I know there are a few posts asking for references about algebraic topology textbooks. Still I have decided to open another one as I would like to ask a slightly different question: which textbook would you read to complement Hatcher's one?
I am more interested in texts which give a different insight or perspective on the same material (or a similar selection of material), rather than in maximally reducing the overlapping with Hatcher. Please give some reasons for your suggestions.
Moreover, beyond algebraic topology textbooks, are there any textbooks in related areas which would be profitable to read at this level – that is the level at which Hatcher's book is written?
In particular I would be interested in a text covering the algebraic material needed for algebraic topology. To give some examples of what I would like to see covered, the exercise on finding the Abelian groups $A$ which fit in the exact sequence $0\rightarrow \mathbb{Z}_{p^n}\rightarrow A\rightarrow \mathbb{Z}_{p^m}\rightarrow 0$ give me headaches (I am not asking for how to solve it here, but rather what to study to be able to confidently solve it). Or the remark "Since $\mathrm{Hom}(H,\mathbb{Z})$ is isomorphic to the free part of $H$ if $H$ is finitely generated" is not obvious to me (it would be in the case of vector spaces where I would translate it to a finite dimensional vector space and its dual are isomorphic).
EDIT: In response to an answer by Daniel Rust here are some more details on which parts of Hatcher I am interested in. I would like to get a better understanding of homology and cohomology (starting from the kind of presentation level you find in more elementary texts like Croom or Naber or Nakahara) and the way they are related. So in particular topics like the Universal Coefficients Theorem, Cup Product and Poincaré Duality.
Apart from my general interest in the topic, one of my aims is to better understand how the intersection matrix of a manifold is defined and practical way of computing it – in particular for 2-cycles inside a 4-manifold. While I know that in the smooth category many issues simplify, my feeling is that it pays off to learn about these concepts in their proper setting, i.e. the topological one.
Best Answer
It's my experience that one doesn't usually 'read through Hatcher' at least not in the traditional sense of reading a book cover to cover. The material in Hatcher is simply too broad, ranging from basic introduction to algebraic topology all the way to very abstract homotopy theory. In my experience it's more of a reference book - after you've read the first chapter or two, you pick the right bits to read when you need them, often not in the order they're written.
The question in your last paragraph is more manageable as you're asking for extra material on the homological algebra covered in the text. My suggestion would be any of the classical texts on homological algebra such as Cartan-Eilenberg for a classical reference or any of a good range of modern texts which may be gentler introductions. There's a helpful thread here which covers books which might be of use.
I think if you are looking for complementary texts for other parts of Hatcher, you would get a better response if you made a separate question for each specific part. For instance "I'm looking for a more in-depth introduction to similar material on the surface theory covered by Hatcher - especially with regard to covering spaces." or "Hatcher glosses over some of the details in the section on Postnikov towers. Is there a reference with more details and possibly some examples?"