Timeline for In Diff, are the surjective submersions precisely the local-section-admitting maps?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 9, 2018 at 12:59 | answer | added | Praphulla Koushik | timeline score: 3 | |
| Sep 29, 2010 at 0:48 | comment | added | Georges Elencwajg | I had a look at the book by Lang you mentioned, Harry: thanks for the reference. It is indeed interesting and idiosyncratic. For example he gives an argument for the local integrability of some function by invoking Hironaka's resolution of singularities (in an introduction to differential geometry!) But, as usual, we can appreciate Lang's gift for detecting the formal aspects of his subject and axiomatizing them. | |
| Sep 28, 2010 at 21:20 | vote | accept | David Roberts♦ | ||
| Sep 28, 2010 at 18:25 | comment | added | Georges Elencwajg | Dear Tim, you are completely right and what you write is not noise but a very pertinent comment. I have edited my answer accordingly. Thank you for your vigilance. | |
| Sep 28, 2010 at 15:34 | comment | added | Tim Perutz | @Georges. At the risk of adding to the noise: if you use the definition you gave in your comment, your answer of "yes" is not right: non-submersions can admit local sections (but the sections can't hit the critical points). | |
| Sep 28, 2010 at 14:43 | comment | added | Harry Gindi | @Georges: Lang's book Fundamentals of differential geometry (which is expanded from his earlier DG book) was partially an attempt to expand the Fascicule des Résultats out to a full book. | |
| Sep 28, 2010 at 14:18 | comment | added | Ryan Budney | @David, in summary, once you settle on a precise definition the answer to your questions is going to follow from the implicit function theorem. | |
| Sep 28, 2010 at 14:15 | comment | added | Georges Elencwajg | That's funnny, without thinking twice, I would have said that $f:X \to Y$ admits local sections if for every $y\in Y$ there is a neighbourhood V of y and a morphism $g:V \to X$ with $f \circ g=Id_V$. Surprisingly there is no consensus, judging from the reactions by Tim, Mariano and André. Where is Bourbaki when you need him to establish the definitive terminology? [I know he wrote a "Fascicule des Résultats" on manifolds, but nobody seems to ever have used it] | |
| Sep 28, 2010 at 13:32 | answer | added | Georges Elencwajg | timeline score: 8 | |
| Sep 28, 2010 at 13:32 | answer | added | André Henriques | timeline score: 17 | |
| Sep 28, 2010 at 13:27 | comment | added | Tim Perutz | Given points $x$ in the source and $y$ in the target of $f$, you can find a local section $s$ such that $s(y)=x$ iff $f(x)=y$ and $D_x f$ is onto. (Use the chain rule and the implicit function theorem.) | |
| Sep 28, 2010 at 12:59 | comment | added | Mariano Suárez-Álvarez | By "local-section-admitting" what do you mean, exactly? That locally at each point there is a section? is the map $(x,y)\in\mathbb R^2\mapsto xy\in\mathbb R$ local-section-admitting? | |
| Sep 28, 2010 at 12:49 | history | asked | David Roberts♦ | CC BY-SA 2.5 |