Briefly added converse statements:

a) If the Euler char of a closed mfd vanishes, then a nowhere vanishing vector field exists.

b) For any positive finite number of points in any connected closed mfd, there is a vector field vanishing at most at these points.

Still need to add good citations.

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- John Milnor, Chapter 6 of:
*Topology from the differential viewpoint*, Princeton University Press, 1997. (ISBN:9780691048338, pdf)

ah, here is an MO answer to that question

]]>can one do the following?

For $X$ a closed manifold, and $v$ a vector field with isolated zeros, remove small disks around the zeros and then *glue them back in*, but now with the boundary spheres glued by the negative of the Hopf winding degree – such that the resulting new manifold has vanishing Euler characteristic? Or something like this?

am starting some minimum

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