4 The Hirzebruch Signature Theorem
We now apply our theory to the case of the Hirzebruch signature theorem. Fix a Riemannian manifold . Recall that the Hodge star operator acts on with eigenspaces . The operator
is an elliptic operator whose index is exactly the signature of .
In Section 1, we have established that
for all . In Section 2, we constructed a heat kernel for , and we observe that this commutes with the Hodge star. So we get an asymptotic series
Putting everything together, we can write
However, we also know the left-hand side is a constant. So the non-constant terms must in fact cancel out. So we know that
At this point, you might think it is rather hopeless to trace through the construction to obtain an explicit identification of , and you would be right. We will cheat as Hirzebruch did. However, it turns out for purely formal reasons, there aren't too many possibilities for it.
Recall that is a rational function of the components of the metric and its derivatives. The same is not true for . However, the change of basis matrix to an eigenbasis of can only be a function of , as it is performed fiberwise. So we can write
where are arbitrary smooth functions of the component of the metric, and are monomials in the derivatives of .
Suppose associates to each Riemannian manifold a differential form that, in local coordinates, can be expressed in the form as above.
Moreover, suppose has weight . That is, if we replace the metric by , then . Then is a polynomial function of the Pontryagin forms, and in fact has weight .
We need to calculate the weight of . Remembering that also depends on the metric, careful bookkeeping reveals that is of weight . So we deduce that is the integral over of a polynomial function of the Pontryagin forms!
Once we know that is a polynomial function of the Pontryagin forms, we just do as Hirzebruch did and evaluate both sides on enough spaces to show that it must be the -genus:
If is a closed Riemannian manifold of dimension (with ), then