Prerequisites for Algebraic Topology

Question

I'd like to self-study Munkres's Topology. I'm already comfortable with point-set topology, so the first part of the book will serve as a nice review with some new theorems every now and then. My main interest is the second part, Algebraic Topology. On that, the author says "we do assume familiarity with the elements of group theory."

What exactly are the elements of group theory that are needed? Is there a PDF or concise book that would provide me with the elements needed in order to study the second part in Munkres's book? (Please don't recommend tomes such as Dummit & Foote). Thanks.

Answer 1

While Munkres is arguably self-contained, there are further topics (on top of the above mentioned) you'll soon run into:

(1) Free Groups (abelian and not). (2) Commutator subgroups, and (from memory). (3) Centralizers.

He also uses the 1st isomorphism theorem galore, and I think the others in exercises (which is contained I guess in mentioning quotient groups; but very key to being able to handle on the fly). The algebraic material you need to get comfortably through the first chapter on fundamental groups isn't hard. Once you hit SvK, I (also reading Munkres on my own) found myself at a considerably larger challenge considering my knowledge (also self-learned) of about 1 semester of group theory.

The books I used and quite liked are Armstrong, Groups and Symmetry; and Artin, Algebra. I can recommend them both, but not compare them to the priorly mentioned ones.

Good luck!

Answer 2

I would agree with Henry T. Horton that, while stating that "we do assume familiarity with the elements of group theory...", the material relevant to continuing on in Munkres is listed/reviewed at the beginning of the section on fundamental groups:

• homomorphisms;
• kernels;
• normal subgroups;
• quotient groups;

with much of this inter-related.

Fraleigh's A First Course in Abstract Algebra would be a perfect place to learn these basics of groups and group theory; the text covers most of what is listed above in the first three Sections (Numbered with Roman Numerals) - the first 120 pages or so, and some of the early material you may already be familiar with.

It's a very readable text, lots of examples and motivation are given for the topics, and with very classic sorts of exercises. This should certainly suffice for what you'd like to better your chances of conquering "Part II" of Munkres.

A good resource to have on hand while reading Munkres, and/or to begin to review before proceeding with Munkres, is Michael Artin's Algebra (2nd edition). It does a great job treating groups!

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If cost is an issue, and/or you can't find a copy of Fraleigh's text at a library:

you might also want to check out Beachy and Blair's On-line Abstract Algebra Text (access to the second edition) and focus on the material covered through/including Groups.