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Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do people recommend?

First - some background; I study on my own. I've read Hatcher's Algebraic Topology (all the way to the end of 4.2) and solved about 75% of the exercises. I've also read Tu's Introduction to Manifolds and solved most of the exercises.

I'd like to move forward within Algebraic Topology and Differential Topology. Some of the topics I'd like to learn are spectral sequences, characteristic classes, Cech cohomology.

I decided to read Bott - Tu next as it covers those topics and everyone praises the clarity of this book.

I'm 80 pages into the book and I've found it to omit a lot of important details. For example the introduction to vector bundles is too brief. It states a lot of facts without proof (algebraic operations on bundles, construction from structure group). I'm finding myself constantly hunting other sources to fill in the details. This is a rather time consuming process. It isn't always easy to find notes or books with the right information at the right level.

The book has few exercises. They are either too easy or impossibly difficult unless you look around. For an example of the latter, one exercise expects the reader to come up with the clutching construction on his/her own. This takes several pages on Hatcher's notes on vector bundles.

Is Bott - Tu expected to be a second reading on the topics it covers? To be fair I found the sections that I'm already familiar with very readable but I didn't learn much more either.

What other books do you recommend as the next step for me? Per this answer, I'm tempted to print off Hatcher notes on characteristic classes and spectral sequences and read those instead. My only problem with them is that they don't have many exercises.

I'm sorry for the long post. I'm studying on my own and I need some guidance.

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    $\begingroup$ You might look at Hirsch's Differential Topology and Spivak's Comprehensive Introduction to Differential Geometry, volume 1, for more basics on manifolds and bundles, along with lots of exercises. Generally you will find that once you get to more advanced books/courses, there is sadly a lack of good exercises. But a lot of the material in Bott-Tu is seriously difficult to find anywhere else — e.g., the proof that Poincaré duality corresponds to transverse intersection of submanifolds. $\endgroup$
    – Ted Shifrin
    Jun 2, 2014 at 19:15
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    $\begingroup$ One text intermediate between Tu and Bott & Tu is Madsen & Tornehave From Calculus to Cohomology. It is pretty dense, with a lot of material. But it also has a lot of exercises. $\endgroup$
    – John M
    Jun 2, 2014 at 20:19
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    $\begingroup$ @JohnM The text you suggests sounds like it has the same problems as B&T if one believes the Amazon Reviews. For example "Utter Lack of Detail and Lack of Motivation Render Text Unreadable" -- but most reviews agree that it's not a good book. $\endgroup$
    – Rudy the Reindeer
    Jun 6, 2016 at 9:10
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    $\begingroup$ M&T is much easier than B&T, and much harder than Tu. I think the reviewers probably were reading M&T when they should have been reading Tu as a first exposure. $\endgroup$
    – John M
    Jun 6, 2016 at 15:31

3 Answers 3

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I think Differential Forms in Algebraic Topology is well written. However, it is written in a way a master teaches a student: There are no loss of detail, no fancy language, and explicit examples are everywhere. The student is supposed to emulate Bott's proof writing style, to criticize it, to tease it and understand every detail of it. When I read it I often find questions I would ask myself was discussed by him in very detail. But I am also aware that this is my second or third time reading the book; when I read it the first time my feeling was quite similar - lost, swamped in "tedious" computations and cannot see the big picture. In short the book was difficult.

What happened? The answer might be mathematical maturity. Bott was already at a level that he could freely talk about ideas and save the computational details for his graduate students as homeworks. He had Stephen Smale and Daniel Quillen as his PhD students. But instead he decided to serve an exemplary role and not hiding things under the rug. This is a seemingly "low level" book aiming for an audience that could see through the big picture. I am sure that Bott could offer sweeping quick and slick proofs for most of the contents in the book - a lot of the books written nowadays by fake experts are like that. However the reader may not able to do any actual computation after reading these books. To be unusually honest, sometimes even the book authors cannot do it either. So in a sense Bott's book filled in a vaccuum.

Unfortunately, what Bott did was still simultanously regarded as "too difficult" and "too easy" by different group of math graduate students. I have heard people outside the field claiming "I finished Bott&Tu in a week", "Bott&Tu is really for undergraduate students", "Bott&Tu is not enough to understand Xxx topic properly" etc. In the same time inexperienced amateurs got easily frustrated. It takes time to realize how important of Bott's works are despite its seemingly simplicity. It takes time to appreciate it and use these ideas in one's own research. And this experience ultimately comes from thinking about these things independently. Thus one has to be patient, with Bott and with one's own potential intellectual development.

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    $\begingroup$ I disagree.This is only true if you have access to lectures, at least some classes and are in graduate school with other similar minded classmates and teachers. As a self-study book, this book is horrible. There is no reason not to give more clear and detailed explanations otherwise. This is a second or third book. In fact no truly good graduate level first book exists in Algebraic Topology. Unlike say riemannian geometry ( do carmo) or analysis (Rudin). $\endgroup$
    – alireza
    Jun 7, 2016 at 5:29
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    $\begingroup$ more accurately, the ones I have looked at, most of your list actually. Bredon for example is a dense reference with little care to be pedagogical. Such things are subjective of course, your large reading list sets you apart in perspective from someone not in graduate school to study topology but looking for a book to learn from. One needs a few hard books read with pain, yes that is good but most books must be, at their level transparent. It is the art of pedagogical writing not to constantly pain the reader as most of these books do. $\endgroup$
    – alireza
    Jun 7, 2016 at 21:20
  • $\begingroup$ I miswrote I meant Royden. I find expecting knowing bott hard to digest, of course the topics you mention is not all that book is. It is for sure not a entirely an undergraduate book, not without a teacher. If you cannot agree on that, the questioner and I have vastly divergent understanding and will probably get no where. $\endgroup$
    – alireza
    Jun 7, 2016 at 21:30
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    $\begingroup$ To be frank, I agree with @alireza Bredon is in no way an undergraduate book, much less for freshmen. Maybe if you went to a very special high school where people learn topology and real analysis by the time they graduate (I have heard of such things, but they are extremely uncommon). For most school districts, where getting a diploma and knowing anything at all about calculus (even without the most basic understanding of proof) is considered an accomplishment, the idea that books like Bredon could be profitably read and understood by college freshmen is absurd. $\endgroup$
    – Chill2Macht
    Mar 18, 2017 at 2:58
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    $\begingroup$ @William:To be unusually honest, I am still struggling in graduate school and my other friend who read Bredon in her freshmen year quit academia already. There is a big difference between digesting material at a fast pace and understanding the material properly (which is addressed in my answer), and an even greater gap between having a deep understanding versus creating influential new mathematics. Mathematicans pride themselves in creating new mathematics, not by reciting theorems from decades old yellow covered books. Maybe you are a late starter, but so are Bott and many others. Good luck! $\endgroup$
    – Bombyx mori
    Mar 18, 2017 at 9:06
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It is definitely for someone who already knows the subject and is looking for a different perspective. It is not advanced and it is not introductory, more a supplement. It is also sloppy and very hard to follow for someone who does not know the subject. The praises are from people who know the subject and like the presentation and a few things not easily found elsewhere. Overall, it is not a good book in my opinion.

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    $\begingroup$ Is your answer assuming or not assuming that the reader of Bott and Tu Differential Forms has read something equivalent to Tu Introduction to Manifolds (written as a prerequisite for Bott and Tu Differential Forms)? $\endgroup$
    – user636532
    Jun 2, 2019 at 4:17
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    $\begingroup$ Assuming that reader has read such a text. $\endgroup$
    – alireza
    Jul 16, 2019 at 20:46
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    $\begingroup$ Thanks @alireza! $\endgroup$
    – user636532
    Jul 19, 2019 at 5:23
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For an introduction to manifolds and especially bundles with motivation and examples, why not Milnor & Stasheff? de Rham doesn't show up until the end of the book - differential forms were not in the original lectures - hardly anyone paid attention to de Rham cohomology in those days

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