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Many graphical examples of the hairy ball theorem show the cowlicks at opposite poles. Is this arrangement necessary, or arbitrary?

I believe technically the theorem says how many cowlicks there must be, not where they are. Intuitively, it seems like potentially the cowlicks could be moved slightly from the axis. However the extreme case, where both cowlicks are right next to each other, seems difficult to imagine.

Also, if you could move the cowlicks to one side, and the cut out that part of the sphere, you would end up with an almost sphere that has no cowlicks. However, it seems like there must be at least one cowlick even in a partial sphere (if it's more than a hemisphere at least).

Micah
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Superbest
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    They can be anywhere. Just push it all around a bit. – Qiaochu Yuan Jul 25 '16 at 05:31
  • "push it all around a bit" extends to "apply any continuous function from the 2-sphere to itself". For a specific example draw a small circle around the point you're standing on, then invert with respect to that circle. Assuming you're not standing somewhere very cold right now, this leaves the Earth's North and South poles both inside that circle you drew, and hence not opposite each other. – Steve Jessop Jul 25 '16 at 10:32
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    One fundamental reason why there cannot be such a restriction is that the theorem applies to purely topological spheres, which don't even come with an inherent notion of "opposite points". – Klaus Draeger Jul 25 '16 at 13:00
  • Once you cut a section of the sphere out it is topologically a sheet and you can just comb the cowlick off the edge. – Evan Jul 25 '16 at 14:34

2 Answers2

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It is not necessary. For an extreme counterexample, see this image from the Wikipedia article for the hairy ball theorem:

this picture

This is a picture of a hairy ball where the hairs only vanish in one point — but at the cost of making the behavior in a neighborhood of that point more complicated.

Micah
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    Might be worth adding that you can realize this lovely picture as the vector field with flow lines given by the stereographic projection of parallel straight lines in the Argand plane onto the Riemann sphere. – Selene Routley Aug 09 '16 at 13:47
  • Also see here for pictures: http://www.pitt.edu/~hajlasz/Popularization%20of%20Mathematics/Czesanie/Czesanie.htm – Behnam Esmayli Aug 10 '16 at 01:02
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The arrangement is arbitrary; the "cowlicks" need not be on opposite poles. If you want a nice step-by-step proof/exercise, see Pugh's Book Real Mathematical Analysis.

Moreover, there need not be more than one cowlick in the statement of the assertion - see wikipedia for instance.

Alekos Robotis
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