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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?
ah, here is an MO answer to that question
added this pointer:
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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