The Index Theorem

Abstract

Let X be a smooth compact manifold without boundary, and E, F be smooth complex vector bundles over X. Then if P : Γ(E) → Γ(F) is an elliptic operator, its principal symbol σ(P) is represented as a smooth section of the bundle Hom(π*E, π*F) over T*X $$\sigma \left( P \right):T*X \to Hom\left( {\pi *E,\pi *F} \right)$$, where π : T*X → X is the cotangent bundle of X, such that σ(P)(v) is an isomorphism whenever v lies outside some compact subset of T*X. Therefore σ(P) represents an equivalence class of compactly supported 1-complexes $$\sigma \left( P \right) = \left[ {\pi *E,\pi *F,\sigma \left( P \right)} \right] \in {K_c}\left( {T*X} \right)$$. We shall identify T*X and TX using a Riemannian metric in X. Thus we shall consider σ(P) as an element of K c (TX).