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Before, I scoured stack exchange for the prerequisites to read this book:

(See: Topology Prerequisites for Algebraic Topology, Module theory for chapters 1-3 of Hatcher Algebraic Topology, Learning Roadmap for Algebraic Topology, Algebra prerequisites for Hatcher's Algebraic Topology)

I got 100 on my point set topology course and am very comfortable with groups, rings, and modules so I thought I should be able to comfortably start learning out of Hatcher. I've heard great things about this book - how it's by far the most readable book in introductory algebraic topology, beautiful typesetting, builds up from basics, assumes little of the reader, etc.

But a couple pages in and I'm completely lost!

Hatcher keeps talking about orientable surfaces and genus - both concepts which are not once mentioned anywhere in Munkres, or indeed, in the majority of introductory topology courses. Neither of these concepts are defined in Hatcher either. So I thought - maybe I'll learn these concepts first and come back to the book! But no - every reference on genus I've found is another book on algebraic topology, one that needs prereqs beyond the ones demanded by Hatcher. All mentions of orientable surfaces lead me to differential geometry references - but I know 0 things about differential geometry.

The kicker is the prof I talked to who is running the course based on this book said I'll be fine in the course given my background knowledge. But from what I've skimmed from the book, I'm sure to fail the course at this rate.

I'll give a different example. The first time Hatcher defines the real projective space is here:

enter image description here

But is it just me or is this 'proof' extremely unrigorous and handwavy? I feel like half this proof relies on the reader's intuition about $\mathbb{R}^n$ for $n<=3$ and somehow this translates to general $\mathbb{R}^n$. The other half is massive leaps in logic that took me a long time just to think of a proof sketch. And this was just one example from the book, mind you.

Honestly, I'm thinking I've just hit a wall in my limits in pure math. What am I doing wrong? What makes this text readable to others but not me?

Or rather, my question should be - how do new people read this book in such a way that they don't get hung up in the way I am doing right now?

kyary
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    Try another book. There are plenty of them. – almagest Dec 31 '19 at 09:19
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    @almagest The course is based on Hatcher. For example, I can read Munkres' Algebraic Topology completely fine. – kyary Dec 31 '19 at 09:21
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    If Hatcher is not readable but Munkres is, learn the subject from Munkres (self-study), and perhaps drop the Hatcher-based course. Readability is a very subjective quality. – quasi Dec 31 '19 at 09:27
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    Alternatively, use MSE to ask specific questions about proofs or discussion points in Hatcher that you feel require clarification. Perhaps after a few rough spots, the rest will be normal sailing. – quasi Dec 31 '19 at 09:37
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    I'm thinking I've just hit a wall in my limits in pure math --- Your comment about being "unrigorous and handwavy" is EXACTLY how I felt with much of algebraic topology, and after some very unpleasant experiences I simply stayed away from it. Mathematics is vast and is practiced in many different ways. I originally liked algebra and point-set topology a lot (late 1970s), so it made sense that I'd like algebraic topology. NO, I didn't. I grew up very interested in physics, especially relativity and the idea of higher dimensions, so it made (continued) – Dave L. Renfro Dec 31 '19 at 09:51
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    sense that I'd like differential geometry and general relativity. NO, I didn't. By the very early 1980s, I realized I liked rigorous reasoning and analysis (real and functional, especially) and philosophy and (still) physics, so it made sense that I'd like the mathematical and philosophical foundations of quantum mechanics. No, I didn't (this was 2 years of graduate study in early 1980s). After some more flirtations (ordinal number and ordered set arithmetic issues, general and set-theoretic topology, etc.) I finally found some things that actually worked for me. – Dave L. Renfro Dec 31 '19 at 09:58
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    Just open your mind, and don't be afraid of the unknown. Build your picture about the things you do understand or feel, and don't concentrate on the unknown/scary things.. – Berci Dec 31 '19 at 10:06
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    By the way, I really like this: "I feel like half this proof relies on the reader's intuition about $\mathbb{R}^n$ for $n<=3$ and somehow this translates to general $\mathbb{R}^n$". In particular, the part in your Hatcher quote about "by induction on $n$" especially seems to me like using a micrometer to measure the length and width of a room (part about induction), and visual inspection to measure the height of the room, then recording the volume to the nearest cubic millimeter. – Dave L. Renfro Dec 31 '19 at 10:06
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    I think that the references to genus and orientable surfaces are, in the beginning, not to be taken seriously : they're remarks for your future self who will understand those concepts. Many textbooks have this: they talk about advanced stuff very early on, without introducing the relevant concepts/proofs, relying on some intuition they hope their reader has; or hoping the reader who doesn't know a thing about them will just skim through them. As for the example, it's not a "non rigorous proof" : it's simply not a proof ! (cont.) – Maxime Ramzi Dec 31 '19 at 10:21
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    (cont.) it's an example ! Sometimes examples are very detailed and can be considered as proofs, sometimes they're not detailed. In the latter situation, they can serve various purposes : give some intuition, point out specific phenomena without delving deep into them; or (I think this is probably the point here) they're here for you to complete them : complete the details yourself, prove what has been left unproved : go from the sketch of a proof to a proof. The proofs of Hatcher are (iirc) quite complete, the examples are not, for this reason. – Maxime Ramzi Dec 31 '19 at 10:23
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    I'll blatantly advertise my own book "Topology and groupoids" which was always intended to give the intuitions. It does not do the classification of surfaces as I thought Massey's books were difficult to surpass on that. But there are lots of things you won;t find elsewhere, eg groupoids! As well as its own accounts of covering spaces and orbit spaces. – Ronnie Brown Dec 31 '19 at 11:20
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    Regarding your concerns about Example 0.4, I must ask: how are you on quotient spaces? In my experience, I've found quotient spaces to be a real weak spot in many students' preparation for algebraic topology. Every step of the constructions in Example 0.4 can be justified by theorems on quotient spaces from Munkres, and it would be a fantastic exercise for you to work through those justifications. – Lee Mosher Dec 31 '19 at 17:45
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    @Lee Mosher: (+1) for comment. This is something I realized too late (by then I'd given up on algebraic topology and moved to other things), mainly as a result of my overconfidence in what I knew of general topology and my weakness in quotient space intuition and finesse, probably reinforced by the fact that most of what one sees in later on in books like Munkres, Willard, Kelley, Dugundji, etc. requires very little competence in working with quotient spaces, especially at the intuitive level in non-pathological spaces. – Dave L. Renfro Dec 31 '19 at 18:14
  • @LeeMosher I know quotient spaces from the section in Munkres, but I've never really had the chance to make use of the universal property, which I'm assuming is what you're referring to – kyary Dec 31 '19 at 22:30
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    I'm not particularly referring to the universal property. In Example 0.4, there are, like five or so different quotient space steps following quickly one after the other: first "$\mathbb RP^n$ is topologized..."; then "$\mathbb RP^n$ is also..."; then "This is equivalent to saying..."; then "... we see that $\mathbb RP^n$ is obtained from...". So when I ask "How are you on quotient spaces?", what I'm referring to is whether you can follow these deductions. – Lee Mosher Dec 31 '19 at 23:14

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I highly recommend that you do not start with chapter 0, and if you really want to read Hatcher, just start with chapter 1. Chapter 0 is supposed to be extremely informal in spirit and can be skipped (he says this in the first para), and so it isn't meant to be scrutinized in that way. You are absolutely NOT hitting your limits in pure math; please don't be discouraged. I think a more gentle introduction to algebraic topology is Massey's "Algebraic Topology, an Introduction." It doesn't cover homology or cohomology, but it does the fundamental group very well. There are nice pictures in the book and it is a good continuation from point-set. Then you can pick up Hatcher at chapter 2 and start with homology.

Anubhav Nanavaty
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