I am taking an introductory Algebraic Topology course, and we have just finished talking about the fundamental group/ covering spaces in Munkres' Topology. However, his treatment of simplicial homology in Elements of Algebraic Topology has been particularly opaque to me.

I'm wondering specifically about a text that focuses on Simplicial homology theory [that doesn't assume a lot of requisite algebra], as well as a clear/geometric consideration of what a simplicial complex is in the first place. Some particular problems I'm having include the following:

  1. Abstract Simplicial Complexes

  2. Linear homomorphism/ geometric realizations

  3. [somewhat an aside] free abelian groups/ direct sums/summans and their use in computing homology.

I'm looking for something that doesn't assume too much, but also motivates the definitions, since this is my first time dealing with homology, and I've yet to see how any of this comes together.


  • $\begingroup$ "Basic Topology" by Armstrong ends with a chapter or two on exactly those things (at a rather introductory level) if I recall correctly, but it overlaps quite heavily with Munkres' "Topology" otherwise, so I don't know if that's the best suggestion. Otherwise, Hatcher's book is a general go-to reference. It's freely available on the web, so you could take a look in there. It probably covers those things (haven't had the time to look into the book enough, regrettably)… $\endgroup$ – A.Sh Mar 8 '16 at 21:45
  • $\begingroup$ "A Basic Course in Algebraic Topology" by Massey contains an excellent introduction/motivation to homology by relating it to Stokes'/Green's Theorem (ch. VI, section 3, pp. 149 - 156). It really helped build my (geometric) intuition on "cycles" and "boundaries." I also highly recommend this blog post: http://jeremykun.com/2013/04/03/homology-theory-a-primer/#comment-52564 - it's an extremely accessible/intuitive introduction into simplicial homology (with an explicit example worked out too). $\endgroup$ – user316092 Mar 8 '16 at 23:15
  • $\begingroup$ The blog was very helpful! I have seen hatcher, but for some reason it seems like simplicial complexes don't receive a full treatment there. I have heard very good things about Massey and Armstrong, thank you for the recommendations. Any book with a nice treatment of free abelian groups etc? $\endgroup$ – Andres Mejia Mar 9 '16 at 2:43
  • $\begingroup$ You may want to look at "Abstract Algebra" by Dummit and Foote. It's one of the standard grad level texts and walks you through free abelian groups, short/long exact sequences, etc. (If I remember, the book doesn't discuss homology, but it's not too hard to apply the algebra to the homology context.) I also like "Basic Abstract Algebra" by Bhattacharya, Jain, and Nagpaul. It's more of an undergrad book, so it's quite accessible. $\endgroup$ – user316092 Mar 10 '16 at 12:38
  • $\begingroup$ I found massey's book to largely suit my needs. It actually covers the basics of free abelian groups etc. and indeed offers excellent motivation. If you post your comment as an answer, I would gladly accept it. $\endgroup$ – Andres Mejia Mar 14 '16 at 19:10

In order to make a record of this, I will post recommendations given to me as well as those I found very helpful.

"A Basic Course in Algebraic Topology" by Massey [courtesy of user316092] is a very easygoing text on algebraic topology, reviews some basic algebra of free abelian groups, and has a good (although brief) description of simplicial homology

Jeremy Kun's blog post is highly recommended for a general overview/ informal discussion for intuition.

However, by and far, the most helpful resource for understanding what the heck a simplicial homology group looks like can be found in "Visual geometry and Topology." In particular, the first chapter is lucid, clear, formal, and really has excellent descriptive power.


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