Timeline for Awfully sophisticated proof for simple facts
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 1, 2023 at 20:13 | comment | added | Sridhar Ramesh | Who says $p$ is divisible by $p'$? (And if you don't already know the integers to be a PID or Bezout domain, common assumptions about lowest-terms representations and divisibility and GCDs and so on go out the window.) | |
| Jan 1, 2023 at 15:12 | comment | added | Kenta Suzuki | @SridharRamesh if such $p'$ and $q'$ exist, then $p/p'=q/q'$ will be a common factor of $p$ and $q$, contradicting them being coprime. | |
| Jun 1, 2022 at 23:38 | comment | added | Sridhar Ramesh | How do we know the line from the origin to $(p, q)$ does not pass through any intermediate lattice points; i.e., that there is no $0 < p' < p$ and $0 < q' < q$ such that $\frac{p}{q} = \frac{p'}{q'}$? Any argument I can think of for this would rapidly establish the integers as a PID anyway. | |
| Jun 1, 2022 at 14:09 | comment | added | Ricky Soda | @SridharRamesh I think that the matrix $\in GL_{2}(\mathbb{Z})$ is coming from the (linear) isomorphism of fundamental groups $\mathbb{Z}^2 \rightarrow \mathbb{Z}^2$ induced by the automorphism of the torus. | |
| Jan 24, 2020 at 19:13 | comment | added | Sridhar Ramesh | Why is the automorphism of the torus corresponding to the regluing linear, thus coming from a matrix with integer coefficients? | |
| Jun 25, 2013 at 3:02 | review | Late answers | |||
| Jun 25, 2013 at 13:46 | |||||
| Dec 20, 2012 at 13:08 | comment | added | Vivek Shende | wow that's cool. are there analogous arguments for the other euclidean domains? | |
| Jan 1, 2012 at 15:57 | comment | added | Autumn Kent | I recently discovered that this is exactly how I think about this when I found myself very gingerly giving the algebraic argument in class (which graduate students find obvious) and then cavalierly dispensing the topological argument as the trivial one. | |
| May 6, 2011 at 0:32 | comment | added | Maxime Bourrigan | I think that in the brain of many low-dimensional topologists, rational numbers, Euclid's algorithm and SL(2,Z) are really instantaneously replaced by topological data on the torus (say homotopy classes of essential curves, Dehn twists and mapping class groups). | |
| May 5, 2011 at 20:53 | comment | added | Harry Altman | Strictly speaking this is a bit weaker than Z being a PID, but wow. | |
| May 5, 2011 at 19:37 | history | answered | anonymous | CC BY-SA 3.0 |