This book presents some of the basic topological ideas used in studying differentiable manifolds and maps. Mathematical prerequisites have been kept to a minimum; the standard course in analysis and general topology is adequate preparation. An appendix briefly summarizes some of the background material. In order to emphasize the geometrical and intuitive aspects of differential topology, I have avoided the use of algebraic topology, except in a few isolated places that can easily be skipped. For the same reason I make no use of differential forms or tensors.

Introdttction

Chapter 1: Manifolds and Maps

0. Submanifolds of Rn k

1. Differential Structures

2. Differentiable Maps and the Tangent Bundle

3. Embeddings and Immersions

4. Manifolds with Boundary

5. A Convention

Chapter 2: Function Spaces

1. The Weak and Strong Topologies on Cr M,N

2. Approximations

3. Approximations on -Manifolds and Manifold Pairs

4. Jets and the Baire Property

5. Analytic Approximations

Chapter 3: Transversality

1. The Morse-Sard Theorem

2. Transversality

Chapter 4: Vector Bundles and Tubular Neighborhoods

1. Vector Bundles

2. Constructions with Vector Bundles

3. The Classification of Vector Bundles

4. Oriented Vector Bundles

5. Tubular Neighborhoods

6. Collars and Tubular Neighborhoods of Neat Submanifolds

7. Analytic Differential Structures

Chapter 5: Degrees, Intersection Numbers, and the Euler Characteristic

1. Degrees of Maps

2. Intersection Numbers and the Euler Characteristic

3. Historical Remarks

Chapter 6: Morse Theory

1. Morse Functions

2. Differential Equations and Regular Level Surfaces

3. Passing Critical Levels and Attaching Cells

4. CW-Complexes

Chapter 7: Cobordism

1. Cobordism and Transversality

2. The Thom Homomorphism

Chapter 8: Isotopy

1. Extending Isotopies

2. Gluing Manifolds Together

3. Isotopies of Disks

Chapter 9: Surfaces

1. Models of Surfaces

2. Characterization of the Disk

3. The Classification of Compact Surfaces

Bibliography

Appendix

Index