Transcription of TOPOLOGY FROM DIFFERENTIABLE VIEWPOINT
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TOPOLOGY FROM DIFFERENTIABLE VIEWPOINT
TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT By John W. Milnor Princeton University Based on notes by David W. Weaver The University Press of Virginia Charlottesville PREFACE THESE lectures were delivered at the University of Virginia in December 1963 under the sponsorship of the Page-Barbour Lecture Foundation. They present some topics from the beginnings of TOPOLOGY , centering about L. E. J. Brouwer s definition, in 1912, of the degree of a mapping. The methods used, however, are those of differential TOPOLOGY , rather than the combinatorial methods of Brouwer. The concept of regular value and the theorem of Sard and Brown, which asserts that every smooth mapping has regular values, play a central role. To simplify the presentation, all manifolds are taken to be infinitely DIFFERENTIABLE and to be explicitly embedded in euclidean space. A small amount of point-set TOPOLOGY and of real variable theory is taken for granted.
6 51. Smooth maniJolds smooth map f:M+N with f(x) = y.The derivative dfz : TM, 3 TN, is defined as follows.Since J is smooth there exist an open set W con- taining x and a smooth map F:W4R1 that coincides with f on W (7 M. De&e df,(v) to be equal to dF,(v) for all v e TM,. To justify this definition we must prove that dF,(v) belongs to TN, and that it does not depend on the …
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