Timeline for In Diff, are the surjective submersions precisely the local-section-admitting maps?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 26, 2019 at 4:01 | comment | added | Praphulla Koushik | Thank you sir.. | |
| Feb 25, 2019 at 22:04 | comment | added | André Henriques | @PraphullaKoushik. Yes. You understand correctly. | |
| Feb 25, 2019 at 5:57 | comment | added | Praphulla Koushik | In case when $f$ is surjective, second condition implies first condition.. Suppose we take $f:M\rightarrow N$ is surjective and $n\in N$. As $f$ is surjective, there exists $m\in M$ such that $f(m)=n$.. Second condition says, there exists a section $s$ from a nbd of $f(m)=n$ to $M$.. This is the first meaning,, Am I misunderstanding anything? | |
| Sep 28, 2010 at 22:32 | comment | added | Pietro Majer | On the first Note: indeed, a linear bounded operator between Banach spaces is a linear section iff it is surjective and its kernel splits. Linear sections are an open set in the space of linear operators between two Banach, also. So the precise criterium for "$f:M\to N$ is a submersion locally at $m\in M$" (this is how I express the Meaning 2) is just: $Df(m)$ is a linear section. | |
| Sep 28, 2010 at 21:21 | comment | added | David Roberts♦ | Thanks, Andre. That disambiguated things nicely. I don't think there would be a nice characterisation of maps in Diff with local sections in the sense of Meaning 1 after all. | |
| Sep 28, 2010 at 21:20 | vote | accept | David Roberts♦ | ||
| Sep 28, 2010 at 14:12 | comment | added | David Carchedi | In fact, a map with property "Meaning 2", in the setting of topological spaces, is said to be a "topological submersion". | |
| Sep 28, 2010 at 13:32 | history | answered | André Henriques | CC BY-SA 2.5 |