Customer Reviews

Elements Of Algebraic Topology

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Algebraic topology is a tough subject to teach, and this book does a very good job. Some prerequisites, however, are essential:

* point set topology (e.g. in Munkres' Topology)

* Abstract algebra

* Mathematical maturity to be willing to follow a definition and argument even when it seems like a weird side-track

In addition, this would not be the first book I would recommend to those interested in algebraic topology. First might be Massey's "Algebraic Topology: and Introduction" that introduces the fundamental group (conceptually easier than homology and cohomology).

At some point, however, a prospective student in topology will have to learn homological algebra and this provides the most concrete approach I know to the subject.

Algebraic topology is a lot of fun, but many of the previous textbooks had not given that impression. This one does.

This well written text is one of the standard references in algebraic topology courses because of its conciseness, and I find it very useful as a reference text.

However I think it is a little incomplete because of several reasons.

(1)It pays no attention to one basic concept of algebraic topology: the fundamental group.

(2) It doesn't cover ^Cech homology, important in other areas, like dimension theory for example.

(3) It doesn't stress the most important feature of algebraic topology: its connection to other areas of mathematics (analysis, differential geometry, etc.).

(4) Its list of references is too short, and lacks almost completely HISTORICAL references which are always important to become an expert in any field.

Conclusion: a good reference on homology and cohomology essentials, but not "the" reference on algebraic topology as a whole.

The material in the book (homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, applications to classical theorems of point-set topology) is for the most part solid. However...

- Munkres really belabors the simplicial theory, and it gets to be quite painful (especially the *CHAPTER* on the topological invariance of simplicial homology groups).

- Some very important topics (homotopy theory, fiber bundles) are not at all discussed.

- The book binding is horrible -- my copy is in two pieces, with several loose pages, and I don't think the hardcover edition is still in print.

It's worth noting that there are quite a few in number of books out there on introductory (i.e. a first course in) alg. top.
In particular, I should mention that the book by Rotman and sizeable portions of Bredon, "Geometry and Topology" can serve as good supplementary reading. I still don't think \pi_1 should have been left out; although one *could* refer to the prequel, there's still more to be desired by way of completeness, if anything, as this book is intended for beginners. For instance, the relation between the fundamental group and the first homology group would have certainly shed some light on these seemingly (at first glance, anyway) disparate invariants (as it is heavy-going on the (co)homological apparatus altogether).

Munkres is by no means encyclopaedic, which is good, in opposition to, say, Spanier or Whitehead, and certainly warrants attention to worked-out examples in detail and some (not-so) routine exercises which makes this book accessible to wider mathematical audiences wishing to learn a little about this fascinating subject.