Introduction CHAPTER �� De Rham Theory ��1 The de Rham Complex on Rn The de Rham complex Compact supports ��2 The Mayer-Vietoris Sequence The functor ��* The Mayer-Vietoris sequence The functor ��*c and the Mayer-Vietoris sequence for compact supports ��3 Orientation and Integration Orientation and the integral of a diferential form Stokes'theorem ��4 Poincare Lemmas The Poincare lemma for de Rham cohomology The Poincare lemma for compactly supported cohomology The degree of a proper map ��5 The Mayer-Vietoris Argument Existence of a good cover Finite dimensionality of de Rham cohomology Poincare duality on an orientable manifold The Kinneth formula and the Leray-Hirsch theorem The Poincare dual of a closed oriented submanifold ��6 The Thom Isomorphism Vector bundles and the reduction of structure groups Operations on vector bundles Compact cohomology of a vector bundle Compact vertical cohomology and integration along the fiber Poincare duality and the Thom class The global angular form, the Euler clas, and the Thom clas Relative de Rham theory ��7 The Nonorientable Case The twisted de Rham complex Integration of densities, Poincare duality, and the Thom isomorphism CHAPTER �� The Cech-de Rham Complex ��8 The Generalized Mayer-Vietoris Principle Reformulation of the Mayer-Vietoris sequence Generalization to countably many open sets and applications ��9 More Examples and Applications of the Mayer-Vietoris Principle Examples: computing the de Rham cohomology from the combinatorics of a good cover Explicit isomorphisms between the double complex and de Rham and Cech The tictac-toe proof of the Kinneth formula ��10 Presheaves and Cech Cohomology Presheaves Cech cohomology ��11 Sphere Bundles Orientability The Euler class of an oriented sphere bundle The global angular form Euler number and the isolated singularities of a section Euler characteristic and the Hopf index theorem ��12 The Thom Isomorphism and Poincare Duality Revisited The Thom isomorphism Euler class and the zero locus of a section A tic-tac-toe lemma Poincare duality ��13 Monodromy When is a locally constant presheaf constant? Examples of monodromy CHAPTER �� Spectral Sequences and Applications ��14 The Spectral Sequence of a Filtered Complex Exact couples The spectral sequence of a fltered complex The spectral sequence of a double complex The spectral sequence of a fiber bundle Some applications Product structures The Gysin sequence Leray's construction ��15 Cohomology with Integer Coefficients Singular homology The cone construction The Mayer-Vietoris sequence for singular chains Singular cohomology The homology spectral sequence ��16 The Path Fibration The path fibration The cohomology of the loop space of a sphere ��17 Review of Homotopy Theory Homotopy groups The relative homotopy sequence Some homotopy groups of the spheres Attaching cells Digression on Morse theory The relation between homotopy and homology ��3(S2) and the Hopf invariant ��18 Applications to Homotopy Theory Eilenberg-MacLane spaces The telescoping construction The cohomology of K (Z, 3) The transgression Basic tricks of the trade Postnikov approximation Computation of ��4(S3) The Whitehead tower Computation of ��5(S3) ��19 Rational Homotopy Theory Minimal models Examples of Minimal Models The main theorem and applications CHAPTER �� Characteristic Classes ��20 Chern Classes of a Complex Vector Bundle The first Chern class of a complex line bundle The projectivization of a vector bundle Main properties of the Chern classes ��21 The Splitting Principle and Flag Manifolds The splitting principle Proof of the Whitney product formula and the equality of the top Chern class and the Euler class Computation of some Chern classes Flag manifolds ��22 Pontrjagin Classes Conjugate bundles Realization and complexification The Pontrjagin classes of a real vector bundle Application to the embedding of a manifold in a Euclidean space ��23 The Search for the Universal Bundle The Grassmannian Digression on the Poincare series of a graded algebra The classification of vector bundles The infinite Grassmannian Concluding remarks References List of Notations Index