Another proof that a vector field on the sphere must have a zero?

[mathwonk]

lavinia maybe you are taking for granted all the background in a book like Bott -Tu. If so all these extra matters I mention may seem trivial to you. E.g. on p. 71, they show how to use a "bump" function to make a smooth extension of a radius function. Then in their proof on page 124 of the degree result, they appeal to this construction of a "global angle form". That is why it takes them 129 pages to give the proof. I am just saying none of this machinery is needed for Lefschetz's argument, beyond the idea that if there are no zeroes inside the circle then the degree on the circle itself would be zero. But in his computation it is visibly 2, for any vector field with finitely many zeroes. done.

Everything you are doing looks great and seems correct. I am just pointing out how simple Lefschetz makes this argument look. In his formulation it can be given to almost anyone. (Even I can understand it and i don't even know what a Thom class is.)

 

[lavinia]

lavinia maybe you are taking for granted all the background in a book like Bott -Tu. If so all these extra matters I mention may seem trivial to you. E.g. on p. 71, they show how to use a "bump" function to make a smooth extension of a radius function. Then in their proof on page 124 of the degree result, they appeal to this construction of a "global angle form". That is why it takes them 129 pages to give the proof. I am just saying none of this machinery is needed for Lefschetz's argument, beyond the idea that if there are no zeroes inside the circle then the degree on the circle itself would be zero. But in his computation it is visibly 2, for any vector field with finitely many zeroes. done.

Everything you are doing looks great and seems correct. I am just pointing out how simple Lefschetz makes this argument look. In his formulation it can be given to almost anyone. (Even I can understand it and i don't even know what a Thom class is.)

I just redid the Lefschetz proof for myself in the gym and see its elegance. Take a vector field and rotate the sphere so that the field at the north pole is not zero. Map the field minus a small polar ice cap into the plane by stereographic projection. The zeros are trapped inside a large disc and on the boundary you have the image of an approximately constant vector field. If you normalize the image field to have length one you get a map of the disc into the unit circle. The usual Milner/Hopf argument which I mentioned above now applies and tells you that the sum of the indices of the zeros inside the disc equals the index of a constant vector field at infinity - which is 2. Very cool.

 

[mathwonk]

sounds like you get it more clearly than i do. great! i loved it when i read it sitting in the library, some 40 years ago. It took me a while to visualize it though. Lefschetz was a neat guy, very upbeat, in spite of having lost his hands in an accident as an engineer, nothing to do but become a great pure mathematician. He looked kind of like a cheerful Dr. No with prosthetic hands. There is cool little AMA (or AMS?) movie of him explaining his fixed point theorem.

 

[mathwonk]

of course this proof puts all the heat on the basic theorem that a smooth map of a disc to the circle must have winding number zero on the boundary circle. I had fun making up a proof of this as a young teacher, using stokes theorem, which i now know is the standard proof. at the time i was just trying to think of some way to apply stokes theorem, for my class. i wound up presenting brouwer fixed point, fundamental theorem of algebra, and the vector field theorem, to my advanced calculus class. this sort of thing is standard now too.

in the plane all you have to know is stokes theorem, plus how to integrate dtheta around a circle. (Bott - Tu's global angle form). In three space you use the analogous solid angle form, e.g. in spherical coordinates. I never understood why books like thomas' calculus presented calculations with all these technical integrals and never used them for anything interesting.

One thing kind of interesting here is that in the plane this is a mod 2 problem but in the three space it is subtler. I.e. the fixed point theorem only requires the fact that 1 ≠ 0, while the degree theorem requires that 1 ≠ -1, much more difficult. I.e. you can do the first with mod 2 homology or intersection numbers, but not the second.

 

[mathwonk]

by the way thanks very much for the discussion. i now understand how lefschetz's hands on proof fits into these other ideas MUCH more than I did before.

 

[lavinia]

I.e. you still need to prove such an extension exists, plus you must prove that all vector fields have the same total degree. It seems to me this is far more than needed for the result, as Lefschetz's simple argument shows. I like seeing that they are in essence the same though.

I was not thinking about the extension. The point was that you easily see the attaching map of the two solid tori from this vector field.

 

[lavinia]

by the way thanks very much for the discussion. i now understand how lefschetz's hands on proof fits into these other ideas MUCH more than I did before.

I think this thread was highly instructive. I enjoy talking with you. I always learn something from you.

 

[mathwonk]

thank you lavinia. I too get a lot of enjoyment from your posts. I think you are quite strong and I am sure you have a bright future in any area you choose. Do you get all this from reading? Or have you also taken some good classes?

I suggest you look at mathoverflow if you have not. It is more of a grad student/professional level math site and I have learned a lot just from reading the questions and answers there.

 

[mathwonk]

apparently i have not read your discussion of solid tori. my apologies. i am somewhat slow to learn. i like to think of my own solutions. i don't know a lot, but i understand fairly well the things i do know.

 

[lavinia]

thank you lavinia. I too get a lot of enjoyment from your posts. I think you are quite strong and I am sure you have a bright future in any area you choose. Do you get all this from reading? Or have you also taken some good classes?

I suggest you look at mathoverflow if you have not. It is more of a grad student/professional level math site and I have learned a lot just from reading the questions and answers there.

I am in awe that you knew Lefschetz. He is one of the immortals.

I mostly read - lately papers - but have sat in on a few classes. what do you do mathematically?

 

[lavinia]

Related to all of this is the differential geometry of the sphere. When it has constant curvature,1, the exterior derivative of its connection 1 form on the unit circle bundle is the pullback of its volume form under the bundle projection map

If there were a non zero vector field, we could normalize it to have length 1 to get a non-zero section of the unit circle bundle. Then the volume form of the sphere would have to be exact. This is the same contradiction one gets from knowing that the tangent circle bundle is RP^3.

The same arguments apply to other surfaces except for the torus since they all can be given geometries with constant negative curvature.

It also follows that any metric on the torus must have points of zero curvature.

So the connection 1 form again tells you that the tangent circle bundle is not a trivial bundle. How does this link up with everything else we have been talking about?

Also the connection 1 form reminds me of the differential form approach to linking number.

 

[mathwonk]

I only read his work, and saw a movie featuring him. That movie is still available I would guess, from the MAA?

well i cannot locate a copy of that movie but the library at cornell has the ones of bott and marston morse, but those as i recall are not as entertaining.