This elegant book by distinguished mathematician John Milnor, provides a clear and 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. The author presents proofs of Sard's theorem and the Hopf theorem.
Preface1Smooth manifolds and smooth maps1Tangent spaces and derivatives2Regular values7The fundamental theorem of algebra82The theorem of Sard and Brown10Manifolds with boundary12The Brouwer fixed point theorem133Proof of Sard's theorem164The degree modulo 2 of a mapping20Smooth homotopy and smooth isotopy205Oriented manifolds26The Brouwer degree276Vector fields and the Euler number327Framed cobordism; the Pontryagin construction42The Hopf theorem508Exercises52AppClassifying 1-manifolds55Bibliography59Index63