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Recently, I read a little portion of homotopy theory from Bredon's 'Topology and Geometry' and found that I like it enough to want to continue reading material in Algebraic Topology.

A little digging around on the internet told me that books like the one by Peter May and Tammo tom Dieck are second texts, and that one would do well to start with Hatcher/Bredon/Massey.

Considering that I have only four months in which to know much of the material at the level of Tammo tom Dieck's book, I was wondering if there is any significant disadvantage to working from such a text, rather than an apparently more elementary text such as Hatcher.

To summarize: 1. What, if any, are the significant advantages of studying algebraic topology from the non-categorical viewpoint, before reading a categorical approach to it? 2. Does the categorical approach, as done in tom Dieck, subsume the non-categorical approach in terms of the results provable?

I know some category theory from MacLane's book, and learnt point-set topology from Munkres. My background in algebra is comprised of the sections on groups, rings, fields and Galois theory from 'Abstract Algebra' by Dummit and Foote and some part of modules from Herstein's 'Topics in Algebra'.

David White
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user137677
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    I believe this question is much more suitable for math stackexchange as mathoverflow is for questions regarding mathematics research. – Victoria M Mar 29 '19 at 20:11
  • I will post it there, then. Is there any way I can take it down from here, though? – user137677 Mar 29 '19 at 20:17
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    I think of all of those textbooks as first texts. i.e. they all start at the basics. The kind of language you prefer to use to describe theorems in algebraic topology is largely up to your taste. – Ryan Budney Mar 29 '19 at 20:26
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    None of these books are "advanced" algebraic topology (e.g. they don't talk about spectra, or highly structure multiplicative structures, which are absolutely fundamental to modern homotopy theory, let alone chromatic stuff). That said the earlier you get used to categories, the better. PS: you can flag the question for the moderators to migrate it to [math.SE]. – Denis Nardin Mar 29 '19 at 20:29
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    That said, what do you want to do with algebraic topology knowledge, once you have it? That might help you pick a book. – Ryan Budney Mar 29 '19 at 20:36
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    Possible duplicate of An "advanced beginner's" book on algebraic topology? – David White Mar 29 '19 at 22:13
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    And I would like to say, characterizing Hatcher's book as "elementary" is a huge disservice. There are a good number of fine math students out there that can devour May's book but simultaneously find Hatcher's book extremely difficult. Hatcher's book is rich -- it's choice of a more visual and human language is a reflection of that. – Ryan Budney Mar 30 '19 at 02:55

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Your question is a duplicate of this one. On that thread, I recommended an excellent book by Davis and Kirk, which is written with more of a categorical viewpoint, and aimed more in the direction of the books of tom Dieck and May you recommended (whereas, Hatcher is arguably aimed more in a geometric/low dimensional topology direction). That thread also discusses a book by Strom, an older one by Spanier, and lecture notes by Jacob Lurie. If I were you, I'd start with Davis and Kirk, and see which of the others you can find for free online. There are many good options, so just pick one and get started, rather than waiting for a book to arrive in the mail.

David White
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  • Davis and Kirk is also explicitly a "second course" in algebraic topology. From the preface: "The prerequisites for a course based on this book include a working knowledge of basic point–set topology, the definition of CW-complexes, fundamental group/covering space theory, and the construction of singular homology including the Eilenberg-Steenrod axioms." It doesn't sound like the original poster has that background. – Greg Friedman Mar 30 '19 at 04:01
  • I disagree. If he got point-set topology from Munkres, algebra from Dummit-Foote, and category theory from Mac Lane, then he has enough to read Davis-Kirk. I agree with Ryan Budney's comment that it really matters what audience the book is written for. – David White Mar 30 '19 at 13:25