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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2019

    am starting some minimum

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2019

    can one do the following?

    For XX a closed manifold, and vv 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

    added this pointer:

    diff, v10, current

    • 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.

    diff, v12, current