Curious Question on reading material for Homological Algebra

Question

As a quick question, while I was going over Algebraic Topology and going over some homology I came across Rotman's Homological Algebra and wondered if it would be a very useful book to use alongside Hatcher's and Rotman's algebraic topology book?

Would Rotman's Book on homological algebra be extra useful to my understanding of homological algebra in algebraic topology or is it just something to cover entirely on its own?

As a last question, I'm looking to learn homological algebra and I came across Rotman's book again due to high reviews. Would Rotman's text be a good place to understand generic homological algebra?

Any advice would be appreciated

Answer 1

I'd say it depends on how far - and in what direction - you want to study algebraic topology. If you're trying to understand algebraic topology, then probably picking up a second book on algebraic topology (Arkowitz, Dieck, May, Spanier, Strom) would probably be best. Each book contains a small section on homological algebra, and although its no way to learn the subject, there is enough non-overlapping material to give a good overview. The different views on algebraic topology given by each book will be more beneficial than picking up a second subject.

On the other hand Rotman's "Homological Algebra" is an excellent book, and if you want to seriously study algebraic topology on a level that makes extensive use of homological algebra then it is a great introduction to the subject that I highly recommend.

However, if you want to seriously study algebraic topology on a level that does not make extensive use of homological algebra, then maybe you should dedicate the time to learning another topic that would be more useful to you in the long run.

Answer 2

I think that the basics ($5$-lemma, snake lemma, Ext, Tor) are covered in Hatcher's book, while the lemmas are proved in slightly more generality in Rotman's book.

Of course, my previous statement requires that you go through most of Hatcher, which is quite an endeavor (I certainly haven't done all of it.)

IMO, Rotman's book is the only one on homological algebra that I find really comprehensible-- but it is long and spends a lot of time on commutative algebra topics (valuation rings, nakayama's lemma etc) that are useful, but probably won't aid you in AT. Because of this, you won't get to the heart of the issue nearly as fast by reading (at least linearly) Rotman's book.

I would say, just keep up with Hatcher, and if you hit brick walls in anything that is Algebraic in flavor, Rotman probably has a more detailed exposition, but it might use some language you're unfamiliar with, in which case it would take a bit of extra effort on your part to understand it (this was my experience using the book as a kind of "proof/definition reference."

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Here is an example: After the first chapter on Homology in Hatcher's book, there is exercise 27, that asks you to show that if for $A \subset X$ and $B \subset Y$, we have $f:X \to Y$ and $f:A \to B$ homotopy equivalence, then $f_*:H_n(X,A) \to H_n(Y,B)$ is an isomorphism.

This is kind of cool and an application of the $5$-lemma-- but I don't think (initially at least) that further topics in homological algebra would be illuminating for this problem (and likewise for many such problems in Hatcher.)

I love Rotman's book and use it all the time.