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Fundamentals of Algebraic Topology

Steven H. Weintraub
Publisher: 
Springer
Publication Date: 
2014
Number of Pages: 
163
Format: 
Hardcover
Series: 
Graduate Texts in Mathematics
Price: 
69.99
ISBN: 
9781493918430
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Ittay Weiss
, on
03/14/2015
]

Algebraic topology is a vast ocean of results. What belongs to the fundamentals is, quite expectedly, to some degree a matter of taste. The choice of topics covered in the book under review falls under what one may call classical algebraic topology. The fundamental group, covering spaces, a heavy dose of homology theory, applications to manifolds, and the higher homotopy groups is what the book is all about. Quillen model structures, simplicial complexes, and \(\infty\)-categories are not mentioned.

There are numerous classical books devoted to algebraic topology of which we mention three: Spanier’s Algebraic Topology, May’s A Concise Course in Algebraic Topology, and Hatcher’s Algebraic Topology. The subject matter of algebraic topology, by its very nature, consists of plenty of geometric ideas and a hoard of algebraic structures. Introductory texts must make a choice that basically amounts to deciding how many pictures to draw. The most concise treatments (i.e. May’s book) leave it to the reader to find the geometric interpretation and justification for the definitions and results, while the much more leisurely paced books (i.e., Hatcher’s) dazzle the reader with picture after picture after picture with careful motivation of each and every topic covered. The book under review is more on the concise end of the spectrum; there are not too many pictorial illustrations, but there is ample use of language to appeal to the reader’s geometric intuition.

Being concise, the author manages to cover an impressive amount of material. The reader is expected to know the basics of general topology and elementary group theory (not much more than basic examples and group actions, but those are used freely). A short appendix devoted to homological algebra serves to assist the reader, either as a preliminary reading or to consult alongside (or after) tackling homology. The book simply goes from 0 to 100 (or rather from \(H_0\) and \(\pi_0\) onwards), covering theory and classical applications in a lively fashion. The presentation is quick and to-the-point, offering plenty of exercises for the reader to appreciate the powerful techniques it develops. In the spirit of conciseness the author chose to skip the more technical proofs. This certainly allows for a very smooth exposition, though it might have been a good idea to indicate which omitted proofs are really hiding intricate details (this is done for some of the omitted proofs but still the reader may fail to appreciate that a seemingly obvious result may harbor unseen dangers).

The choice of topics is interesting; without a doubt each and every one belongs to the fundamentals of the subject. The exposition is exquisite, making reading the book very enjoyable. The book certainly has its place among the existing literature, as it offers something different from its peers. There are two missing aspects, however, that I feel would have offered a valuable improvement on an already excellent text. First, a closing short chapter on modern algebraic topology would have been a welcome addition. Such a chapter could have also given at least a taste of the missing subjects mentioned in the opening paragraph of this review.

Second, and more important, I find the absence of the use of category theory a bit mysterious. There is a short appendix devoted to categories and functors, but the main text does not make use of that at all (the appendix explicitly mentions that categories have been avoided). I find that to be unfortunate. Algebraic topology is the birthplace of category theory and the language of category theory is so suitable for algebraic topology that not using it really must be justified. Personally, I see no reason to avoid categories (as Hatcher, for instance, does); the formalism is not in the least hard, certainly not in comparison to the demands imposed by the other algebraic beasts one encounters when studying algebraic topology. Reading an entire book on algebraic topology without encountering the phrase “this is functorial” is a weird experience, and doubly so in a book that constantly emphasizes the role and importance of morphisms.

All in all, what the book does it does very well, and it achieves a lot. The fundamental group, Van Kampen’s theorem, covering spaces and the full Galois correspondence, free homotopy classes, an axiomatic treatment of homology followed by ordinary homology, cellular homology, an exploration of spheres and projective spaces, singular homology, a treatment of coefficients, cohomology, orientability of manifolds, Poincaré and Lefschetz duality, and the higher homotopy groups all make for a fun ride. Certainly a recommended read.


Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.

Preface

​1. The Basics

2. The Fundamental Group

3. Generalized Homology Theory

4. Ordinary Homology Theory

5. Singular Homology Theory

6. Manifolds

7. Homotopy Theory

A. Elementary Homological Algebra

B. Bilinear Forms

C. Categories and Functors

Bibliography

Index.