Mathematician John Milnor provides a succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. Milnor also presents proofs of Sard's theorem and the Hopf theorem Provides a clear introduction to one of the important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, this book goes on to examine tangent spaces, oriented manifolds, and vector fields. It discusses concepts such as homotopy, the index number of a map, and the Pontryagin construction. Preface vii 1. Smooth manifolds and smooth maps 1(9) Tangent spaces and derivatives 2(5) Regular values 7(1) The fundamental theorem of algebra 8(2) 2. The theorem of Sard and Brown 10(6) Manifolds with boundary 12(1) The Brouwer fixed point theorem 13(3) 3. Proof of Sard's theorem 16(4) 4. The degree modulo 2 of a mapping 20(6) Smooth homotopy and smooth isotopy 20(6) 5. Oriented manifolds 26(6) The Brouwer degree 27(5) 6. Vector fields and the Euler number 32(10) 7. Framed cobordism; the Pontryagin construction 42(10) The Hopf theorem 50(2) 8. Exercises 52(3) Appendix: Classifying 1-manifolds 55(4) Bibliography 59(4) Index 63