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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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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. –  Ted Shifrin Jun 2 at 19:15
    
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. –  John M Jun 2 at 20:19

1 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.

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