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I have no doubt that Hatcher's is a great text, but it is not for me. It is clearly written for someone with some prior knowledge of these topics, and I have none. For example, in the first few pages he defines deformation retractions, mapping cylinders, and homotopies. I gained very little insight into what these actually are from reading.

I am looking for an alternative text. To give an idea of my background, I am most comfortable with analysis, moderately comfortable with topology (first five chapters of Munkres's text), and mildly comfortable with algebra (basic group and ring theory). I am hoping for a text that not only defines the terms, but motivates them and helps the reader to understand them. I am most interested in homotopy groups and manifolds, out of the "main branches of algebraic topology" listed on Wikipedia.

I'm aware this question has been asked before, but a cursory search didn't really find any good consensus.

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    $\begingroup$ It’s hard to understand your complaint, because it depends heavily on what “actually are” means to you. Hatcher has formal definitions, pictures, and problems-can you be more clear about what you feel is missing? $\endgroup$ Commented Dec 5, 2019 at 6:01
  • $\begingroup$ If you are most comfortable with analysis then give Fulton's text a go. $\endgroup$ Commented Dec 5, 2019 at 6:03
  • $\begingroup$ @KevinCarlson I work better with explicit examples, i.e. defining the spaces and maps involved in a deformation retraction or homotopy explicitly. Perhaps this is too much to ask for? I wouldn't say I have a "complaint" so much as a feeling that this would not be the best text for me to learn from. $\endgroup$ Commented Dec 5, 2019 at 6:23
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    $\begingroup$ Why not look at part 2 of Munkres? $\endgroup$ Commented Dec 5, 2019 at 7:06
  • $\begingroup$ @JDZ I wasn't aware that Part 2 of Munkres was algebraic topology. Thanks, I will give that a try! $\endgroup$ Commented Dec 5, 2019 at 7:09

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If you want a more rigorous book with geometric motivation I recommend John M. Lee`s topological manifolds where he does a lot of stuff on covering spaces homologies and cohomologies. As a supplement you can next go to his book on Smooth Manifold to get to the differential case. I especially like his very through and rigorous introduction of quotient spaces/topologies and so on which are used very heavily and which Hatcher explains mostly in a very pictorial and unsatisfying way.

Try Tammo Dieck's Algebraic Topology. It is very detailed and a good supplement to Hatcher.I got my first exposure to algebraic topology from that book.

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  • $\begingroup$ Thank you, I think I will try Dieck. I have seen it recommended before in other posts and it includes an introductory chapter on topological spaces before diving into fundamental groups, which seems useful. $\endgroup$ Commented Dec 5, 2019 at 20:39
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I was in a similar situation. I sat in on a geometric topology class and we were using A Basic Course in Algebraic Topology, by Massey. I wanted other material to look at, since my background was lacking, so I picked up Differential Forms in Algebraic Topology, by Raoul Bott and Loring Tu and then Algebraic Topology A First Course, by Fulton.

I can't recommend them too much because I've decided a better way (for me) to get motivation and some background material would be to work up through complex analysis, Riemann surfaces, and differential geometry to complex projective varieties (here I'm looking toward Mumford's Algebraic Geometry I, his Red Book of Varieties and Schemes, and then his Alg Geometry II book with Oda), supplementing with as much commutative algebra as I need to fill in places. This has turned into a major project but I will come back to the topology stuff after it's all said and done.

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    $\begingroup$ Thank you for the recommendations. I cannot sympathize with your motivation to work up through complex analysis as I have absolutely no background in the subject.Best of luck to you though. $\endgroup$ Commented Dec 5, 2019 at 7:09

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