Why post this question again

I don't think earlier posts reviewed a number of the textbooks I'm interested in, and haven't considered my criteria.

My background

At the graduate level, I've self-studied the material below. Note that I do zoom-meet with a post-doc analyst math tutor once a week, and I typically take a full exam and review it with him before moving to the next course.

  1. Abstract Algebra (Dummit). I covered 2 semesters worth of material. This was early in my graduate studies, and I think my study habits were much weaker/my math maturity level was lower.
  2. Topology (Gamelin). First 2 sections (Chapters 1-13). Fairly strong grasp of material and proofs, good result on the exam.
  3. Real Analysis (Stein and Shakarchi). I followed a Cornell course in terms of reading and homework. I didn't spend enough time understanding some proofs, and I didn't do enough problems on important topics (I'm working on Measure Theory by Axler now to remedy this, see below).
  4. Introduction to Manifolds (Tu). I read this cover-to-cover, had a good grasp of the proofs and exposition in general, and worked many problems per section. I like how this book has short sections with exercises at the end. The exposition/ease of understanding the proofs was excellent.
  5. Measure, Integration & Real Analysis (Axler). This is a model graduate level book for me. It has the perfect balance of clarity in the proofs, concision, but also intentionally leaving some gaps for the reader to complete ("as you should verify.."). I'm only in to the second section, but the proof style/exposition are slightly more challenging than Loring Tu's book -- but this level is ideal for me. Like in Tu, this book has short sections with many problems after each short section section.

My criteria for an Algebraic Topology text

A ranked list by importance is below (some of the rankings were difficult).

  1. The right level of difficulty for exposition/proofs (see Axler and Tu for examples, above).
  2. Sections aren't longer than about 10 pages, with exercises at the end. Books with 20-30+ pages without exercises are much less readable for me. It's harder to focus on learning 1-2 important results with this kind of book format.
  3. Hints/solutions in the back of the book. Axler is possibly changing my mind about this, but it was helpful to have hints in the Tu and Gamelin books (especially when it's only a hint). But I'm possibly gaining more insights with Axler, because the cost of finding solutions is higher (you can still look on stackexchange, but may not find anything).
  4. PDF/electronic version. This is almost a requirement, but all of the books I'm interested in have this anyway.

Possible Algebraic Topology texts.

  1. Hatcher. This book doesn't seem to have my 1-3 criteria. I know it's the main recommendation for an introductory text. From my brief review, I noticed that the sections are very long, the style of exposition possibly wasn't to my liking. The overall difficulty level may be a bit too high for my first foray into the subject.

  2. Shastri. What do people think about the proof difficulty/exposition/overall quality of this book? I haven't seen any posts that specifically address this. I notice that the sections are short, there's hints in the back of the book. The proofs have some "as you should verify" in them like Axler. Is this a good self-study book for my level?

  3. Bredon. This one looks fairly readable and approachable. How does the difficulty level compare to other books? Exposition?

  4. Rotman. This one seemed approachable and had very short sections

  5. Fulton. This one looked possibly too easy and appeared to have less depth/breadth than other texts.

  6. Massey (A Basic Course in Algebraic Topology). This book looks a more advance, is likely less accessible than what I need. His easier book (the Introduction book) seems to leave out cohomology theory and other items I want to have covered.

How to helpfully respond

When you review/recommend a book, please explain how it fits in with my criteria. My initial guess is that Shastri, Bredon, or Rotman might be the best fit for me. I'd really like to get people's feedback on these books especially.

  • $\begingroup$ I'm not very knowledgeable about algebraic topology (see my comments to Hatcher Algebraic Topology: I have all the prereqs, so why is this book unreadable for me? for instance) so I can't offer advice, but since Massey has at least 4 books on algebraic topology (2 of them are on my bookshelves, in fact), you should probably state which book you mean. This might also apply to one or more of the other authors, although I haven't investigated the other authors. $\endgroup$ Aug 10, 2022 at 13:29
  • $\begingroup$ I think (and I thought this was also the common belief) that Hatcher is easier and less precise than Massey (any of the four books) and Fulton. $\endgroup$
    – carciofo21
    Aug 10, 2022 at 14:31
  • $\begingroup$ @DaveL.Renfro, I just saw the suggestion in your post to not read Chapter 0 of Hatcher first. This idea seems helpful. But like you, at first glance Hatcher's style seemed a bit less precise than I'm accustomed to. carciofo: I added the title to Massey as a "Basic Course", and now see that this text doesn't look easy. "An Introduction" is the easier Massey text I think, but it seems to leave out things I'm interested in (cohomology, etc). $\endgroup$
    – IsaacR24
    Aug 10, 2022 at 15:36
  • $\begingroup$ Just to be clear, neither the question nor any answers (if they exist; I forgot if there were any answers) were by me, only a few comments. And the comment (or answer, as the case may be) about not reading Chapter 0 of Hatcher first was not made by me. BTW, I'm reminded of Chapter 0: Preliminaries in Kelley's "General Topology" and Chapter 1 in Munkres' "Topology. A First Course", each of which I recall (mid to late 1970s) graduate students and faculty often saying to others not to worry about if you felt you didn't have the background to read! $\endgroup$ Aug 10, 2022 at 17:29
  • $\begingroup$ A missing criterion is what material you want to see. Munkres has a solid book on homology and cohomology, for example. $\endgroup$ Aug 11, 2022 at 3:23

1 Answer 1


I am fan of old books because they are tied with the history of mathematics and when I read old book I understand that what was behind the theorems as I always ask myself where these beautiful and powerful theorems come from!

  • Do you easy with category theory? (i.e. by language of spaces and arrows) If you are not easy with it, I don't recommend Massey's Book. But if you know what they are and what they do, then I choose Massey GTM 127! It has no extra explanation.
  • Fulton's Book is a nice for those who want to know where these stuff come from. e.g. Fulton force readers to examine that why an integral is independent from loop on some spaces. Then he introduces winding number. (I like his approach. going from question to theorems!)
  • Shastri has tried his book to be comprehensive. IMO most of famous theorems are there.
  • As I tried to read Hatcher's Book for first time, it was confusing for me. But perhaps for those who want to read a second book may be a good one.
  • I like Rotman also. It explains without any extra words.
  • Bredon I think is a little technical book.

If you want short books, I recommend

Prasolov, V. V., Elements of homology theory. Transl. from the Russian by Olga Sipacheva, Graduate Studies in Mathematics 81. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3812-9/hbk). ix, 418 p. (2007). ZBL1120.55001.

You can ignore last two chapters.

Or even

Lee, John M., Introduction to topological manifolds, Graduate Texts in Mathematics. 202. New York, NY: Springer. xvii, 385 p. (2000). ZBL0956.57001.

or 180 pages (150 pages) Undergraduate Texts in Mathematics:

Croom, Fred H., Basic concepts of algebraic topology, Undergraduate Texts in Mathematics. New York - Heidelberg - Berlin: Springer-Verlag. X, 177 p. DM 36.00; $ 18.00 (1978). ZBL0378.55001.

or the 270 pages book

Kosniowski, Czes, A first course in algebraic topology, Cambridge etc.: Cambridge University Press. VIII, 269 p. hbk: \textsterling 18.00; pbk: \textsterling 6.96 (1980). ZBL0441.55001.


Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian, Computational homology, Applied Mathematical Sciences 157. New York, NY: Springer (ISBN 0-387-40853-3/hbk). xvii, 480 p. (2004). ZBL1039.55001.

  • 3
    $\begingroup$ I can't recommend May's book as a first encounter. It's too advanced and terse for an introduction and not even intended as a textbook (the author says so in the preface). It's an amazing book if you already know some algebraic topology and want to see the big picture, but as a primer it's overwhelming. $\endgroup$ Aug 10, 2022 at 17:30
  • $\begingroup$ @LukasHeger tnx. I've edited the post. $\endgroup$
    – C.F.G
    Aug 11, 2022 at 3:26
  • $\begingroup$ @C.F.G: can you rank each of the books you listed based on my criteria? In particular, I'm curious which ones are most accessible in terms of proof difficulty/readability (where Axler and Tu are the gold standard for me. Is Shastri accessible? When you say Bredon is "technical", do you mean hard to understand? $\endgroup$
    – IsaacR24
    Aug 11, 2022 at 12:52

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