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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 17th 2019

am starting some minimum

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 23rd 2019

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?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 23rd 2019

ah, here is an MO answer to that question

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 23rd 2020

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2021

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.