Algebraic Topology by Edwin H. Spanier (PDF)

Description

This book surveys the fundamental ideas of algebraic topology. The first part covers the fundamental group, its definition and application in the study of covering spaces. The second part turns to homology theory including cohomology, cup products, cohomology operations and topological manifolds. The final part is devoted to Homotropy theory, including basic facts about homotropy groups and applications to obstruction theory.

User’s Reviews

Editorial Reviews: From the Back Cover The reader of this book is assumed to have a grasp of the elementary concepts of set theory, general topology, and algebra. Following are brief summaries of some concepts and results in these areas which are used in this book. Those listed explicitly are done so either because they may not be exactly standard or because they are of particular importance in the subsequent text.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book was an incredible step forward when it was written (1962-1963). Lefschetz’s

⭐was the main text at the time. A large number of other good to great books on the subject have appeared since then, so a review for current readers needs to address two separate issues: its suitability as a textbook and its mathematical content.I took the course from Mr. Spanier at Berkeley a decade after the text was written. He was a fantastic teacher – one of the two best I’ve ever had (the other taught nonlinear circuit theory). We did NOT use this text, except as a reference and problem source. He had pretty much abandonded the extreme abstract categorical approach by then. The notes I have follow the topical pattern of the book, but are so modified as to be essentially a different book, especially after covering spaces and the first homotopy group. His statement was that his treatment had changed since the subject had changed significantly. So much more has changed since then that I would not recommend this book as a primary text these days. Bredon’s

⭐is much better suited to today’s student.So, why did I give it four stars? First, notice that it splits stylewise into three segments, corresponding the treatment of its material in a three quarter academic year. The first three chapters (intro, covering spaces, polyhedral) have really not been superceded in a beginning text. Topics are covered very thoroughly, aiding the student new to the subject. The next three chapters (homology) are written much with much less explanation included – indeed, some areas leave much to the reader to discover and, consequently, aren’t very helpful if the instructor doesn’t fill in the details (the text expects a rather rapid mathematical maturation from the first part – too much of a ramp in my opinion), but the text is comprehensive. The last section (homotopy theory, obstruction theory and spectral sequences) should just be treated as a reference – it’d be hard to find all this material in such a compact form elsewhere and the obstruction theory section has fantastic coverage of what was known as of the writing of this book. It’s way too terse for a novice to learn from and there are some great books out there these days on the material.

⭐Spanier’s book is a wonderful treatment of many important ideas in algebraic topology, from covering spaces to Cech Cohomology. However, the going is difficult for those not initiated into the basic ideas. The proofs are correct, but often too terse for graduate students. It would also be nice if there were more explicit examples in the text, as in Greenberg and Harper’s book, for example. There are plentiful exercises that go deep into the theoretical aspects of the subject. I use this book as a reference for my research and for that purpose it is excellent!

⭐good

⭐This book is a highly advanced and very formal treatment of algebraic topology and meant for researchers who already have considerable background in the subject. A category-theoretic functorial point of view is stressed throughout the book, and the author himself states that the title of the book could have been “Functorial Topology”. It serves best as a reference book, although there are problem sets at the end of each chapter. After a brief introduction to set theory, general topology, and algebra, homotopy and the fundamental group are covered in Chapter 1. Categories and functors are defined, and some examples are given, but the reader will have to consult the literature for an in-depth discussion. Homotopy is introduced as an equivalence class of maps between topological pairs. Fixing a base point allows the author to define H-spaces, but he does not motivate the real need for using pointed spaces, namely as a way of obtaining the composition law for the loops in the fundamental group. By suitable use of the reduced join, reduced product, and reduced suspension, the author shows how to obtain H-groups and H co-groups. The fundamental group is defined in the last section of the chapter, and the author does clarify the non-uniqueness of the fundamental group based at different points of a path-connected space. Covering spaces and fibrations are discussed in the next chapter. The author does a fairly good job of discussing these, and does a very good job of motivating the definition of a fiber bundle as a generalized covering space where the “fiber” is not discrete. The fundamental group is used to classify covering spaces. In chapter 3 the author gets down to the task of computing the fundamental group of a space using polyhedra. Although this subject is intensely geometrical. only six diagrams are included in the discussion. Homology is introduced via a categorical approach in the next chapter. Singular homology on the category of topological pairs and simplicial homology on the category of simplicial pairs. The author begins the chapter with a nice intuitive discussion, but then quickly runs off to an extremely abstract definition-theorem-proof treatment of homology theory. The discussion reads like one straight out of a book on homological algebra. This approach is even more apparent in the next chapter, where homology theory is extended to general coefficient groups. The Steenrod squaring operations, which have a beautiful geometric interpretation, are instead treated in this chapter as cohomology operations. The logic used is impeccable but the real understanding gained is severely lacking. General cohomology theory is treated in the next chapter with the duality between homology and cohomology investigated via the slant product. Characteristic classes, so important in applications, are discussed using algebraic constructions via the cup product and Steenrod squares. Characteristic classes do have a nice geometric interpretation, but this is totally masked in the discussion here. The higher homotopy groups and CW complexes are discussed in Chapter 7, but again, the functorial approach used here totally obscures the underlying geometrical constructions. Obstruction theory is the subject of Chapter8, with Eilenberg-Maclane spaces leading off the discussion. The author does give some motivation in the first few paragraphs on how obstructions arise as an impediment to a lifting of a map, but an explicit, concrete example is what is needed here. The last chapter covers spectral sequences as applied to homotopy groups of spheres. More homological algebra again, and the same material could be obtained (and in more detail) in a book on that subject.

⭐A classic book on algebraic topology! But very important reference book. For first reading, it’s too dry, but this book deals with the subject in completely rigorous manner.

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