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An Illustrated Introduction to Topology and Homotopy

Sasho Kalajdzievski
Chapman & Hall/CRC
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
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Some time ago I tried to acquire a copy of the book under review from the CRC Press representative to my department. I’d seen some advertisements for the book that looked extremely tantalizing; what particularly attracted me, whose exposure to topology has been, shall we say, eccentric, was the prospect of being able to visualize topological processes and properties, i.e. the proposition that Kalajdzievski’s illustrations would be remarkably enlightening to me. Unfortunately, the folks at CRC Press, or at least a positive number of them, were under the impression that An Illustrated Introduction to Topology and Homotopy is somehow just a topological picture book, i.e. an art book void of real mathematical or theoretical content, and so I didn’t pursue the matter even though I admit I was dubious about their characterization. Well, they were wrong. Now that I have a copy of this Illustrated Introduction in my hands, it is clear as vodka that the book is far, far more than a picture book: it is a serious textbook at the advanced undergraduate and beginning graduate levels, and is obviously also of great use to people like me: fellow travelers with spotty topology backgrounds.

All right, let’s start with the pedagogy angle. As I just indicated, Kalajdzievski makes it clear in the book’s Preface that this two-part book is meant to cover advanced undergraduate and beginning graduate level topology, even without “any pretense to go very deeply into these subjects.” Well, that’s just Kalajdzievski the topologist talking, I think: are any of us happy with what we get to cover in any of the beginning courses, given what we would want to do if we were preaching only to the choir? No: this is a problem even with lower division material, and it’s doubly the case for the book under review. As far as topology curricula go, the Illustrated Introduction to Topology and Homotopy is very good on all counts, and Kalajdzievski’s disclaimer about relative depth notwithstanding, it will serve well in the courses he is aiming at.

Part 1 is devoted to topology at the indicated levels and contains the standard fare, including topological spaces (metric spaces prominently featured), subspaces and quotient spaces, manifolds, CW-complexes, connectedness, compactness, and separation properties. The coverage is good and comes supplemented with beefy problem sets in one-to-one correspondence with the book’s subsections. The problem sets appear to cover a decent spectrum without going into particularly austere directions. Kalajdzievski has evidently resisted the temptation to foist sticky wickets on the reader in the form of suggestions in the text proper, to be followed up zealously in the exercises (and before you know it you have Atiyah-MacDonald: wonderful in its own magical way, but anything but a classroom text for a typical group of relatively inexperienced majors). However, what is most distinctive about An Illustrated Introduction to Topology and Homotopy is the abundance of well-constructed (computer-drawn) pictures of all sorts of things topological. These illustrations serve very effectively to confirm (or correct) intuitions and nebulous mental pictures that arise as one makes one’s way through the subject, and they aim at developing the almost ineffable “visual sense” that topologists and geometers lay claim to (and just think: Pontryagin was blind!).

Part 2 is devoted to homotopy, and Kalajdzievski states that it is pitched “mostly at the early graduate level,” and so it is. I am struck by the happy fact that right off the bat, isotopy precedes the introduction of homotopy: a very sound move. And there are pictures galore, of course (I am reminded of another very well illustrated approach to this material, namely, Hajime Sato’s Algebraic Topology, An Intuitive Approach). There is a marvelous rendering of the Brouwer fixed point theorem, the Jordan curve theorem, and more: all very visually evocative as well as pedagogically sound (in the old sense: no kumbayah, just sold mathematics presented well).

We go on to meet Seifert-Van Kampen, torus knots and Alexander’s horned sphere (see pp. 320–322), classification of low-dimensional manifolds, covering spaces, Borsuk-Ulam, and finally the happy marriage of finite group theory and topology spawning, for example, Cayley graphs and topographs. What’s a topograph? Well (cf. p. 436), it’s “a graph … together with a mapping [from its set of vertices into R] which we call the topography of [the graph]. Cool.

The book provides no coverage of homology or cohomology, and one might argue that beginning graduate students should see that material pretty early on. But that is a choice to be made by the professor (I happen to love that stuff, so I have a bias — but this is not meant as a criticism of Kalajdzievski), and, after all, there are many excellent texts available on algebraic topology properly so-called, as distinct from the broader-based themes Kalajdzievski deals with.

This is a wonderful book. I am much too selfish to give my copy away to one of the two knot theorists neighboring me in my department, but I’ll tell them about the book: p. 311 ff. (starting off with “Seifert-Van Kampen theorem and knots”) should be enough to move them to tears of joy. The only way the book could be improved is if the pictures would be in many colors rather than the chosen shades of grey. I am going to use Kalajdzievski’s book often, seeing that I do get to teach topology with some frequency and my own work in number theory has begun to exhibit serious topological (in fact, homotopical) tendencies. And perhaps I’ll tell my CRC representative that they have to rethink their sales-pitch.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Sets, Numbers, Cardinals, and Ordinals
Sets and Numbers
Sets and Cardinal Numbers
Axiom of Choice and Equivalent Statements


Metric Spaces: Definition, Examples, and Basics
Metric Spaces: Definition and Examples
Metric Spaces: Basics


Topological Spaces: Definition and Examples
The Definition and Some Simple Examples
Some Basic Notions
Dense and Nowhere Dense Sets
Continuous Mappings


Subspaces, Quotient Spaces, Manifolds, and CW-Complexes
Quotient Spaces
The Gluing Lemma, Topological Sums, and Some Special Quotient Spaces
Manifolds and CW-Complexes


Products of Spaces
Finite Products of Spaces
Infinite Products of Spaces
Box Topology


Connected Spaces and Path Connected Spaces
Connected Spaces: Definition and Basic Facts
Properties of Connected Spaces
Path Connected Spaces
Path Connected Spaces: More Properties and Related Matters
Locally Connected and Locally Path Connected Spaces


Compactness and Related Matters
Compact Spaces: Definition
Properties of Compact Spaces
Compact, Lindelöf, and Countably Compact Spaces
Bolzano, Weierstrass, and Lebesgue
Infinite Products of Spaces and Tychonoff Theorem


Separation Properties
The Hierarchy of Separation Properties
Regular Spaces and Normal Spaces
Normal Spaces and Subspaces


Urysohn, Tietze, and Stone-Čech
Urysohn Lemma
The Tietze Extension Theorem
Stone-Čech Compactification


Isotopy and Homotopy

Isotopy and Ambient Isotopy
Homotopy and Paths
The Fundamental Group of a Space


The Fundamental Group of a Circle and Applications
The Fundamental Group of a Circle
Brouwer Fixed Point Theorem and the Fundamental Theorem of Algebra
The Jordan Curve Theorem

Combinatorial Group Theory
Group Presentations
Free Groups, Tietze, Dehn
Free Products and Free Products with Amalgamation


Seifert–van Kampen Theorem and Applications
Seifert–van Kampen Theorem
Seifert–van Kampen Theorem: Examples
The Seifert–van Kampen Theorem and Knots
Torus Knots and Alexander’s Horned Sphere


On Classifying Manifolds and Related Topics
Compact 2-Manifolds: Preliminary Results
Compact 2-Manifolds: Classification
Regarding Classification of CW-Complexes and Higher Dimensional Manifolds
Higher Homotopy Groups: A Brief Overview


Covering Spaces, Part 1
Covering Spaces: Definition, Examples, and Preliminaries
Lifts of Paths
Lifts of Mappings
Covering Spaces and Homotopy


Covering Spaces, Part 2
Covering Spaces and Sheets
Covering Trans formations
Covering Spaces and Groups Acting Properly Discontinuously
Covering Spaces: Existence
The Borsuk–Ulam Theorem


Applications in Group Theory
Cayley Graphs and Covering Spaces
Topographs and Presentations
Subgroups of Free Groups
Two Subgroup Theorems