using a trick from complex analysis called the residue theorem.