Differential Topology | Morris W. Hirsch

§"There has long been a need for introductory text on differential topology, and the reviewer must be one of many who have contemplated writing such a book. The appearance of a book by such an appropriate author as Morris Hirsch promises to fill this gap. The philosophy, well expressed in the introduction, emphasizing the origins and applications of smooth manifolds, augurs well, as does the list of contents which proceeds through the commonly agreed key ideas of the subject. After the two introductory chapters, we are taken reasonably thoroughly through transversality, vector bundles and tubular neighbourhoods, degrees and the Euler characteristic, Morse theory, cobordism and isotopy; and the book closes by applying these basic ideas to classify compact surfaces up to diffeomorphism (unfortunately there is a slip in the statement of the final result: one needs the surfaces to be both orientable or both nonorientable). §It is an attractive book to read, through the absence of presision will irritate some; and in some ways goes further than one might expect (a nice proof of Sard's theorem, and non-trivial discussion of the real analytic case). " C. T. C. Wall 1 "... a most readable book. The author has taken considerable trouble with the exposition and has improved on previous accounts in many ways. There are some good diagrams and plenty of exercises ... " Bulletin of the American Mathematical Society 2