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…)