Differential Geometry and Topology

The present set of notes was developed during a course given in the 1965-66 academic year. It is hoped that, in spite of the rather fragmentary character of the notes, they will be of use to graduate students and others wishing to survey the material with which they are concerned. Our emphasis lies on the development and application of intersection theoretic methods for the calculation of various interesting topological invariants. Chapter 1 gives a summary of the usual basic generalities of ditferential topology. The fundamental lemma of Sard is proved and yields an elementary proof for the Brouwer fixed point theorem. Chapter 2 uses Sard's lemma, and the transversality arguments originally developed by Rene Thorn, to derive the classical connections between geometric intersection theory and algebraic homology on a rigorous basis. In the following chapter we use these intersection theoretic results to calculate the cohomology ring of the Grassmann spaces; the facts derived in this way form the basis for our subsequent discussion of.Whitney and Chem classes. In the second part of Chapter 3 we use intersection theoretic arguments, combined with arguments taken from Morse theory, to prove the Poincare duality theorem for differentiable manifolds. Chapter 4 summarizes various basic facts concerning fiber bundles, especially linear bundles. Chapter 5 gives an outline of the algebraic theory of spectral sequences. In Chapter 6 we combine the general principles discussed in the two preceding chapters with the intersection theoretic methods developed in Chapter 3 and discuss the characteristic classes of linear bundles. In the following chapter we develop various fundamental formulae of Riemannian geometry; then, combining these with the topological material developed in the preceding chapters, we derive the very interesting generalization of the Riemann-Roch theorem due to Chem.