I learned basic algebraic topology before, contents including homotopy lifting lemma and covering space and Van Kampen theorem and etc. (The class used Hatcher's Algebraic Topology when I took this course)
However, I have never learned things related to homological algebra. I'm taking algebraic topology dealing with homological algebra this semester, so I wonder if there is a rigorous text on this side of algebraic topology.
Personally, I do not like Hatcher's style of text. It is not that much formal in my sense and materials are not well-ordered. (For example, he proves a very special case then after few pages, he states a theorem which is more general, then just notes there that copy the idea of the proof before he handed) I personally like rigor, formal, and axiomatic approaches even if text may seem dry. For example, I like Rudin's and Folland's and Mukres' and Dummit's and Rotman's styles of texts and etc, but I dislike Stein's and Hatcher's styles of texts.
Moreover, I have seen a post saying that "Hatcher uses $\Delta$ complexes, which are rarely used". So what would be the standard complexes? And what text develops theory using that complexes?
Thank you in advance! 🙂
Best Answer
tom Dieck's Algebraic Topology is great. It is very rigorous, presents an incredibly wide range of topics, uses (admittedly minimal) categorical language, and gives a much more homotopical perspective on many things.
A small but large detail I always remember is that Hatcher's proof of homotopy invariance of singular homology is not very useful for generalization, but tom Dieck's, which inductively constructs a natural homotopy, leads to the acyclic models theorem, from which several other (otherwise tough) theorems follow.