Lectures on Algebraic Topology
At a first glance, this nice, short book is comparable to other brief texts of a similar vein. Its main purpose is to introduce the reader to the basics of algebraic topology and in particular to homology theory and its applications (which is described in depth - about three-quarters of the book is devoted to it) and homotopy theory.
The book begins with homology theory, which is introduced from the geometric approach of simplicial homology. In fact, whenever it is possible within the text, results and proofs are presented through a geometric vision and intuition. Basic and classical properties of simplicial homology together with some applications are then presented. After that, the axiomatic approach to homology theory is described together with a sketch of the proof of uniqueness of homology theory on polyhedra. Then, the author introduces cellular homology and highlights the computational properties of this approach. The chapter also contains the Lefschetz fixed point theorem, a brief introduction to homology with coefficients, and a quick description of some elements of cohomology theory. This introduction to homology theory finishes with a description of Poincaré duality on manifolds.
The basics of homotopy theory are then presented in very brief terms. Starting with the definition of the fundamental group, its computation is described via the Van Kampen theorem applied in different situations. A very short introduction to higher homotopy groups and the action on them by the fundamental group is then followed by an equally extensive presentation of the long exact sequence of a fibre bundle. The chapter is completed with a compendium of the basic properties of covering spaces.
But, as mentioned above, all of this follows from a quick first look at the book under review. After a careful reading, though, this short piece reveals many insights from which different readers may benefit.
The mathematical beginner, perhaps an undergraduate, will find a very intuitive and geometrical, yet formal and rigorous, approach to homology theory. The classical combinatorial difficulties to understanding simplicial homology (a good example being the proof of the simplicial approximation theorem) are overcome with a coherent, geometrical and logical exposition.
The smooth and clear explanation at the beginning of the text of how the abstract concepts of categories and functors are going to be used later on, as well as the geometrical sketch of the proof of the uniqueness of homology theory on polyhedra and the introduction to cellular homology for computational purposes, are good examples of this attempt to bring non-trivial concepts to beginners.
For the graduate student, or the outsider to algebraic topology with some mathematical knowledge and with interests in other branches, this book is also of help. On one hand, some aspects of algebraic topology are presented in a not commonly used approach (for instance, the computation of the fundamental group of the complement of some knots). On the other hand, the use of the basics of simplicial homology theory to attain deep results on more geometrical environments are given in the same geometrical but rigorous language. Here follows an example of this:
After the introduction of the degree of a self map on a manifold, and with the few tools developed at that point, the author readily presents the homotopy classification of immersions of the circle into the plane and the fundamental theorem of algebra, and shows that a vector filed on the 2-sphere has a singular point.
Finally, this text could also be of use for the expert from a teaching point of view. I am sure that any experienced mathematician could find here new ways and tips, at least of an expository nature, of presenting these classical topics in the classroom.
Having said that, there are two more aspects of the text that should be remarked upon, as the author gives them special attention. That is, the exercises and computability.
In fact the whole book is scattered with many exercises of differing scales of difficulty and whose purposes vary. Some of them are provided to encourage and puzzle the independent reader as well as to make them evaluate their knowledge. Others are included to fill gaps left on purpose either to complete proofs of stated theorems or to establish results needed to make the book self-contained. In all cases hints and/or answers to the problems are presented at the end of the text.
On the other hand, concerning computability, by giving explicit and far from complicated algorithms the author points out that homology groups and fundamental groups can be explicitly calculated for many spaces found in nature.