Let be a compact manifold, vector bundles over . Last time, I sketched the definition of what it means for a differential operator

to be *elliptic*: the associated symbol

was required to be an isomorphism outside the zero section. The goal of the index theorem is to use this symbol to compute the *index* of , which we saw last time was a well-defined number

invariant under continuous perturbations of *through elliptic operators* (by general facts about Fredholm operators).

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

**Observation:** The symbol of an elliptic operator determines an element in . (more…)