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I am an graduate student interested in fluid dynamics and have almost zero background in differential and algebraic topology. I must say that I do know some analysis (Lebesgue integration plus basics of functional analysis) and some solid linear algebra (including canonical forms etc) plus the rudiments of Abstract Algebra (not including modules or Galois theory). But I don't know any differential or algebraic topology. (i.e I don't know anything about fundamental groups or homology)

However, I need to make a presentation about the Poincare- Hopf theorem in about two months. Is there any reference that contains a fairly elementary discussion of this theorem along with the relevant concepts and perhaps the complete proof which can be followed by someone with no background in these things.

I was told to look at the textbook by Guillemin and Pollack on Differential Topology. Any additional references which are simple and intuitive and present a complete mathematical proof without skipping steps and assuming too much of a mathematical maturity would be gratefully appreciated.

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Guillemin and Pollack is a very nice reference for this. You don't need to know any homology, or anything about fundamental groups. It's largely just basic calculus + Sard's theorem. G&P don't take a geodesic to the proof, but it's fairly efficient. I think you could condense a proof for a presentation quite easily, focusing on the basics of the tangent bundle $TN$ and how it sits naturally in $N \times N$. –  Ryan Budney Feb 25 '12 at 1:00
Milnor's "Topology from a differentiable viewpoint" also does the job. –  Ryan Budney Feb 25 '12 at 1:08
Thanks a lot for your answer. –  Shibi Vasudevan Feb 25 '12 at 1:41
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up vote 4 down vote accepted

The key geometric idea in the proof is to consider the manifold $M$ as sitting inside the tangent bundle $TM$. You perturb that embedding to some new one, $M'$ which is transverse to $M$, and take that (signed) intersection number. So a low-tech analogue to this is to think of the "central core" $C$ of the Moebius band, and think through how, regardless of how you deform $C$ in the Moebius band, there is generically an odd number of points in $C \cap C'$.

The perturbation can be done via the flow along a vector field, so all you need to do is cook up an appropriate vector field where you can pull-off the computation of the local degrees. A popular way to finish the argument is to adapt a vector field to a triangulation (or any other appropriate stratification of the manifold) and compute all the local degrees that way. With the triangulation at least, it's clear they add up to the Euler characteristic.

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Thanks a lot for your answer. –  Shibi Vasudevan Feb 25 '12 at 1:42
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