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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.


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

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    $\begingroup$ Did you come across Stillwell's book Classical Topology and Combinatorial Group Theory? I couldn't get in to it, but it felt it was at a good level for me (I am a geometric group theorist with, like you, poor topology...). $\endgroup$ – user1729 Mar 21 '18 at 10:31
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I assume you're familiar with the standard books in combinatorial group theory.

For combinatorial group theory and topology, try these books:

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.

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  • $\begingroup$ Yes, but what topological prerequisites are there for each? I found those books via Google. $\endgroup$ – Shaun Mar 21 '18 at 13:49
  • $\begingroup$ I see that the Stilwell link has a review that says the book introduces topology, so I guess that's good enough. $\endgroup$ – Shaun Mar 21 '18 at 13:53
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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.)

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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.

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  • $\begingroup$ What has this got to do with combinatorial group theory? $\endgroup$ – Shaun Mar 21 '18 at 14:56
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    $\begingroup$ The main application of the topology developed in this book is to group theory, specifically to topics which naturally arise in the study of combinatorial group theory, e.g. free products, Schreier transversals, Kurosh subgroup theorem, Grushko's theorem. See chapters 3, 6, and 7. After working through much of this book, you will be well-prepared to attempt more advanced works (e.g. Trees by Serre or the article Topological Methods in Group Theory by Scott & Wall). I think you'll appreciate the time invested when ideas in classical combinatorial group theory texts start to seem natural. $\endgroup$ – Robert Bell Mar 21 '18 at 19:59

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