# Examples of Computations in Algebraic Topology

I have started reading "Differential Forms in Algebraic Topology" by Bott, Tu, recently. While I'm quite happy with the exposition of the theorems and explanation of theoretical results, I'm missing examples in many places.

There are some exercises in the text, which for example ask one to compute cohomology of the Möbius band or the Euler class of the tautological bundle on $\mathbb C P^n$. However, I'm having some difficulty with this, in particular I'm never really sure, whether I'm doing things the right way.

I would therefore like to take a look at some examples of concrete computations, so I could get some kind of guidance on how to do it.

So I was wondering whether there is a book which contains many examples or maybe there are some ressources on the internet that I've been unable to find?

Thanks in advance for any pointers!

Best regards,

Sam

Edit: Maybe I should add that the topic of the book is not so much general algebraic topology.

• De Rham Theory
• Cech-de Rham Complex
• Spectral Sequences and Applications
• Characteristic Classes

Throughout the book one is working with differentiable manifolds.

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This isn't a specific question, it's a request for examples. You can get plenty of homology/cohomology example computations from on-line textbooks, like Hatcher's "Algebraic Topology". –  Ryan Budney Apr 21 '11 at 4:29

Try "Geometry of Differential Forms" by Shigeyuki Morita. It will help you build intuition for Differential Forms, the de Rham Cohomology, as well as Characteristic Classes. After reading Morita, you would still have to read the more specialized topics of Bott and Tu like the chapter on Spectral Sequences.

If on the other hand you are only stuck with the Algebraic Topology stuff in Bott and Tu, I recommend reading one of the standard introductory textbooks. Some of them (like Hatcher) are also available as ebook, as Ryan Budney has already pointed out.

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Thanks for your suggestion. I have had a look at it and it seems to be rather a good book. –  Sam Apr 22 '11 at 19:35

Well, I think that Dubrovin, Fomenko, Novikov's Modern geometry 2,3 will be helpful.

They are volume 104, 124 in Graduate texts in mathematics.

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Thanks, these books seem worth having a look at as well. In general, GTM books are great. –  Sam Apr 22 '11 at 19:54