From: rusin@washington.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: A combed hairy ball has AT LEAST TWO bald spots!! Date: 3 Mar 1995 21:47:37 GMT In article <3j7ts4\$tsj@bubba.ucc.okstate.edu>, John Chandler wrote: >There have been several posts in sci.math recently >saying, in effect, > A combed hairy ball has at least one bald spot. >A much more difficult and beautiful result was proved by Brouwer: > A combed hairy ball has at least two bald spots!! Well, since I gave an exercise to show this is NOT true I guess it's up to me to make the correction. This is true _if_ the "bald spots" are counted according to multiplicity, but not otherwise. Here's a solution with one bald spot: Since the sphere with one point removed is homeomorphic to the plane, use the fact that a constant (and thus non-zero) vector field on the plane exists, so one gets a vector field on the once-punctured sphere. In pictures, do this. Have the vector field point along a great circle from the North pole, through Greenwich, the Southpole, the dateline, and back. The rest of the vector field also points through circles beginning at the north pole. Each circle departs a little more from the first one, with the worst deviation being at its southernmost point. Eventually the circles look like small circle centered at points a little away from the north pole lying on the 90-degree and 270-degree lines of longitude. You can recognize the bald spot at the north pole as a double point by thinking of the vector field I just described as the limit of the following ones: Begin with a vector field which points along circles which are centered on the sphere's, uh, east-west axis. Then, gradually drag both the bald spots (at the East and West poles, I guess) until they both reach the north pole; at each stage, the vector field should consist of a bunch of parallel lines lined up between the bald spots, then swung around the earth (spreading out a little as they head south). These lines point in the direction of the magnetic field caused by a current running in a wire passing through the bald points and the earth's center (I think). Well, I guess this proves one thing: a picture's worth at least 300 words. :-) dave ============================================================================== Date: Fri, 3 Mar 1995 16:42:26 -0800 From: [Permission pending] To: rusin@washington.math.niu.edu Subject: Re: A combed hairy ball has AT LEAST TWO bald spots!! In article <3j82pp\$f69@watson.math.niu.edu> you write: >In article <3j7ts4\$tsj@bubba.ucc.okstate.edu>, >John Chandler wrote: >>There have been several posts in sci.math recently >>saying, in effect, >> A combed hairy ball has at least one bald spot. >>A much more difficult and beautiful result was proved by Brouwer: >> A combed hairy ball has at least two bald spots!! > >Well, since I gave an exercise to show this is NOT true I guess it's up >to me to make the correction. This is true _if_ the "bald spots" are >counted according to multiplicity, but not otherwise. I would have omitted this last sentence, lest it confuse the bewildered. Here's a solution > >dave ------------------------------------------------------------- I would have just said: Consider a line L tangent to the unit sphere S in R^3 at the point N. For every plane P contining L, consider the intersection of P with S. These create a family of circles C on S all disjoint from each other, except that they all contain the point N. Now define a vector field on S by at each point x, use the tangent direction of the unique circle C through x, with magnitude equal to the distance from the point N. Except if x = N, in which case let it be the zero vector. [Permission pending] ============================================================================== Date: Mon, 6 Mar 1995 09:00:45 -0800 From: [Permission pending] To: rusin@math.niu.edu (Dave Rusin) Subject: Re: A combed hairy ball has AT LEAST TWO bald spots!! Actually my example works fine for a C^0 example, but it's not clear in what differentiability class it lies, due to the fact I used "distance to N" as the magnitude of the field. Probably (distance to N)^2 would lead to a C^oo field, or maybe even C^w. [Permission pending] ============================================================================== From: jpc@a.cs.okstate.edu (John Chandler) Newsgroups: sci.math Subject: > A combed hairy ball has AT LEAST TWO bald spots!! Date: 6 Mar 1995 20:05:42 GMT Last week I paraphrased from memory L. E. J. Brouwer's Hairy Ball Theorem. Unfortunately I may have simplified and/or generalized it to a form that has the disadvantage of not being true. Pardon me. The correct statement of the theorem is: Theorem 8: A reduced distribution on a sphere possesses at least two radiating points. -- L. E. J. Brouwer, II KNAW Proc. 12 (1910) 716-734 This can also be found on page 301 of L. E. J. Brouwer -- Collected Works Volume 2: Geometry, Analysis, Topology, and Mechanics Edited by Hans Freudenthal North-Holland Publishing Company, 1976 -- John Chandler jpc@a.cs.okstate.edu