An introduction to Characteristic Classes 1

Preface Introduction Manifolds and pseudomanifolds Orientation Orientation of pseudomanifolds Orientation of manifolds Oriented double covering Poincaré isomorphism (manifolds) Boundary The Gauß map(s) Generalisation of the Gauß map Fibre bundles. Fibre bundles Vector bundles Examples of fibre bundles – real case Examples of fibre bundles – complex case The ``Euler–Poincaré'' characteristic The Greek period Maurolico – Descartes – Euler Maurolico (1494–1575) Descartes (1596–1650) Euler (1707–1783) Descartes' Theorem is equivalent to Euler Formula Proofs of Euler Formula The generalization: Euler–Poincaré characteristic Poincaré–Hopf Theorem The index of a vector field. Relation with the Gauß map Proof of Poincaré–Hopf Theorem The smooth case without boundary The smooth case with boundary Characteristic classes : smooth case General obstruction theory Index of an r-frame General obstruction theory Applications Stiefel–Whitney classes Chern classes Singular varieties Stratifications Angles Whitney stratifications Fundamental properties of Whitney stratifications Poincaré homomorphism Alexander isomorphism Cellular tubes Poincaré–Hopf Theorem (singular varieties) Introduction Why the radial vector fields ? Why the dual cells decomposition ? Radial vector fields Radial vector fields – Local construction Radial vector fields – Global construction Poincaré–Hopf Theorem for singular varieties. Schwartz classes Radial extension of frames Local radial extension of r-frames Global radial extension of r-frames Obstruction cocycles and classes MacPherson classes Nash transformation local Euler obstruction Constructible sets and functions Mather classes MacPherson classes Developments and perspectives Remarks and complements. About the fundamental Chern article. The polar varieties and Mather classes More developments of Chern classes for singular varieties Bivariant classes Other generalizations of classes in the singular case. Hirzebruch formalism The Euler local obstruction Some applications in other mathematical domains and in theoretical physics. Bibliography Index of Authors Index