I recently read the following question about the Poincare-Hopf theorem in a compact submanifold. All the answers were very satisfactory to me. Is there any reference where I can look for more details of the proof given on the first answer?
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$\begingroup$ Could you be a little more specific on what you would like details on? This argument about adapting vector fields to triangulations is a standard one. You can likely find it in a few differential topology textbooks, for example I think Guillemin and Pollack has a version of this kind of argument. $\endgroup$– Ryan BudneyCommented Sep 5, 2022 at 18:22
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$\begingroup$ I am trying to understand how one can adapt the Poincare-Hopf index theorem to a non-compact manifold. In particular, I am considering a vector field that does not vanish at the infinity of my non-compact manifold. The answer of Professor Bill Thurston is exactly what I am looking for. I would like to read a reference to improve my knowledge (I am self studying the Poincare-Hopf theorem and this topic is new to me). $\endgroup$– NinpouCommented Sep 5, 2022 at 19:03
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$\begingroup$ For non-compact manifolds you have the problem that the Euler characteristic may not be defined. Are you restricting to some class of manifolds where it is defined? $\endgroup$– Ryan BudneyCommented Sep 5, 2022 at 19:24
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$\begingroup$ The manifold is the infinity cylinder $M=S^1\times \mathbb{R}$. I have a vector field $V$ defined all over $M$. I know, from the context of my problem, that this vector field does not have isolated zeroes for high values of height $H$. I want to compute some topological invariant related to the zeros of $V$. $\endgroup$– NinpouCommented Sep 6, 2022 at 0:46
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$\begingroup$ I was wondering if I can choose a compact submanifold $N, \partial N \subset M$ where V does not vanish on the boundary $\partial N$. Then I would compute the Poincare-Hopf theorem for the submanifold $N, \partial N$. Since I know the behavior of the vector field at infinity, it would be sufficient for me to know the index of $V$ in a compact region $S^1\times[-H, H]$ where H is a larger number. $\endgroup$– NinpouCommented Sep 6, 2022 at 0:49
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