Differential Topology

• 1 : Manifolds and Maps.- 0. Submanifolds of ?n+k.- 1. Differential Structures.- 2. Differentiable Maps and the Tangent Bundle.- 3. Embeddings and Immersions.- 4. Manifolds with Boundary.- 5. A Convention.- 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.- 3 : Transversality.- 1. The Morse-Sard Theorem.- 2. Transversality.- 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.- 5 : Degrees, Intersection Numbers, and the Euler Characteristic.- 1. Degrees of Maps.- 2. Intersection Numbers and the Euler Characteristic.- 3. Historical Remarks.- 6 : Morse Theory.- 1. Morse Functions.- 2. Differential Equations and Regular Level Surfaces.- 3. Passing Critical Levels and Attaching Cells.- 4. CW-Complexes.- 7 : Cobordism.- 1. Cobordism and Transversality.- 2. The Thorn Homomorphism.- 8 : Isotopy.- 1. Extending Isotopies.- 2. Gluing Manifolds Together.- 3. Isotopies of Disks.- 9 : Surfaces.- 1. Models of Surfaces.- 2. Characterization of the Disk.- 3. The Classification of Compact Surfaces.

M.W. Hirsch Differential Topology "A very valuable book. In little over 200 pages, it presents a well-organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology. Newly introduced concepts are usually well motivated, and often the historical development of an idea is described. There is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text. "-MATHEMATICAL REVIEWS