I'm taking a graduate sequence alg. topology course based on Hatcher and I've already learned point-set topology, groups, and rings. I scanned the first three chapters of Hatcher and saw modules appearing multiple times.

I'm planning on covering enough module theory over the break from Dummit and Foote so that I can comfortably take this course without struggling to make up for my knowledge deficit:

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How much of this appears in Hatcher (chap 1-3)? In the interest that I want to do well in this course, should I read chapters 11 and 12 or can I just skip them for now?

  • 2
    $\begingroup$ It depends what parts you are reading. Definitely $10.1-10.4$, $11.1-11.13$, and $12.1$. If you are applying to grad school, these are basically all things you will need to learn at some point $\endgroup$
    – pancini
    Dec 9, 2019 at 8:10
  • $\begingroup$ oh I definitely agree. I plan on learning these things, but since the course is next semester I want to know what parts are course-specific. $\endgroup$
    – kyary
    Dec 9, 2019 at 8:19

1 Answer 1


You will need all 3 chapters.

Chapter 10 is extremely important (you can't expect to follow an algebraic topology course if you're not comfortable with exact sequences, quotients, direct sums, etc. the basics of homological algebra)

Chapter 11 is extremely important too (in any case, I don't understand how you can be at a stage where you learn algebraic topology and not know about matrices and determinants, but if you are well you need to learn about them as well, because they come up extremely often as well)

Chapter 12 might be the least important of the 12, because the PID's that appear in practice in early algebraic topology are essentially $\mathbb Z$ and fields (and sometimes some localisations of $\mathbb Z$, but the theory for them follows from that for $\mathbb Z$, so the general theory is not needed specifically); so you can skip chapter 12 if you know or can learn someplace else (some of) the theory of finitely generated modules over $\mathbb Z$ and the basic properties of free modules over $\mathbb Z$ (otherwise known as abelian groups)


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