This is the first in a series of posts about the Atiyah-Singer index theorem.

Let be finite-dimensional vector spaces (over , say), and consider the space of linear maps . To each , we can assign two numbers: the dimension of the kernel and the dimension of the cokernel . These are obviously nonconstant, and not even locally constant. However, the difference is constant in .

This was a trivial observation, but it leads to something deeper. More generally, let’s consider an operator (such as, eventually, a differential operator), on an infinite-dimensional Hilbert space. Choose separable, infinite-dimensional Hilbert spaces ; while they are abstractly isomorphic, we don’t necessarily want to choose an isomorphism between them. Consider a bounded linear operator .

Definition 1isFredholmif is “invertible up to compact operators,” i.e. there is a bounded operator such that and are compact.

In other words, if one forms the category of Hilbert spaces and bounded operators, and quotients by the ideal (in this category) of compact operators, then is invertible in the quotient category. It thus follows that adding a compact operator does not change Fredholmness: in particular, is Fredholm if and is compact.

Fredholm operators are the appropriate setting for generalizing the small bit of linear algebra I mentioned earlier. In fact,

Proposition 2A Fredholm operator has a finite-dimensional kernel and cokernel.

*Proof:* In fact, let be the kernel. Then if , we have

where is a “pseudoinverse” to as above. If we let range over the elements of of norm one, then the right-hand-side ranges over a compact set by assumption. But a locally compact Banach space is finite-dimensional, so is finite-dimensional. Taking adjoints, we can similarly see that the cokernel is finite-dimensional (because the adjoint is also Fredholm).

The space of Fredholm operators between a pair of separable, infinite-dimensional Hilbert spaces is interesting. For instance, it has the homotopy type of , so it is a representing space for K-theory. In particular, the space of its connected components is just . The stratification of the space of Fredholm operators is given by the *index*.

Definition 3Given a Fredholm operator , we define theindexof to be .

This is equivalently , so we see that the index is always zero for a self-adjoint operator. One can check various formal properties of the index, for instance that it is additive: .

In fact, the index turns out to be a *homotopy invariant.* We have:

Proposition 4The index is locally constant on the space of Fredholm operators.

A consequence is that changing a Fredholm operator by a compact operator doesn’t affect the index, i.e. for compact. That is because of the path connecting the two operators.

**Elliptic operators**

The important case of Fredholm operators relevant to the index theorem will be given by elliptic operators on a compact manifold. Let be a compact manifold, and vector bundles on . Let be a *differential operator* of some order on sections of to sections of . In local coordinates, this means that is just a matrix of (linear) partial differential operators.

Given , we can associate a *symbol* or linearization

where is the projection from the cotangent bundle. The idea is that the symbol of a differential operator should be , but stating in terms of cotangent bundles is the invariant way of doing so. In order to do this, let’s choose a cotangent vector lying over . To define the map

associated to , pick and extend to a section defined in a small neighborhood of . Choose a smooth function such that , and consider

One can check that this is well-defined and gives a homomorphism of bundles. The idea is that to a differential operator of degree , the symbol is some sort of linearization dependent on a cotangent vector (but homogeneous of degree in that cotangent vector ).

Definition 5The operator isellipticif is an isomorphism outside the zero section.

The idea is that an elliptic operator is something which looks like the Laplacian on , whose symbol is multiplication by and that is nonzero except at the origin.

The idea is that ellipticity is going to impose very strong properties on the operator . For instance, elements of the kernel of are going to be sections; this is a special case of elliptic regularity. I don’t have a great explanation for this, but essentially the main point seems to be that one can choose a pseudo-inverse—a “parametrix”—for . This will be an operator such that and are somehow close to the identity.

Now, of course, this is a bit tricky because would then presumably have to be a differential operator of negative order; how can you invert something like the Laplacian with another differential operator? One has to enlarge the space of operators and consider more generally *pseudodifferential* operators, which are allowed to have negative order. I don’t really want to get into all this here. However, here’s the idea:

**Intuition:** The map is something like a Fredholm operator, in that it admits a pseudoinverse such that are “close” to the identity.

Of course, are not Hilbert spaces, and one has to be more precise about one is actually talking about here. For our purposes, though, the intuition makes plausible the following facts.

**Fact 1:** The elliptic operator has a finite-dimensional kernel and cokernel, and thus an **index**.

**Fact 2:** The index is invariant under continuous perturbations and thus by homotopies of elliptic operators.

There is this general idea in mathematics that invariants which are discrete and locally constant should be given by topological data. For instance, in the theory of algebraic curves, there is essentially one discrete invariant: the genus (and then a continuous family of curves within that genus). But the genus is of course purely topological. So, now we have this invariant of elliptic operators on manifolds given by the index, and we’ve seen that it is discrete and locally constant. One might thus wonder whether using *topological* data, one might compute the index.

The Atiyah-Singer index theorem states that this is true, and that in fact, to compute the index, one has to take the element in K-theory (of ) defined by the symbol , take its Chern character, take a characteristic class of the manifold , multiply, and evaluate on the fundamental class. More transparently, the index theorem defines two homomorphisms

from the -theory of for a compact manifold to , one in terms of the indices of elliptic operators and one in terms of topological data (the Thom isomorphism, essentially), and states that they are equal.

September 24, 2012 at 12:15 pm

Nice post! I took me a moment’s reflection to be sure of this, so you may wish to make explicit that in the equation defining the index map, right before definition 5, e is a section of E that equals the given element on the fiber over x.

September 25, 2012 at 8:21 pm

Thanks for the correction!