Am I reading Bott - Tu right?

Question

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.

Answer 1

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.

Answer 2

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

Answer 3

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.