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Topology: A Geometric Approach

Terry Lawson
Oxford University Press
Publication Date: 
Number of Pages: 
Oxford Graduate Texts in Matheamtics 9
[Reviewed by
P. N. Ruane
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Here’s a book whose main intentions are:

  1. To introduce advanced undergraduates and beginning graduate students to topology.
  2. To focus upon low-dimensional spaces existing in 3-space (in keeping with the title).
  3. To be compatible with the Moore method of teaching topology.

And, although there is no such claim made by the author, it is obviously based upon many years experience in the teaching of topology. But how would I know this?

Well, there are over 520 exercises dispersed throughout 350 pages of richly varied text, and solutions for about 40% of these are provided in an appendix, and readers may apply to the publisher for the remaining 60%. Obviously, all this material must have been tried and tested over many years of classroom use, and the accuracy of the dialogue and the confidence of the author most probably emanates from such experience.

As for the intended readership, I would say that Terry Lawson’s expectations are too modest, since the early chapters could serve as a valuable source of directed reading in a course for second year undergraduates who have a good grounding in analysis. Then there are all those exercises, which are ideally suited to courses based upon any method of teaching.

Speaking of the Moore approach, most readers will know why it is so strongly associated with topology, but student-centred learning can be applied to any area of mathematics and, indeed, to any academic discipline. In fact, this philosophy was at the heart of British primary school teaching in the 1960s and its history predates Moore’s work by some distance (heuriskein Gk). Of course, such an educational ideal is one thing, but courses must be carefully planned to allow the time necessary for exploratory activities, which may mean reducing the amount of material to be covered (a problem which was not anticipated by educationists in Britain).

To the point: I really like this book. Ideas are introduced in a geometric context and the development of material always maintains its excitement. Enough point set topology is introduced in the first forty pages to commence an early investigation into spatial ideas (isometries in metric spaces, Jordan curve theorem and geometric figures as quotient spaces). Chapter 2 deals with the classification of surfaces (handles, isotopy, Euler characteristic etc) and algebraic topology takes its bow in chapter 3, with the main players being the fundamental group, vector fields on a surface, homotopy equivalence etc. In the final three chapters the learning curve becomes considerably steeper as the reader is invited to scale the rugged heights of Mt. Homology. But just think of the reward for going the whole distance; homotopy equivalence leads to isomorphism of homology groups! Long live mathematics!

Peter Ruane is retired from university teaching and now dabbles in as many creative diversions as possible.

Part I: A Geometric Introduction to Topology
1. Basic point set topology
2. The classification of surfaces
3. The fundamental group and its applications
Part II: Covering Spaces, CW Complexes and Homology
4. Covering spaces
5. CW complexes
6. Homology
Selected solutions