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I got really stuck to the end of Guillemin and Pollack (in particular, here) and plan to give up.

Give up Guillemin and Pollack, not math though.

It seems John Milnor's classic little book topology from the differential view point is an ideal alternate for the first three chapters. So I am thinking of starting that one - good idea? If so, how about the forth chapter on calculus and cohomology in Guillemin and Pollack? What would be an more comprehensible alternative? A reference of a book directing to the specific chapter(s) would be really appreciated.

And also I can just restart Guillemin and Pollack.

So I am wondering the best move at the moment?

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WishingFish
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    For me, the best move would be finding somebody to talk through the ideas I were having trouble with. Such a person might also know you well enough to give better next-step advice than a stranger on Math.SE. – Brett Frankel Aug 05 '13 at 17:22
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    Guillemin and Pollack's book is a softer version of Milnor's book. So it's not going to be any easier reading. You might find it refreshing to read another perspective on very similar material, though. – Ryan Budney Aug 05 '13 at 17:24
  • The point of Guillemin and Pollack's book is that much of basic algebraic topology has very simple and easy-to-digest interpretation in the world of smooth manifolds. So perhaps one way to get a feel for the ambient world G&P is working in would be to study algebraic topology, perhaps out of either the Bredon or Hatcher textbooks, or both. You'll want to study all the basics: fundamental groups, covering spaces, homology, cohomology, Poincare duality and Serre's interpretation of cohomology as maps to Eilenberg-Maclane spaces. As you digest that, G&P will become easier. – Ryan Budney Aug 05 '13 at 17:30
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    I've looked at a few of your G&P problem threads. It seems to me you're dwelling quite a bit on the calculus and analysis end of things. Perhaps read a G&P-friendly calculus book, like Hubbard's calculus book? http://www.math.cornell.edu/~hubbard/vectorcalculus.html – Ryan Budney Aug 05 '13 at 17:46
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    Don't read just one book at a time. – Neal Aug 06 '13 at 04:45
  • Got it, thanks Neal. Seems people are giving me a collection now...:) – WishingFish Aug 06 '13 at 04:46

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My advice (which is something generic to life) is that things do get easier. You might struggle with a particular topic in mathematics for a while but if you keep working at it, then eventually you will improve and your previous mathematical self will be a shadow of your current mathematical self. Of course, sometimes it's just not productive to read a particular textbook anymore and it's better to switch references.

I think as Ryan Budney hints at, it might not be the actual differential topology that is challenging but rather the underlying concepts from multivariable calculus. Do you think that would be an accurate description of things? If so, then one thing that's advisable to do is to go to the bare minimal definitions of what you're reading, e.g., differential forms, pullbacks, the de Rham differential, integration on manifolds. You might re-read this material but when you see a new definition, make it a habit to compute something with it. I can see that's what you're doing based on your questions here and it's excellent. The wonderful thing is that whenever you're stuck, you can just ask a question here.

Milnor's book is a condensed treatment (roughly 60 pages?) of differential topology. However, one book that you might like is Differential Forms in Algebraic Topology by Bott and Tu. It's an excellent textbook that gives a different perspective of differential forms via their use in algebraic topology. You might find it interesting.

I hope this helps!

Amitesh Datta
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  • That's really helpful, thanks Amitesh. Interestingly, James Cook also recommended me Tu's book, http://www.amazon.com/An-Introduction-Manifolds-Universitext-Loring/dp/1441973990. It does make sense that both are good and to the topic. But at least I would like to know which one to start? – WishingFish Aug 06 '13 at 04:36
  • @WishingFish The book you linked is more elementary, I think, than Bott and Tu's textbook. However, in theory, they are independent of one another (Bott and Tu claim in the preface that only a knowledge of multivariable calculus, linear algebra, and point-set topology is necessary to read their textbook). In saying that, I think James Cook's recommendation is great given your background and what you're thinking about right now. Why not read a little bit of the beginnings of both books and decide which is best for you? Bott and Tu's book has an algebraic topology flavor. – Amitesh Datta Aug 06 '13 at 07:11