Topology from the Differentiable Viewpoint by John Willard Milnor

By John Willard Milnor

This dependent e-book via amazing mathematician John Milnor, offers a transparent and succinct advent to at least one of crucial topics in sleek arithmetic. starting with simple thoughts corresponding to diffeomorphisms and tender manifolds, he is going directly to research tangent areas, orientated manifolds, and vector fields. Key recommendations similar to homotopy, the index variety of a map, and the Pontryagin development are mentioned. the writer provides proofs of Sard's theorem and the Hopf theorem.

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Rt/(a - a ) , the required difieomorphism h:S'-+M is defined by the formula h(cos 0, sin 0) = f(t) for a = g(t) for c < t < d, < t < 8. which is maximal in the sense that f cannot be extended over any larger interval as a parametrization by arc-length: it is only necessary to extend f as far as possible to the left and then as far as possible to the right. If M is not diffeoniorphic to A", we will prove that f is onto, and hence is a diffeoniorphism. For if the open set f ( I ) were not all of M , there would be a limit point z of f ( I ) in M - f ( I ) .

B) If y and z are regular values of g also, where Ilf(4 - 9(”)1/ < IIY APPENDIX CLASSIFYING 1-MAN1F O LD S - 41 for all x, prove that K f 1 ( Y ) , f-’(zN = Vg-l(Y), f-’W = G’(Y), g-‘(4). c) Prove that l(f-’(y), f - ’ ( z ) ) depends only on the honlotopy class of f , and does not depend on the choice of y and z. This integer H(f) = l(f-’(y), f - ’ ( z ) ) is called the Hopf invariant of f. ) PROBLEM 15. If the dimension p is odd, prove that H(f) a composition S2P-l = W E WILL prove the following result, which has been assumed in the text.

F : I U L-'(J) + f ( I ) U g ( J ) . + 57 Classifying I-manifolds * See Markov [19]; and also a forthcoming paper by Boone, Haken, and Poknaru in Fundamenta Mathematicae. BIBLIOGRAPHY following is a nliscellaneous list consisting of original sources and of reconimended textbooks. For the reader who wishes to pursue the study of differential topology, let me recommend Milnor [22], Alunkres [25], and Pontryagin [as]. The survey articles [23] and [32] should also prove useful. For background knowledge in closely related fields, let me recommend Hilton and Wylie [ 1I], Hu [16], Lang [18], de Rham [29], Steenrod [34], and Sternberg [35].

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