Textos De Matemática Introduction to Characteristic Classes and Index Theory

This book is based on a course given by the author at the university ofLisbon during the academic year 1997–1998. Its aim is to give the readeran idea of how the theory of characteristic classes can be applied to solveindex problems. Starting from the Lefschetz fixed point theorem and itsapplication to the computation of the Euler-Poincaré characteristic of acompact orientable manifold, we first develop the theory of Euler classesof orientable manifolds and real vector bundles. Then, we study Stiefel-Whitney classes and the general modulo 2 characteristic classes of real vectorbundles. Similar considerations for complex vector bundles lead us to theChern classes. We conclude the part devoted to characteristic classes by astudy of global and local Chern characters. The rest of the book is thencentered around the Riemann-Roch theorem. We present first a very simpleproof which works for compact complex curves and allows us to make linkswith the original results of Riemann and Roch. Then, we treat in details thecase of compact complex projective manifolds by more advanced methods. Faculdade de Ciências — Departamento de MatemáticaRua Ernesto de Vasconcelos Bloco C1, 3 Piso1749-016 LISBOA — PORTUGAL