This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré-Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.

Table of Contents

Chapter 1 Differential Manifolds and Differentiable Maps

Chapter 2 The Derivatives of Differentiable Maps

Chapter 3 Fibre Bundles

Chapter 4 Differential Forms and Integration

Chapter 5 The Exterior Derivative

Chapter 6 De Rham Cohomology

Chapter 7 Degrees, Indices and Related Topics

Chapter 8 Lie Groups

chapter 9 A Rapid Course in Differential Analysis