Front Cover; Vector Bundles: Foundations and Stiefel-Whitney Classes; Copyright Page; Contents; Preface; Introduction; Chapter I. Base Spaces; 0. Introduction; 1. The Category of Base Spaces; 2. Some Simplicial Spaces; 3. More Simplicial Spaces; 4. Weak Simplicial Spaces; 5. CW Spaces; 6. Smooth Manifolds; 7. Grassmann Manifolds; 8. Some More Coverings; 9. The Mayer-Vietoris Technique; 10. Remarks and Exercises; Chapter II. Fiber Bundles; 0. Introduction; 1. Fibre Bundles and Fiber Bundles; 2. Coordinate Bundles; 3. Bundles over Contractible Spaces; 4. Pullbacks along Homotopic Maps

5. Reduction of Structure Groups6. Polar Decompositions; 7. The Leray-Hirsch Theorem; 8. Remarks and Exercises; Chapter III. Vector Bundles; 0. Introduction; 1. Real Vector Bundles; 2. Whitney Sums and Products; 3. Riemannian Metrics; 4. Sections of Vector Bundles; 5. Smooth Vector Bundles; 6. Vector Fields and Tangent Bundles; 7. Canonical Vector Bundles; 8. The Homotopy Classification Theorem; 9. More Smooth Vector Bundles; 10. Orientable Vector Bundles; 11. Complex Vector Bundles; 12. Realifications and Complexifications; 13. Remarks and Exercises; Chapter IV. Z/2 Euler Classes

Chapter VI. Unoriented Manifolds0. Introduction; 1. Z/2 Fundamental Classes; 2. Z/2 Poincaré-Lefschetz Duality; 3. Multiplicative Z/2 Classes of RPn; 4. Nonimmersions and Nonembeddings; 5. Stiefel-Whitney Numbers; 6. Stiefel-Whitney Genera; 7. Z/2 Thom Forms; 8. The Thom-Wu Theorem; 9. Remarks and Exercises; References; Glossary of Notation; Index; Pure Applied Mathematics