# Algebraic topology textbook

## Main Question or Discussion Point

I see that there are four different GTM textbooks on the subject. Which one of these is the most suitable for self-study?

GTM 56: Algebraic Topology: An Introduction / Massey
GTM 127: A Basic Course in Algebraic Topology / Massey
GTM 153: Algebraic Topology / Fulton

I want to pick up enough background material to move on to knot theory (GTM 175?) and topological QFT.

## Answers and Replies

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DarMM
Gold Member
To be honest probably the best way to learn algebraic topology for physics is to read:
Charles Nash and Siddhartha Sen, Topology and Geometry for Physicists, Academic Press
in conjunction with
Mikio Nakahara, Geometry, Topology, and Physics 2nd edition, A. Hilger, New York

Those will cover most topics. However for very advanced stuff, like you might need for topological field theory, read the above texts and then read:
Charles Nash, Differential Topology and Quantum Field Theory, Academic Press

That's an interesting idea, but Amazon reviews of Nash/Sen aren't exactly glowing. And there's one review there that accuses Nakahara of having on average one misprint per page. (See Maxwell's equations here.) My own inspection of the first two chapters revealed at least one missing "*" on page 9 and a $\hbar$ out of nowhere on page 17.

Last edited:
DarMM
Gold Member
That's an interesting idea, but Amazon reviews of Nash/Sen aren't exactly glowing. And there's one review there that accuses Nakahara of having on average one misprint per page. (See Maxwell's equations here.) My own inspection of the first two chapters revealed at least one missing "*" on page 9 and a $\hbar$ out of nowhere on page 17.

Nakahara is mainly read for its sections on algebraic topology and differential geometry and is one of the best books for dealing with the Atiyah-Singer index theorem from a physicists viewpoint. It is one of the most commonly used textbooks for topology and geometry in physics. It is also an excellent reference work. There is a few mistakes, but they are mostly confined to the early chapter on QM.

Nash and Sen will give you a good example of the homotopy and homology groups. If you are concerned with reviews this is Professor Tom Kibble of Imperial College London on the book:
Professor Kibble said:
"One of the most remarkable developments of the last decade in the penetration of topological concepts into theoretical physics. Homotopy groups and fibre bundles have become everyday working tools. Most of the textbooks on these subjects were written with pure mathematicians in mind, however, and are unnecessarily opaque to people with a less rigorous background. This concise introduction will make the subject much more accessible. With plenty of simple examples, it strikes just the right balance between unnecessary mathematical pedantry and arm-waving woolliness...it can be thoroughly recommended.
Unfortunately outside these books most "Topology and Geometry" books for physicists are concerned mainly with differential geometry not algebraic topology.

For Fiber bundles, Knot theory and simple Topological QFT you might try:
John Baez and Javier Muniain, Gauge Fields, Knots, and Gravity, World Scientific Press.
Which will really help with knot theory and topological field theory.

Even if you don't try Nash and Sen, you probably will have to read Nakahara, as it is the standard reference.

If your ultimate aim is to learn knot theory and topological field theory solely I would first read Baez's book and especially try the excercises, they're great. Then read Nakahara for some general stuff on the Atiyah Singer index theorem and algebraic topology. Finally for the really serious stuff read the last book by Nash.