Topology from the Differentiable Viewpoint

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.

<TR>Preface<TR>1Smooth manifolds and smooth maps1<TR>Tangent spaces and derivatives2<TR>Regular values7<TR>The fundamental theorem of algebra8<TR>2The theorem of Sard and Brown10<TR>Manifolds with boundary12<TR>The Brouwer fixed point theorem13<TR>3Proof of Sard's theorem16<TR>4The degree modulo 2 of a mapping20<TR>Smooth homotopy and smooth isotopy20<TR>5Oriented manifolds26<TR>The Brouwer degree27<TR>6Vector fields and the Euler number32<TR>7Framed cobordism; the Pontryagin construction42<TR>The Hopf theorem50<TR>8Exercises52<TR>AppClassifying 1-manifolds55<TR>Bibliography59<TR>Index63