Lectures on the H-Cobordism Theorem (Princeton Legacy Library) by John Milnor (PDF)

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These lectures provide students and specialists with preliminary and valuable information from university courses and seminars in mathematics. This set gives new proof of the h-cobordism theorem that is different from the original proof presented by S. Smale.Originally published in 1965.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

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⭐Milnor’s “Lectures on the h-cobordism theorem” consists of a proof and applications of the h-cobordism theorem, an important technical result that, among other things, leads immediately to a proof of the Poincare conjecture for smooth manifolds of dimension >= 5. The theorem was originally proved by Smale in 1962 (part of the basis for his Fields Medal) using handlebody techniques, but in this book Milnor presents a (partially) different proof using Morse theoretic lemmas due to Morse and Barden. Along the way, a number of theorems and techniques in differential topology are used or derived (including the Whitney trick, finger moves, bicollarings, surgery, extensions of embeddings and isotopies, smooth structures on unions, Poincare duality), which makes this valuable as much for the methods as for the final result. Definitely it is one of the best books for learning how to actually prove things in topology, mixing in Morse theory, Riemannian geometry, and algebraic topology, too.The h-cobordism theorem states that an n-dim simply connected cobordism W between 2 simply connected (n-1)-dim manifolds V,V’ that are each homotopy equivalent to W (which is the definition of an h-cobordism) is isomorphic to a product, with the isomorphism depending on the category of manifolds with which one is working. In particular, if such a W exists, then V and V’ must be diffeomorphic. This book, as well as Smale’s original proof, only deals with the smooth case, where the theorem is shown to be true for all n except possibly 4 or 5 (with the n=4 case being equivalent to the original Poincare conjecture that has now been proved using different techniques). PL and topological versions of the theorem also has been proved, as well as one with somewhat relaxed requirements (the s-cobordism theorem), but none of these extensions are explored in the book, although there is some mention of them. The proof basically proceeds by finding a Morse function on the cobordism for which it can be decomposed into a union of “elementary cobordisms,” each with a single critical value and index. The Morse function is then perturbed in a series of steps so that the critical points can be cancelled, eventually leaving a cobordism with no critical points, which is diffeomorphic to a product. In the handlebody version of the proof (not presented here, but see Kosinski’s

⭐), this decomposition is equivalent to a handlebody decomposition, which is then manipulated by sliding and cancelling handles and until they are all removed. The dimensional restriction (n>=6) primarily comes from the application of the Whitney trick – in lower dimensions there is not enough room to remove algebraically cancelling intersections between embedded spheres – and it is this limitation that accounts for the difficulties in 4-dimensional smooth topology.The book is accessible to graduate students who have taken 1st-year courses in differential and algebraic topology. He cites such texts as Munkres’

⭐, his own lectures notes (which appear in

⭐), and de Rham’s

⭐for the differential topology, but the reader may prefer more modern treatments, such as Guillemin & Pollack’s

⭐or Hirsch’s

⭐. For the algebraic topology he often mentions results (such as Alexander duality) without citing any reference, so you’ll need a good background in homology theory (I’d recommend Dold’s

⭐or Greenberg & Harris’s

⭐). One definite prerequisite is the first part of Milnor’s own

⭐, which provides the foundation for the proof. In fact, this book is good companion to “Morse Theory,” in the sense that many important and basic results in the subject that were surprisingly left out of “Morse Theory” are proved here.Overall the book is relatively clean, with only some typos, chiefly in the form of missing or mistaken subscripts and indices. This only really becomes a problem at the end of Chapter 5, where a key subscript 1 is omitted in the last equation of page 65 (should be E_{1} instead of E) and then repeatedly for the constant k on the next page (i.e., k_{1} instead of K). On pg. 101 he states that he has proved the case index = 1 when he means index = 0 (but this should be obvious), and in a number of places a theorem is cited by the wrong number, in particular, 3.14 and 3.4 are switched a lot.Despite its age this remains one of the most significant books in the theory of manifolds. The aforementioned book by Kosinski, which you should certainly read as well, is the only other one that I can think of that contains the complete proof of the h-cobordism theorem, but the method is different. More recently, Matsumoto’s

⭐copies much of the material from this book, including identical proofs and diagrams for Theorem’s 4.1 and 5.6, and mixes handlebody arguments with the analytic ones, but stops well short of even stating the h-cobordism theorem. The final few pages contain applications of the theorem to prove various results about smooth manifolds that are contractible or homotopy spheres; cf. Kosinski for more extensive results of this type.

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