On a closed Riemannian manifold, there are three closely related objects we can study:
Topological invariants such as the Euler characteristic and the signature;
Indexes of differential operators; and
Differential forms on the manifold.
(i) and (ii) are mainly related by the Hodge decomposition theorem — the signature and Euler characteristic count the dimensions of the cohomology groups, and Hodge theory says the cohomology groups are exactly the kernels of the Laplacian. This is explored briefly in Appendix A.
(i) and (iii) are related to each other via results such as the Gauss–Bonnet theorem and the Hirzebruch signature theorem. The Gauss–Bonnet theorem says the Euler characteristic of a surface is the integral of the curvature; The Hirzebruch signature theorem says the signature is the integral of certain differential forms called the -genus, given as a polynomial function of the Pontryagin forms.
(ii) and (iii) can also be related directly to each other, and this connection is what I would call “index theory”. The main theorem is the Atiyah–Singer index theorem, and once we have accepted the connection between (i) and (ii), we can regard the Gauss–Bonnet theorem and the Hirzebruch signature theorem as our prototypical examples of index theory.
I wish to make the point that (i) and (ii) are purely global invariants, in the sense that we can only make sense of them when we are given the manifold as a whole. On the contrary, the construction in (iii) is purely local. As long as we construct the Pontryagin form using Chern–Weil theory, the differential forms are globally defined up to equality, and not just up to exact forms.
Thus, it makes sense to ask the following question — if we are given a compact Riemannian manifold with boundary, we can still integrate the differential form we had above. The signature still makes sense, and we can ask how these two relate.
We do not expect them to be equal. In the case of a surface with boundary, Gauss–Bonnet says
where is the Gaussian curvature and is the geodesic curvature of the boundary. However, if the neighbourhood of the boundary is isometric to the product , then the boundary term vanishes, and so the Euler characteristic is just the integral of the curvature.
For the signature, even if we assume the boundary is isometric to a product, the integral of the -genus still does not give the signature. However, the difference between the two is entirely a function of the Riemannian manifold . Indeed, if we have two manifolds with the same boundary, and both are isometric to products near the boundary, then we can glue them together along the boundary. Since both the signature and the integral of the -genus add up when we glue manifolds along boundaries, the error terms of and must be the same.
It turns out this error term is a spectral invariant. We can define an elliptic self-adjoint operator on which, up to some signs, is . We can define
where we sum over the eigenvalues of with multiplicity. This converges when is large, and has a meromorphic continuation to all of . We will show that the error term is then .
Our strategy to understanding this is to employ the heat kernel, which is a way of understanding the connection between (ii) and (iii). In Section 1, we discuss the classical heat equation and construct the heat kernel on a general closed manifold. From that particular construction, it will be evident how the heat kernel is related to the index of a differential operator. In Section 2, we provide an alternative construction of the heat kernel of the Laplace–Beltrami operator, which gives us some precise estimates. In Section 3, we use the estimates to construct a heat kernel for the Laplacian on open manifolds. This section is merely an example of what one can do with the estimates, and the results are not used anywhere else. In Section 4, we use our results in Section 1 and 2 to prove Hirzebruch's signature theorem.
In Section 5, we begin to study the signature on a manifold with boundary and connect this to the index of an operator, which we of course hope to calculate using the heat kernel. This is rather more complicated than what one might initially think. In Section 6, we study the heat kernel on a manifold of the form , which we think of as the collar neighbourhood of a manifold. In Section 7, we glue the results of the previous chapter with results on the interior established early on to understand the heat kernel on manifolds with boundary, and finally apply this theory to prove the theorem stated above.
To give credit where credit is due, Section 2 is essentially lifted straight out of [Pat71, Section 4]; Section 3 out of [Dod83]. The rest of the work is cherry-picked out of [ABP73] and [APS75] (see also [ABP75] for an errata to [ABP73]).