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Topology and geometry. (English) Zbl 0791.55001
Graduate Texts in Mathematics. 139. New York: Springer-Verlag. xiv, 557 p. (1993).
My education in topology was informed by conversations and lectures. In this informal learning process there were continual surprises: The Alexander hornes sphere, the subtle differences between topological and smooth manifolds, the old method of intersection numbers, the fact that the Lefschetz number had a local sum formula, coincidence theory, etc. The book under review puts these key examples and points of view in a single textbook, and succeeds in conveying the intricate relationships between the algebraic machines of homology, cohomology and homotopy groups with the geometric and topological spaces and maps which are the main objects of study in Topology and Geometry.
The book throughout is carefully written. Most statements are proved, but with varying levels of sketchiness. Key central results have proofs with clear descriptions of the strategy of the proof. Many of these are the best I have seen and have given me deeper understanding, even though I have presented these proofs many times before. The proof of the braid exact sequence is a model for writing down diagram chasing. The cellular homology description is very clear, especially the discussion of the boundary operator.
The author has several different proofs of certain theorems such as the Lefschetz fixed point theorem or the Borsuk-Ulam theorem. This has the effect of making the various chapters semi-independent of each other, so the reader can dip into the middle of the book and still understand the arguments. Another effect is that the different proofs give a sense of stability to the results and illustrate the relationship between various points of view.
For example, take the Lefschetz fixed point theorem. In chapter IV, section 23, the author first proves it via the Hopf proof for finite simplicial complexes. Then he introduces ENR’s, which are developed in an appendix. A short proof extends the theorem to compact ENR’s and as corollaries he shows the theorem holds for compact CW complexes and also for topological manifolds. A corollary gives the fact that a vector field on a smooth manifold has a zero if the Euler characteristic of the manifold is not zero. The converse is true and is proved in Chapter VII using obstruction theory. Then in Chapter VI, section 12, the author takes up the Lefschetz fixed point theorem again using Lefschetz’s intersection number approach for manifolds. Not only does the fact that the graph of the function intersecting with the diagonal give new insight, but the other shoe is dropped, the Lefschetz number is a sum of local indices, one for each fixed point. Finally, in section 14, the Lefschetz coincidence theorem is proved. This result generalizes the fixed point theorem in section 12 and allows a very careful and useful discussion of umkehr maps, which the author calls transfer maps. These umkehr maps are very important, and this is by far the most complete and accessible discussion of them in any textbook.
The Hopf and the Lefschetz versions of the fixed point theorem are more than two different proofs, they are two different theorems, and yet they are the same theorem. This phenomenon occurs throughout topology, and the author has recorded it very well for his readers.
The book gives a good lean account of general topology in Chapter I. Differential manifolds are exposed in Chapter II. Included in this is transversality, fiber bundles and vector bundles. Chapter III discusses the fundamental group and covering spaces. Chapters IV, V, VI describe homology, cohomology, and Poincaré duality and their applications. It is here that the many different ways of constructing or defining spaces mentioned earlier in the book play an important role in examples and problems. These include Lie groups, homogeneous spaces, covering spaces, vector bundles, CW complexes and simplicial complexes. Chapter VII discusses homotopy theory and includes the Hurewicz theorem and the Whitehead theorem. Appendices deal with the additivity axiom, which is added to the usual Eilenberg-Steenrod axioms so that uniqueness holds for CW complexes instead of only finite CW complexes; set theory; Sard’s Lemma; direct limits; and ENR’s.
Not mentioned are spectral sequences, classifying spaces (except for Eilenberg-MacLane spaces), and the Puppe sequence for fibrations. De Rham cohomology and the attendent discussions of forms is included in Chapter V. The equivalence of de Rham and rational cohomology is proved as is Stokes theorem. So it is remarkable that no mention whatever is made of the integral formula for the degree of a map. This theorem, which Guillemin and Pollack call largely underrated, could be used to state the Gauss-Bonnet theorem and thus strengthen the ‘Geometry’ in the title of the book.
I predict that this book will be the standard text for Algebraic Topology for the next twenty years. For the expert it offers new clear arguments and new points of view, for the professor it gives foundations for many examples. For the beginner, it is a good book to own. It is probably too difficult for a beginner to read alone, but in combination with an instructor, or a less challenging book, the book will begin to grow on him. The exercises are fair ones, that is they are not large tracts of undigested theory. They are more difficult than they should be because sometimes the terms in the problem are not found in the text, for example “three torus”. But otherwise they are the best collection of problems I have seen for an Algebraic Topology text. It is clear that a student who can use this book effectively will be able to use Topology effectively.

55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes