Timeline for What's wrong with compact-open topology on the space of maps?
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 14, 2010 at 16:01 | vote | accept | Igor Belegradek | ||
| May 14, 2010 at 14:10 | answer | added | Andrew Stacey | timeline score: 5 | |
| May 14, 2010 at 13:48 | history | edited | Igor Belegradek | CC BY-SA 2.5 |
added 844 characters in body; deleted 7 characters in body
|
| May 14, 2010 at 13:13 | comment | added | Andrew Stacey | It would help if you could focus your question more precisely, then. Can you describe an example that you want to know about? As I said, "compact-open" can mean lots of different things and in infinite dimensions one can often play a little fast-and-loose with the actual topology, so if you could be more specific in your question, we might be able to be more specific in our answers. In that section, KM had a specific purpose in mind: finding manifold structures. To do that, they found that the Whitney topology was better. If you have a different application, something else may work better. | |
| May 14, 2010 at 12:54 | comment | added | Igor Belegradek | @Willie, I mistakenly typed $C^\infty(M,N)$ instead of $C^\infty(M, \mathbb R^n)$; the latter is a vector space. As for your example with $M=N=S^1$, I want $M$ be non-compact. @Petya, I think if $s$ is any section then $ts$ with $t\in [0,1]$ is a continuous path joining $s$ to the zero section. | |
| May 14, 2010 at 12:49 | comment | added | Igor Belegradek | @Andrew, the phrase "$C^\infty(M,\mathbb R^n)$ is a Banach space" came from a typo in my Russian edition of Hirsch's book; it should be $C^r$ with $r$ finite and $>1$. You are right that I lack understanding of these matters, and this is exactly why I am asking the question. I find the viewpoint of KM book disappointing: the way I see it they say that "compact-open topology" whatever it means is no good, and one needs a modification of Whitney topology to make things work. This is no use to me because for my geometric purposes topology of convergence on compact sets is best. | |
| May 14, 2010 at 12:38 | history | edited | Igor Belegradek | CC BY-SA 2.5 |
added 15 characters in body; edited body
|
| May 14, 2010 at 12:11 | history | edited | Igor Belegradek | CC BY-SA 2.5 |
added 10 characters in body
|
| May 14, 2010 at 11:43 | comment | added | Andrew Stacey | Comments on the added quotes: The first is taken from the introduction to a chapter and so should be taken as such. You should read the whole chapter to discover the meaning behind those words. I don't have the source for the second quote to hand, but there's still a mistake there: C^\infty(M,R^n) is not a Banach space. Generally, "compact-open" is a bit misused and so one should always ask "what exactly do you mean?". In particular, the KM book has quite an extensive discussion on the different topologies and I recommend that you read that and then ask a more focussed question. | |
| May 14, 2010 at 11:19 | history | edited | Igor Belegradek | CC BY-SA 2.5 |
added 1110 characters in body
|
| May 14, 2010 at 9:17 | comment | added | Andrew Stacey | As stated, the rumours that you have heard are false. The simplest case is C^\infty(R,R) - this is sections of the trivial R-bundle over R. As Sergei hints in his "answer", this is a locally convex topological vector space, and hence contractible. I suggest that you track down the source of the rumour and get some clarification and then ask a more focussed question. | |
| May 14, 2010 at 9:07 | answer | added | Sergei Ivanov | timeline score: 8 | |
| May 14, 2010 at 4:33 | comment | added | Harry Gindi | And an arXiv paper giving an overview/introduction: arxiv.org/abs/math/0510097 . I hope it's relevant. | |
| May 14, 2010 at 4:29 | comment | added | Harry Gindi | Andrew Stacey has started to write up some stuff on the nLab about the "differential topology of mapping spaces" based on a series of lectures he gave at NTNU. It seems like it might be relevant: ncatlab.org/nlab/show/… | |
| May 14, 2010 at 4:07 | comment | added | user1835 | This is not an answer, but Hirsch's Differential Topology book has a nice chapter on function space topologies. | |
| May 14, 2010 at 2:32 | history | asked | Igor Belegradek | CC BY-SA 2.5 |