Timeline for A certain kind of proof of the Hairy Ball Theorem
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 12, 2022 at 11:11 | comment | added | HJRW | Looks good. Thanks! | |
| Dec 12, 2022 at 5:38 | comment | added | David Feldman | But otherwise, in at least one direction, the integral curve of $V_1$ starting at $w$ can never cross any $c\in C'$, and thus never escape $W$. Again, the argument above yields a curve in $W$ transverse to $V_2$, and that contradicts the lack of a lower bound. | |
| Dec 12, 2022 at 5:37 | comment | added | David Feldman | Otherwise, some chain $C$ has no lower bound. For each $c\in C$, $c$ sits transverse to $V_1$, or if not, to $V_2$, and said transverse vector field either flows into or out of the disk $c$ bounds. At least one of the four possibilities determines a subchain $C'$ cofinal in $C$, so assume without loss of generality that $V_1$ flows into every disk bounded by some curve $c\in C'$. The closed disks bounded by curves in $C'$ have a non-empty intersection $W$. If $V$ vanishes at some $w\in W$, done. | |
| Dec 12, 2022 at 5:37 | comment | added | David Feldman | BFPT itself admits a proof along similar lines, as follows. Write $D$ for either disk that $S$ bounds. Order, by containment of the closed disks they bound within $D$, the family ${\cal F}$ of all simple closed curves that sit in $D$ and sit transverse to either $V_1$ or $V_2$ (or both). Should every descending chain of such curves have a lower bound, Zorn's Lemma would yield a minimal such curve, $S'$ say. From a point on $S'$ and the corresponding transverse vector field, the argument above yields a fixed point or exposes a violation of the minimality of $S'$. | |
| Dec 11, 2022 at 21:15 | comment | added | David Feldman | Can you iterate the argument to obtain a nested sequence of closed discs, and maybe get a zero that way? Thinking about it a little more, I so see a Zorn's Lemma phrasing that's quite simple. Will write it up. As Brouwer FPT is not intuitionistically valid (indeed Brouwer invented intuitionistic logic to capture this insight), there's got to be something wrong with any proof that looks too constructive. So an appeal to something like ZL seems natural. | |
| Dec 11, 2022 at 18:34 | comment | added | David Feldman | Can you iterate the argument to obtain a nested sequence of closed discs, and maybe get a zero that way? Yes, I had exactly that thought after I posted (but didn't edit just to keep things simple). The iteration will have to go on transfinitely, so there will be things to check at limit ordinals, but not bad. | |
| Dec 11, 2022 at 14:24 | comment | added | HJRW | It seems a bit of a shame to invoke "the vector field version of the Brouwer FPT", which other proofs of the hairy ball theorem will give you for free. (I have Thurston's in mind.) Can you iterate the argument to obtain a nested sequence of closed discs, and maybe get a zero that way? | |
| Dec 11, 2022 at 13:59 | history | edited | Sam Nead |
edited tags
|
|
| Dec 11, 2022 at 12:14 | comment | added | Sam Nead | Nice. Your technique to obtain the simple closed curve $S$ is often called a "shadowing lemma" or a "closing lemma" in dynamics. | |
| Dec 11, 2022 at 7:02 | history | asked | David Feldman | CC BY-SA 4.0 |