This is migrated by math.stackexchange as I did not receive an answer. I do not know if it is too naive for this site.
I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing pseudodifferential operators), I am wondering how practical this theorem is. The following question is general in nature:
Suppose we have a PDE given by certain elliptic differential operator, how computable is the topological index of this differential operator? If we give certain boundary conditions on the domain (for example, the unit circle with a point removed, a triangle, a square, etc), can we extend the $K$-theory proof to this case? I know the $K$-theory rhetoric proof in literature, but to my knowledge this proof is highly abstract and does not seem to be directly computable. Now if we are interested in the analytical side of things, but cannot compute the analytical index directly because of analytical difficulties, how difficult is it to compute the topological index instead? It does not appear obvious to me how one may compute the Todd class or the chern character in practical cases.
The question is motiviated by the following observation:
Given additional algebraic structure (for example, if $M$ is a homogeneous space, $E$ is a bundle with fibre isomorphic to $H$) we can show that Atiyah-Singer can be reduced to direct algebraic computations. However, what if the underlying manifold is really bad? What if it has boundaries of codimension 1 or higher?How computable is the index if we encounter an analytical/geometrical singularity?(which appears quite often in PDE).
On the other hand, suppose we have a manifold with corners and we know a certain operator's topological index. How much hope do we have in recovering the associated operator by recovering its principal symbol? Can we use this to put certain analytical limits on the manifolds(like how bad an operator on it could be if the index is given)?
Best Answer
As Johannes Ebert said, it's best if at first you stay away from boundary value problems. For some elliptic operators there may not even exist local boundary conditions satisfying the conditions guaranteeing Fredholmness; the Dolbeault operator is such an example. Therefore often one has to deal with pseudo-local boundary value problems such as the Atiyah-Patodi-Singer boundary condition.
A pseudo-diff operator on a closed manifold is Fredholm iff it is elliptic and the index is determined by the principla symbol, which is an element in the $K$-theory of a commutative algebra. For a boundary value problem Fredholmness is a much more subtle issue. It imposes restrictions on the type of boundary value conditions allowed (think Lopatinskii-Schapiro) and as Boutet de Monvel has shown almost four decades ago, the index is determined by the symbol of the problems which is an element in the $K$-theory of a certain non-commutative algebra; see e.g. this paper and the references therein.
The index of an operator on a closed manifold is eminently computable. In most geometric applications it can be reduced to the computation of the indices of a few classical operators: the spin and spin-c Dirac operators, the Hodge-de Rham operator (leading to the Gauss-Bonnet and the Hirzebruch signature operator), Dolbeault operator (leading to the Riemann-Roch-Hirzebruch formula).
The reduction to these cases requires good knowledge of representation theory, differential geometry and extensive familiarity with the theory of characteristic classes.
For manifolds with corners things are even more nebulous; same for most noncompact manifolds. In any case, to paraphrase one of my former professors, if you can describe a PDE problem explicitly, and you can prove its Fredholmness, then the index theorem will give you an answer as explicit as your question.