# Reference request : Intro to geometry/topology for beginners

I am looking for an introductory book in Topology, but one meant for a younger audience than is usually the case :

I am planning to do a reading course with some sophomore students and I would like to motivate them to take up mathematics as a major. They have seen Apostol's Calculus, some Linear Algebra (a la Hoffman & Kunze), and some Group theory (a la Herstein). ie. They are not unused to rigour, but might be a little daunted by too much of it.

Therefore, I would like to do some serious mathematics with them, but with a view to get them interested in geometry/topology (They chose this topic based on advice from some older students).

Some topics I had in mind were :

a) Wallpaper patterns/Platonic solids and their symmetry groups

b) Euler's theorem for polyhedra ($v-e+f = 2$)

c) A gentle introduction to the fundamental group

Could anyone suggest a good book to follow? I can only think of Armstrong's Basic topology, which I will follow in the absence of anything better, but some of you out there probably have some experience doing this (in summer math camps and such). Any suggestions?

Thanks.

• I added geometry to your title, because I was misled by the original version. What you want is at best peripheral to what I think of as topology. If you had wanted general topology, I’d have suggested the out of print Theory and Examples of Point-Set Topology by John Greever, provided that you could find copies: it’s a Moore method text, leaving most of the proofs to the student, which makes it very good for a readings course. (As a freshman I took that course from the author.) – Brian M. Scott Sep 3 '13 at 19:57
• You have students who have gone through Apostol's Calculus, Hoffman and Kunze's Linear Algebra and some group theory, but are not sure they want to major in mathematics? Where are you teaching?? I'm sorry to say that at my institution this is essentially the material we would like our graduating majors to know (and I'm sure that some get degrees knowing less). – Pete L. Clark Sep 3 '13 at 20:28
• @PeteL.Clark: They take up Mathematics after 4 semesters of Calculus/Linear Alg/Probability here - this is common in India, although the level of rigour varies from place to place. That said, these students are bright, and I am excited to do something interesting with them. – Prahlad Vaidyanathan Sep 4 '13 at 1:45
• @BrianM.Scott: My apologies for the inaccurate title. I will look at Greever's book. Thanks! – Prahlad Vaidyanathan Sep 4 '13 at 1:47
• As a student I enjoyed reading Basic Topology by M. A. Armstrong, and since you're aware of it I firmly recommend it. One thing I would shy away from is choosing a book that tries too hard to make itself interesting at the cost of rigor, like Henle's A Combinatorial Introduction to Topology. Topology is hard, and an appropriate amount of rigor keeps the subject organized for the student. Also, you may want to check out Armstrong's Groups and Symmetry - won't do c), but it gives a great treatment of a) and b). It goes great with his topology book. – Gyu Eun Lee Sep 4 '13 at 13:12

Intuitive Combinatorial Topology by V.G. Boltyanskii, V.A. Efremovich covers these topics.

Also there is First Concepts of Topology by William G. Chinn, Norman Earl Steenrod.

• Ok - I haven't seen either of these. Will go find them tomorrow :) – Prahlad Vaidyanathan Sep 3 '13 at 17:53
• Actually, they are (were) intended for high school students in former USSR. But they cover a lot of things. I added Google Books links, you can see previews. – njguliyev Sep 3 '13 at 17:55
• Intuitive Combinatorial Topology it is! Thanks. It's a great book. – Prahlad Vaidyanathan Sep 4 '13 at 13:44

If you really mean general-topology, than I would recommend Set Theory and Metric Spaces, by Kaplansky. This is an evergreen classic, and it is well-suited for a bright student to read mostly on her own and then discuss with others. The exposition is extremely lucid and complete, and is usually followed by some interesting and challenging problems. In fact I led an informal course on this with two friends of mine (physics / computer science types who were past college age but still very interested in mathematics) and it worked nicely.

On the other hand, none of your examples are in general topology. The first is geometry, and the latter two are algebraic topology. In my experience if you compromise too much on "rigor" or "seriousness" in algebraic topology then you wind up staring at pictures and talking about glue. On the other hand, a lot of the main texts on algebraic topology will be a bit forbidding for most undergraduates.* One book which I enjoyed as an undergraduate and still looks very nice to me now is John Stillwell's Classical Topology and Combinatorial Group Theory. In contrast to Kaplansky's text, most undergraduates would want some help and direction with this book, which would give you something to do.

*: "For a long time, books on algebraic topology were of two kinds: either they ended with the Klein bottle, or they were written in the style of a letter to Norman Steenrod." -- Giancarlo Rota

• +1 for "you wind up staring at pictures and talking about glue". I will definitely keep this in mind. There is a thin line between motivation and glue :) – Prahlad Vaidyanathan Sep 4 '13 at 3:20

For point set topology, by far the most popular book is Topology by Munkres.

It is a very easy read. Other reference is Basic topology by Jones and bartlet publisher. I forgot the author's name.

Yet another book is essential topology published by springer in its undergraduate series. Again I forgot the author.

• Yes, Munkres is standard, but a little pedantic for what I had in mind. Good book, but I want something more "flavourful". I will look at Jones though. Thanks! – Prahlad Vaidyanathan Sep 3 '13 at 17:51

I haven't taught from it, but Topology Now! by Robert Messer and Philip Straffin seems like a nice introduction. They bring in knots and links right from the start, which should appeal to many students.

This one may be a bit more than what you need; but is a superb book regardless: Singer and Thorpe's "Lecture notes on topology and geometry". It starts with basic general topology and goes on to fundamental group and simplicial complexes and later to manifolds and de Rham's theorem and ends with a small dose of Riemannian geometry(geodesics, etc). This can be a 1 year reading course for a bright undergraduate.

Next is in regards to:

if you compromise too much on "rigor" or "seriousness" in algebraic topology then you wind up staring at pictures and talking about glue.

If the rigorous definitions of deformation retract, homotopy equivalence etc. are learned from somewhere appropriate(such as Armstrong), then V. V. Prosolov's "Intuitive Topology" is an excellent book to motivate oneself into topology.