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 ${K(TX)}$. (more…)

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

Let ${V, W}$ be finite-dimensional vector spaces (over ${\mathbb{C}}$, say), and consider the space ${\hom_{\mathbb{C}}(V, W)}$ of linear maps ${T: V \rightarrow W}$. To each ${T \in \hom_{\mathbb{C}}(V, W)}$, we can assign two numbers: the dimension of the kernel ${\ker T}$ and the dimension of the cokernel ${\mathrm{coker} T}$. These are obviously nonconstant, and not even locally constant. However, the difference ${\dim \ker T - \dim \mathrm{coker} T = \dim V - \dim W}$ is constant in ${T}$.

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 ${V, W}$; while they are abstractly isomorphic, we don’t necessarily want to choose an isomorphism between them. Consider a bounded linear operator ${T: V \rightarrow W}$.

Definition 1 ${T}$ is Fredholm if ${T}$ is “invertible up to compact operators,” i.e. there is a bounded operator ${U: W \rightarrow U}$ such that ${TU - I}$ and ${UT - I}$ 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 ${T}$ is invertible in the quotient category. It thus follows that adding a compact operator does not change Fredholmness: in particular, ${I + K}$ is Fredholm if ${V = W}$ and ${K: V \rightarrow V}$ is compact.

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

Proposition 2 A Fredholm operator ${T: V \rightarrow W}$ has a finite-dimensional kernel and cokernel.

Proof: In fact, let ${V' \subset V }$ be the kernel. Then if ${v' \in V'}$, we have $\displaystyle v' = UT v' + (I - UT) v' = (I - UT) v'$

where ${U}$ is a “pseudoinverse” to ${T}$ as above. If we let ${v'}$ range over the elements of ${v'}$ 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 ${V'}$ is finite-dimensional. Taking adjoints, we can similarly see that the cokernel is finite-dimensional (because the adjoint is also Fredholm). $\Box$

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

Definition 3 Given a Fredholm operator ${T: V \rightarrow W}$, we define the index of ${T}$ to be ${\dim \ker T - \dim \mathrm{coker} T}$. (more…)