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I am taking a course in Algebraic Topology next semester, so I thought of starting to read about it on my own.

My concern: I have read Topology by Munkres, and read the first chapter on algebraic topology from that book, so I have an idea about the fundamental group and covering spaces. Later I read the book Topology by Klaus Janich till the chapter on CW complexes. I really liked the book: it was not rigorous (somewhat unclear at some places) but enough to draw my interest in the subject. I was planning to start reading algebraic topology more seriously, so a senior suggested me the book Algebraic Topology: A First Course by William Fulton. I have read the first chapter, but I am having some problems. I actually don't have a good background in multivariable calculus. The author gives an overview of calculus in the plane and he introduces some basic concepts about multivariable calculus. I really like algebra; I have a good background in fields and Galois theory, ring and group theory (don't know free abelian groups yet). I am reading commutative algebra side by side. So I would really like to learn algebraic topology in an 'algebraic way', and I am concerned that Fulton in this book does this by another way (a differential approach or something). But at the same time I want to learn algebraic topology in an intuitive way (with pictures) and I've been told that this is the most expository book available on algebraic topology. Am I right about this book, I mean does it approach algebraic topology using category theory and other algebraic tools or in some other way? If so please tell me is it more important for intuition, so that I can read more on multivariable calculus and then start again?

I know this kind of question is not supported in this community, but I do need help about this. My course will be starting from next month, and I want to get a good concept about this subject. And I want to take up a course about algebraic geometry too. That's why I don't want to spend time experimenting. Thanks in advance.

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  • $\begingroup$ Why switching authors? Just read Elements of Algebraic Topology by Munkres himself. $\endgroup$ – TheGeekGreek May 23 '18 at 21:43
  • $\begingroup$ @TheGeekGreek I will check that for sure. I just didn't feel the motivation while reading Topology by Munkres. One the other hand I really like the writing style of Fulton (his book Algebraic Curves), that's why I was wanting to switch books. $\endgroup$ – user398623 May 24 '18 at 3:33
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A cursory glance through the book suggests it is has a strong differential flavour. If you’re familiar with multivariable calculus especially as formulated through differential forms and exterior derivatives, this isn’t necessarily a bad thing, and does help in understand what it means for two cycles to be homologous. The book presents de Rham cohomology even before homology, and restricts itself to the 0th and 1st (co)homology groups. It also briefly discusses vector fields which are more likely to be covered in a differential topology course. It includes Čech cohomology which is not something you meet in a first algebraic topology course usually.

So, I wouldn’t think that this book matches your background or aims. I’d recommend Topology & Geometry by Bredon as one that I liked, but there are multiple books out there that would serve you better! Feel free to reach out to me for more suggestions or recommendations on books.

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  • $\begingroup$ What about Introduction to algebraic topology by Rotman? Actually Bredon isn't available in my library. $\endgroup$ – user398623 May 24 '18 at 3:30
  • $\begingroup$ I don’t really have access to Rotman so I have no idea what his exposition is like. I’ll see if I can find a table of contents and previews to gauge this. On a side note, are you taking an undergraduate or graduate course in algebraic topology because their emphasis tends to be different and different books would be useful. $\endgroup$ – Osama Ghani May 24 '18 at 7:36
  • $\begingroup$ Thanks a lot. Have you read any books other than Bredon? $\endgroup$ – user398623 May 24 '18 at 7:39
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    $\begingroup$ I’ve read Basic Topology by Armstrong. It covers the fundamental group, classification of surfaces and simplicial homology and applications. It’s nice enough, aimed at undergraduates. There’s also Introduction to Topological Manifolds by Lee. It covers the fundamental group, covering spaces and homology but very thoroughly and in a sophisticated manner. Are you taking a grad or undergrad course? $\endgroup$ – Osama Ghani May 24 '18 at 7:49
  • $\begingroup$ I am a last year undergraduate student taking a graduate course. And I will take a look at Armstrong and Lee's books. Thanks. $\endgroup$ – user398623 May 24 '18 at 11:39
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I have a few suggestions:

1) Algebraic Topology by Hatcher is a very readable book that explains things moderately well in more of an informal manner - lots of diagrams for low dimensional things. If you already know about covering spaces and fundamental groups this book will be easily accessible.

2) My personal favourite is A Concise Course in Algebraic Topology by May. This is quite a short book - gets straight to the point and is quite algebraic. It's also quite easy to read. When you finish that you can even move onto its sequel, More Concise Algebraic Topology by May and Ponto.

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  • $\begingroup$ I would love to start with Hatcher and May, but aren't these really terse for a first read? I have read only the chapter of Munkres containing the fundamental group and covering spaces, not much. $\endgroup$ – user398623 May 24 '18 at 3:27
  • $\begingroup$ Chapters 0, most of 1 and parts of 2 are readable by Hatcher, but I agree it’s a way easier read once you’re taught the material. May’s book is very category theoretic and is well, very concise, making it (in my opinion) not a very good initial resource. $\endgroup$ – Osama Ghani May 24 '18 at 7:31
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    $\begingroup$ @user398623, Hatcher is a great place to start. I would suggest not expecting to master the subject on first take, and be prepared to draw from various sources, and to return to them at a later point, when you have improved your general understanding. One of Hatcher's good points is its wealth of examples, which will be of great use even if you feel slightly uneasy with its presentation of the subject matter. I agree with Osama; May's book is too concise for a learner, and is best used as a reference text. FWIW, I dislike Hatcher, and recommend Arkowitz's "Introduction to Homotopy Theory". $\endgroup$ – Tyrone May 24 '18 at 9:20

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