Book Recommendations: Combinatorial Group Theory and Topological Prerequisites.

Question

I'm doing a PhD in combinatorial group theory and I can't help but notice that topology is adjacent to the research I'm doing. (In particular, I black box the combinatorial asphericity of certain presentations.)

My topology isn't very good: off the top of my head, I can't remember the definition of a topology${}^1$ - that's how bad it is.

Do you have any book recommendations for the topology of combinatorial group theory and what are each book's topological prerequisites?

A simple Google search produces a number of books but no review I've found of any of them states the topological prerequisites of the book at hand.

Simple bullet points of topic titles would be satisfactory.

Ideally, I'd like a book that introduces the very basics of topology alongside its applications to combinatorial group theory in depth.

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[1] It is 2:38 am . . . now . . . where I am, so, yeah, that's my excuse.

Answer 1

I assume you're familiar with the standard books in combinatorial group theory.

For combinatorial group theory and topology, try these books:

• Classical Topology and Combinatorial Group Theory by John Stilwell

• Combinatorial Group Theory: A Topological Approach by Daniel E. Cohen

Stilwell says in the preface:

The only prerequisites are some familiarity with elementary set theory, coordinate geometry and linear algebra, $\epsilon$-$\delta$ arguments as in rigorous calculus, and the group concept.

Cohen assumes familiarity with point-set topology.

Answer 2

As for topology, you'll need algebraic topology, not point-set topology.

As a first step, you might want to look at the Springer GTM by Rotman on Group theory. After introducing finitely presented groups he does a teeny-weeny bit on topology which might get you sufficiently off the ice so that you can evaluate topology books better. (Rotman also has a book in the GTM series on algebraic topology, which might be a good follow-up.)

Answer 3

I like Massey's an Algebraic Topology: An Introduction (not the similarly titled, Basic Introduction to Algebraic Topology). While it does not treat homology, which you will need to learn at some point, it gives a careful treatment of fundamental groups, covering spaces, and applications to group theory, e.g. Stallings' proof of Grushko's Theorem. The book starts with a classification of triangulable surfaces, which is a fun way to start. And this starting point is good place to gain some familiarity with cutting and pasting arguments, which can seem rather informal to someone coming from algebra. But you've got to start to somewhere!

As to point-set topology, probably the best thing to do is look up things and ask questions when you want to learn more.