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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 17th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexam starting some minimum <a href="https://ncatlab.org/nlab/revision/Poincar%C3%A9%E2%80%93Hopf+theorem/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Poincar%C3%A9%E2%80%93Hopf+theorem">current</a>

   am starting some minimum

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 23rd 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexcan 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 23rd 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexah, here is an [MO answer](https://mathoverflow.net/a/285200/381) to that question

   ah, here is an MO answer to that question

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded this pointer: * [[John Milnor]], Chapter 6 of: _Topology from the differential viewpoint_, Princeton University Press, 1997. ([ISBN:9780691048338](https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/Poincar%C3%A9%E2%80%93Hopf+theorem/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/Poincar%C3%A9%E2%80%93Hopf+theorem/10">v10</a>, <a href="https://ncatlab.org/nlab/show/Poincar%C3%A9%E2%80%93Hopf+theorem">current</a>

   added this pointer:

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

   diff, v10, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 23rd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexBriefly 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. <a href="https://ncatlab.org/nlab/revision/diff/Poincar%C3%A9%E2%80%93Hopf+theorem/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Poincar%C3%A9%E2%80%93Hopf+theorem/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Poincar%C3%A9%E2%80%93Hopf+theorem">current</a>

   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