🧵 rate my proof
Anonymous at Tue, 2 Jul 2024 18:22:04 UTC No. 16264651
Every vector field on a sphere must have a singularity. One can prove this by considering the integral lines of the vector field. If there are no singularities then the integral lines define a retraction onto a greater circle which would force the fundamental group of the sphere and the greater circle to be the same. The fundamental group of the greater circle is the integers whereas the sphere's is trivial. This is a contradiction so every vector field on the sphere must have a singularity.
Anonymous at Tue, 2 Jul 2024 18:38:04 UTC No. 16264676
>>16264651
>the integral lines define a retraction onto a great circle
This isn't obvious to me. Can you elaborate?
Anonymous at Tue, 2 Jul 2024 18:51:39 UTC No. 16264695
>>16264651
It's dumb, because of what >>16264676
Why do you lie about having a proof when all you have is a vague (and false) intuition?
Anonymous at Tue, 2 Jul 2024 18:53:00 UTC No. 16264697
>asks to rate their proof
>proof: it was revealed to me in my dream
Anonymous at Tue, 2 Jul 2024 19:04:40 UTC No. 16264718
>>16264676
the field is essentially a differential equation on a compact manifold so there is a solution. the solution is the flow lines which must intersect the greater circle at a unique point. this defines the retraction
Anonymous at Tue, 2 Jul 2024 19:05:47 UTC No. 16264720
>>16264718
> the solution is the flow lines which must intersect the greater circle
Doesn't follow.
Anonymous at Tue, 2 Jul 2024 19:06:10 UTC No. 16264724
>>16264695
>>16264718
which part is incorrect?
Anonymous at Tue, 2 Jul 2024 19:07:40 UTC No. 16264726
Anonymous at Tue, 2 Jul 2024 19:07:54 UTC No. 16264727
>>16264720
do you have a counter-example? if there was no unique intersection then two flow lines would meet at the same point and have different vectors at the point. this obviously can't happen. the flow lines do not intersect and foliate the manifold
Anonymous at Tue, 2 Jul 2024 19:08:09 UTC No. 16264728
>>16264718
Explain why such a great circle intersecting all flow lines must exist. Why is it not possible that there are flow lines never intersecting the great circle?
Anonymous at Tue, 2 Jul 2024 19:08:54 UTC No. 16264729
>>16264726
provide a counter-example
Anonymous at Tue, 2 Jul 2024 19:09:55 UTC No. 16264735
>>16264728
because the field is non-zero at every point. standard argument from ODEs guarantees a solution for all time
Anonymous at Tue, 2 Jul 2024 19:10:49 UTC No. 16264739
>>16264727
> do you have a counter-example?
No, because there are no nonsingular vector fields on a 2-sphere. You're claiming to prove a true theorem but that doesn't automatically make your "proof" correct.
> if there was no unique intersection then two flow lines would meet at the same point
Prove it.
Anonymous at Tue, 2 Jul 2024 19:11:50 UTC No. 16264741
>>16264729
see >>16264739
>>16264735
So you're just trolling now ig
Anonymous at Tue, 2 Jul 2024 19:12:23 UTC No. 16264743
>>16264735
That's not what I asked. This only shows that flow lines exist. Not that they have to intersect the same great circle.
Anonymous at Tue, 2 Jul 2024 19:19:40 UTC No. 16264771
>>16264743
oh i see. ok i know how to fix the argument
Anonymous at Tue, 2 Jul 2024 19:21:17 UTC No. 16264776
>>16264771
Does your fix involve using algebraic topology?
Anonymous at Tue, 2 Jul 2024 19:23:59 UTC No. 16264782
>>16264776
i'm using the fundamental group so yes
Anonymous at Tue, 2 Jul 2024 19:27:05 UTC No. 16264788
>>16264776
What's the lamba in that screenshot?
Anonymous at Tue, 2 Jul 2024 19:30:53 UTC No. 16264795
Anonymous at Tue, 2 Jul 2024 19:31:54 UTC No. 16264797
>>16264782
Yeah that's not going to work.
Anonymous at Tue, 2 Jul 2024 19:35:03 UTC No. 16264800
>>16264797
i'm pretty sure it will. i just have to show that all flow lines have finite length and are closed
Anonymous at Tue, 2 Jul 2024 19:38:55 UTC No. 16264804
>>16264800
>i just have to show that all flow lines have finite length and are closed
Suppose they were. How would the theorem follow?
Anonymous at Tue, 2 Jul 2024 19:43:36 UTC No. 16264812
>>16264804
standard bisection argument and uniqueness of flows splits the sphere into two parts. looking at the boundary from each side means we can pretend to be on a disk and then the result follows by brouwer's fixed point theorem
Anonymous at Tue, 2 Jul 2024 19:45:58 UTC No. 16264816
>>16264812
How do you obtain a map without fixed points?
Anonymous at Tue, 2 Jul 2024 19:47:03 UTC No. 16264819
>>16264816
it's the standard argument for showing there is no retraction from the disk to its boundary
Anonymous at Tue, 2 Jul 2024 19:49:43 UTC No. 16264823
>>16264819
anyway, i'm pretty happy with this. the only hard part is showing the flow lines are of finite length and closed but i can do this with another infinite bisection argument and reach a contradiction.
Anonymous at Tue, 2 Jul 2024 19:55:35 UTC No. 16264833
>>16264819
1. The Jordan curve theorem gives you two components whose boundary is the curve. But how do you know that these two components are homeomorphic to disks? AFAIK Jordan-Brouwer curve theorem doesn't give you this.
2. Assuming you've got a disk bounded by a flow curve in your sphere, how do you obtain the retraction to the boundary?
Anonymous at Tue, 2 Jul 2024 19:57:38 UTC No. 16264836
>>16264833
Oh just looked it up, so nevermind about the 1. Apparently it's called the Jordan-Schoenflies theorem. Question 2 still stands though.
Anonymous at Tue, 2 Jul 2024 19:59:04 UTC No. 16264837
>>16264833
standard argument. like i said, the hard part is showing finite length and closed. the finite length and closed means i can always perturb it to a smooth curve with a standard compactness argument and the rest follows by non-existence of retraction to the boundary using fundmamental groups
Anonymous at Tue, 2 Jul 2024 20:01:43 UTC No. 16264840
>>16264837
Can you just sketch out the construction of the retraction or point to the name of the theorem?
Anonymous at Tue, 2 Jul 2024 20:02:44 UTC No. 16264843
>>16264837
> rest follows by non-existence of retraction to the boundary using fundmamental groups
That's exactly what I keep asking you about. What's the retraction?
Anonymous at Tue, 2 Jul 2024 20:05:58 UTC No. 16264847
>>16264840
i'm pretty sure you can figure it out if you try
Anonymous at Tue, 2 Jul 2024 20:09:20 UTC No. 16264850
>>16264847
Given that you've repeatedly made false claims ITT and demonstrate no understanding of what a proof is, you being sure doesn't mean much to me.
Anonymous at Tue, 2 Jul 2024 20:10:07 UTC No. 16264851
>>16264850
and yet it looks like i have found a novel proof
Anonymous at Tue, 2 Jul 2024 20:11:33 UTC No. 16264853
>>16264851
Will eagerly await the publication on vixra.
Anonymous at Tue, 2 Jul 2024 20:13:37 UTC No. 16264858
>>16264851
but let's spell it out. we have reduced to the case of a disk and it's boundary from the sphere by using the existence of a closed curve. since there are no singularities we get a mapping from the disk to the boundary which gives us a map of fundamental groups. the disk has trivial fundamental group so this map doesn't exist which means there must be a singularity in the disk
Anonymous at Tue, 2 Jul 2024 20:14:47 UTC No. 16264862
>>16264837
That sounds extremely sus. Repeatedly appealing to alleged "standard arguments" makes it only worse. I doubt that finite length + closed will suffice to construct your retraction. In fact I'm fairly certain that three closed flow lines cleverly placed on a sphere will make a choice of a great circle intersecting all of them impossible.
Anonymous at Tue, 2 Jul 2024 20:14:56 UTC No. 16264863
>>16264853
no need. this was fun. i knew it could be done. now i just need to prove every flow line has finite length and is closed
Anonymous at Tue, 2 Jul 2024 20:15:38 UTC No. 16264866
>>16264862
Standard argument, trust.
Anonymous at Tue, 2 Jul 2024 20:16:16 UTC No. 16264868
>>16264862
no the great circle was a red herring but the general idea is correct. there's no reason to intersect great circles. all that's needed is an existence of one closed and finite integral curve
Anonymous at Tue, 2 Jul 2024 20:16:39 UTC No. 16264869
>>16264858
>since there are no singularities we get a mapping from the disk to the boundary
This is your answer to the question
>how do you get a retraction from the disk to the boundary
LMAO you are delusional. Probably a /pol/fag as well.
Anonymous at Tue, 2 Jul 2024 20:16:53 UTC No. 16264870
>>16264866
Okay Sir I will do the needful and redeem the standard argument
Anonymous at Tue, 2 Jul 2024 20:17:17 UTC No. 16264871
>>16264866
i think you're just jelly bro. this is a legit novel proof of existence of singularities on a sphere for smooth and continuous vector fields
Anonymous at Tue, 2 Jul 2024 20:18:18 UTC No. 16264874
>>16264871
but it's surprising no one else had figured this out before me. i really am an ultra genius
Anonymous at Tue, 2 Jul 2024 20:20:05 UTC No. 16264878
>>16264874
You don't have a retraction onto the boundary.
Anonymous at Tue, 2 Jul 2024 20:20:47 UTC No. 16264879
>>16264878
i do because if i don't then i can find nested closed curves which must converge to a singularity
Anonymous at Tue, 2 Jul 2024 20:21:51 UTC No. 16264881
>>16264879
actually, this means the existence of any finite closed curve must bound a singularity.
Anonymous at Tue, 2 Jul 2024 20:44:47 UTC No. 16264906
>>16264881
Now I understand why the lefschets number and orthognal flows create singularities. This all fits together very nicely