Textbook to Self-Study Algebraic Topology / Rating them by Readability

Question

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.

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.

and

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.