Let ${M}$ be a compact manifold, ${E, F}$ vector bundles over ${M}$. Last time, I sketched the definition of what it means for a differential operator

$\displaystyle D: \Gamma(E) \rightarrow \Gamma(F)$

to be elliptic: the associated symbol

$\displaystyle \sigma(D): \pi^* E \rightarrow \pi^* F, \quad \pi: T^* X \rightarrow X$

was required to be an isomorphism outside the zero section. The goal of the index theorem is to use this symbol ${\sigma(D)}$ to compute the index of ${D}$, which we saw last time was a well-defined number

$\displaystyle \mathrm{index} D = \dim \ker D - \dim \mathrm{coker} D \in \mathbb{Z}$

invariant under continuous perturbations of ${D}$ through elliptic operators (by general facts about Fredholm operators).

The main observation is that ${D}$, in virtue of its symbol, determines an element of ${K(TX)}$. (Henceforth, we shall identify the tangent bundle ${TX}$ with the cotangent bundle ${T^*X}$, by choice of a Riemannian metric; the specific metric is not really important since ${K}$-theory is a homotopy invariant.) In fact, we have that ${K(TX)}$ is the (reduced) ${K}$-theory of the Thom space, so it is equivalently ${K(BX, SX)}$ for ${BX}$ the unit ball bundle and ${SX}$ the unit sphere bundle. But we have seen that to give an element of ${K(BX, SX)}$ is the same as giving a pair of vector bundles on ${BX}$ together with an isomorphism on ${SX}$, modulo certain relations.

Observation: The symbol of an elliptic operator determines an element in ${K(TX)}$.

1. The analytical index

There is still a problem, though. The symbol of an elliptic operator corresponds to a map ${\pi^* E \rightarrow \pi^* F}$, but on each fiber it is a homogeneous polynomial. By contrast, if we are interested in the ${K}$-theory ${K(BX, SX)}$, we should really be interested in maps which are not given by homogeneous polynomials.

This means that one has to extend the class of operators. Rather than simply considering differential operators, one needs to consider pseudodifferential operators. These pseudodifferential operators can still be given a well-defined symbol, so it makes sense to talk about ellipticity. But the benefit is that the symbol doesn’t have to be polynomial. As a result, any element in ${K(BX, SX)}$ comes from a pseudodifferential operator.

So, the main observation is:

Observation: The symbol of a pseudodifferential operator defines an element in ${K(BX, SX)}$. All elements in ${K(BX, SX)}$ come from some pseudodifferential operator.

A pseudodifferential operator which is elliptic still satisfies the regularity theorem. In fact, the regularity theorem can be proved by constructing a “quasi-inverse” or parametrix (itself a pseudodifferential operator). So an elliptic pseudodifferential operator should be thought of as defining a Fredholm operator ${\Gamma(E) \rightarrow \Gamma(F)}$ (again, in a sense which can be made precise using Sobolev spaces). In particular, we can associate to an elliptic pseudodifferential operator an “index” as before.

Here is the main result that motivates the index theorem:

Theorem 1 The index of an elliptic pseudodifferential operator depends only on the associated element in ${K(BX, SX)}$, so we get a map, called the analytical index

$\displaystyle K(BX, SX) \rightarrow \mathbb{Z}.$

The idea is that the class in ${K(BX, SX)}$ itself determines the homotopy class of the symbol in the space of elliptic symbols, by a general construction in K-theory. However, a homotopy class of symbols can be lifted to a homotopy of operators, and a homotopy of operators leaves invariant the index because the index is locally constant on the space of Fredholm operators.

2. The topological index

Since the analytical index defines a map

$\displaystyle K(TX ) = K(BX, SX) \rightarrow \mathbb{Z} ,$

for any compact manifold ${X}$, one might attempt to construct it purely topologically. In fact, this can be done, using the Thom isomorphism.

The starting point is to make ${K(TX)}$ into a covariant functor for imbeddings. Let ${i: X \rightarrow Y}$ be a closed imbedding of manifolds. We will define a map

$\displaystyle i_!: K(TX) \rightarrow K(TY).$

At the level of elliptic operators, we are “pushing forward” an elliptic operator from ${X}$ to ${Y}$; the crux of Atiyah-Singer’s proof is that this push-forward operation preserves the analytical index.

In order to do this, we can use the tubular neighborhood theorem to find a neighborhood ${N \subset Y}$ of ${X}$. Here ${N}$ is the normal bundle to ${X}$ in ${Y}$. Consequently, we have a map

$\displaystyle j_!: K(TX) \rightarrow K(TN)$

which is given by the Thom isomorphism. In fact, let ${\pi: TX \rightarrow X}$ be the projection. Then the inclusion ${TX \rightarrow TN}$ can be identified with the inclusion of the zero section in the complex vector bundle ${\pi^*N \oplus \pi^*N \rightarrow TX}$. As a result, we have a Thom pushforward map as claimed. Next, we use the open imbedding ${N \hookrightarrow Y}$ to push-forward ${K(TN) \rightarrow K(TY)}$; compactly supported K-theory is covariantly functorial for open imbeddings, so this can be done. Together, we have defined the map ${i_!: K(TX) \rightarrow K(TY)}$.

Now, we have the push-forward maps, and we can use them to construct the index. If ${X}$ is any compact manifold, the Whitney imbedding theorem allows us to find an imbedding

$\displaystyle i: X \hookrightarrow S^N$

for some large ${N}$. As a result, we get a map

$\displaystyle i_!: K(TX) \rightarrow K(TS^N).$

But on the other hand, the inclusion ${P \hookrightarrow S^N}$ for ${P}$ a point defines equally a map

$\displaystyle j_!: K(\ast) \rightarrow K(TS^N)$

which is an isomorphism in view of the Bott periodicity theorem. Note that ${K(\ast)}$ is canonically ${\mathbb{Z}}$. If we take the composite, we can define:

Definition 2 The topological index is the map ${K(TX) \rightarrow \mathbb{Z}}$ given by ${j_!^{-1} \circ i_!}$.

It requires a bit of checking that this is actually well-defined and independent of the imbedding; this follows without too much trouble from the transitivity of the Thom isomorphism, though.

3. The index theorem

The index theorem can now be stated very simply:

Theorem 3 The topological and analytical indices coincide as maps ${K(TX) \rightarrow \mathbb{Z}}$.

This gives a recipe for calculating the index of an elliptic differential (or pseudodifferential) operator ${D: \Gamma(E) \rightarrow \Gamma(F)}$ on a compact manifold ${X}$. Form the associated symbol homomorphism ${\sigma(D): \pi^*E \rightarrow \pi^* F}$ on the (co)tangent bundle ${\pi: TX \rightarrow X}$. Use the “difference bundle” construction to obtain an element in ${K(TX) = K(BX, SX)}$. Then, compute the push-forward of this to ${K(T \mathbb{R}^n)}$ for some imbedding ${X \hookrightarrow \mathbb{R}^n}$, and then figure out which multiple of the (Bott) generator of ${K(T \mathbb{R}^n)}$ it is. That number is the index of your operator.

Unfortunately this recipe (though elegant) might be a bit complicated, and one probably wants something easier to compute with: for instance, one might have given the characteristic classes of ${E}$ and ${F}$, and one might wish to obtain the index as a number from them. In the third paper in the IEO series, Atiyah and Singer translated this into a formula in cohomology instead of K-theory. Here the point is essentially to use the Chern character map

$\displaystyle K(X) \rightarrow H^{**}(X; \mathbb{Q})$

for a space ${X}$, and then to keep straight how the Thom isomorphisms in K-theory and cohomology compare to each other. This will be the subject of a future post.

4. The proof (very sketchy)

I don’t want to present anything near the complete proof on this blog, as there a number of technical details in the analysis (which I don’t understand too well). Let me try to sketch the approach.

The topological index is a homomorphism

$\displaystyle t: K(TX) \rightarrow \mathbb{Z}$

defined for every compact manifold. Essentially from its definition, one can quickly check that if ${i: X \rightarrow Y}$ is an imbedding, then there is a commutative diagram

Moreover, the topological index when ${X = \ast}$ is just the dimension map ${K(T\ast) \rightarrow \mathbb{Z}}$. These two properties essentiallycharacterize the topological index. If we had another transformation

$\displaystyle \widetilde{t}: K(TX) \rightarrow \mathbb{Z}$

which commuted with the push-forward maps ${i_!}$ above, and was the identity when ${ X = \ast}$, it would have to be the topological index. This is essentially formal because ${\widetilde{t}}$ is then determined by its values on the spheres, and on those it is determined by its action on ${\ast}$ in view of Bott periodicity.

So most of IEO 1 is spent on establishing that the analytical index satisfies these two properties.

5. Equivariance

The index theorem as in IEO 1 is actually stated in equivariant generality. Let ${G}$ be a compact Lie group acting on a compact manifold ${X}$. Then ${TX}$ is also a ${G}$-space, and one can define the topological index

$\displaystyle K_G(TX) \rightarrow R(G)$

to the representation ring ${R(G)}$ of ${G}$. This is done in the same way, except now one imbeds ${X}$ inside an equivariant euclidean space (that is, a ${G}$-representation) using the equivariant version of the Whitney imbedding theorem. Then, one needs to invoke the equivariant version of Bott periodicity, which is actually quite difficult in general. (Here one only needs it when ${G}$ is abelian, which is easier.)

Similarly, one can define the analytical index

$\displaystyle K_G(TX) \rightarrow R(G)$

which is given as follows. An element of ${K_G(TX)}$ can be represented (via the symbol) by an equivariant elliptic pseudodifferential operator ${D}$. Given ${D}$, one can define the equivariant index ${[\ker D ] - [\mathrm{coker} D]}$; since ${\ker D, \mathrm{coker} D}$ are ${G}$-representations by equivariance of ${D}$, this makes sense.

As before, the main result is:

Theorem 4 The analytical and topological indices coincide as maps ${K_G(TX) \rightarrow R(G)}$.

In IEO 1, the equivariance is built into the approach from the ground up, and in fact it is crucial to the proof. See this MathOverflow question. The reason that in the paper, equivariance is important is precisely establishing that the index commutes with the lower shriek maps ${i_!}$ above. In fact, the idea is to interpret ${i_!}$ as a suitable “product” with an equivariant K-theory class on the sphere. Since (as Atiyah-Singer show), the analytical index preserves products, one thus just has to compute some analytical indices on the sphere.

In most of the common applications, it seems that the non-equivariant version is all that one needs, though.