The index theorem

This chapter presents the proof of the geometrical index theorem for compatible elliptic complexes of Dirac type. It uses the Chern isomorphism between K – theory and cohomology to give a proof of the Atiyah-Singer index theorem in general. The chapter introduces the material concerning Clifford modules, and discusses the twisted spin complex, the twisted de Rham complex, the Yang-Mills complex, and the geometric index theorem. It also discusses the Riemann-Roch theorem and Lefschetz fixed point formulas. The chapter gives the preliminaries people shall need to prove the Atiyah-Singer index theorem in general, and shows that the eta function is regular at the origin. It proves the geometrical index theorem for manifolds with boundary and boundary conditions of Atiyah Patodi Singer type; the eta invariant enters in a crucial fashion. The chapter discusses the applications of the eta invariant when one takes coefficients in a flat bundle.