Viewed as a whole, this is a very valuable book. In little over 200 pages, it presents a well organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology. Newly introduced concepts are usually well motivated, and often the historical development of an idea is described. There is an abundance of exercises which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text. As is stated in the preface, ``mathematical prerequisites have been kept to a minimum; the standard course in analysis and general topology is adequate preparation''. Still, beginning graduate students might find it hard to work their way through some of the more condensed proofs (e.g., that of Theorem 2.1.5). Another difficulty for the beginner may be caused by misprints or slightly confusing wording, which sometimes occur even in the final statement of results (e.g., in Theorems 2.2.3, 4.4.5, 9.3.3 and 9.3.11; also, Thom's theorem 7.2.4(c) is not quoted correctly). The book starts with an introduction which sketches some of the history and of the main ideas of differential topology. After first discussing submanifolds in Euclidean space, differentiable maps between them, and their tangent bundles and maps, Chapter 1 proceeds to introduce these and related basic concepts in full generality. Among other results one can already find a Whitney embedding theorem for compact manifolds here. Chapter 2 deals with the space Cr(M,N) of Cr-maps between two given manifolds M and N (with or without boundaries), its standard topologies (mainly the strong one), and with conditions that ensure that (closed) embeddings, immersions, submersions, proper maps, diffeomorphisms, Cs-maps etc., 0≤r<s≤∞, form open or dense subsets; in the process, bump functions, partitions of unity, and the technique of convolution are introduced. As an application, the existence and uniqueness of compatible Cs-structures on Cr-manifolds, 1≤r<s≤∞, are obtained. An alternate definition of the weak and strong topology on Cr(M,N) is given via jets, and the Baire property is established for weakly closed subspaces endowed with the strong topology. In a final section, various results on analytic approximations are discussed without proof. In Chapter 3, the Morse-Sard theorem is proved, and several strong transversality theorems are deduced. Applications include Brouwer's fixed point theorem and the high connectivity of Stiefel manifolds. Chapter 4 first develops the basic facts about vector bundles and their classification. Then, a section on orientations discusses the connections between (co-) orientibility of hypermanifolds and their separation behaviour, and deduces a non-embedding result. Existence and uniqueness of tubular neighborhoods (and collars) are proved without any use of sprays. Finally, the author establishes Whitney's theorem that every closed smooth manifold is diffeomorphic to an analytic submanifold of Euclidean space. It is a stated intention of the author to avoid the machinery of algebraic topology. In turn, he analyses some of its central concepts in great detail from the differential viewpoint. In Chapter 5 he discusses mapping degrees (and their classifying property for maps into the sphere), intersection numbers, Euler numbers and characteristics and their importance for normal and tangent vector fields. As has become traditional in this context, the chapter also contains a geometric proof of the fundamental theorem of algebra. Chapter 6 deals with Morse theory. A treatment of flows is also included, as well as a short digression into algebraic topology (to obtain the Morse inequalities and the full Hopf theorem on vectorfields). Chapter 7 introduces cobordism groups and gives their homotopy-theoretic description. Isotopy and diffeotopy questions are discussed in Chapter 8. The results then are applied to the question of the existence of a differential structure on the union of two smooth manifolds which have been glued together along boundary components. Finally, Chapter 9 presents a welcome treatment of compact surfaces and their classification via a Morse-theoretic approach. An extremely brief appendix summarizes a few basic facts of analysis and point set topology. Throughout the whole book, one idea is stressed very explicitly: the passage from local to global; it is systematized by various globalization theorems. Approximation theorems involving higher derivatives form another subject which gets more detailed attention here than in most other general books on differential topology.
Reviewer: Koschorke, Ulrich [form MathSciNet]