Timeline for CW-presentation of configurations of points in plane and space
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 8, 2020 at 19:18 | answer | added | Nicholas Kuhn | timeline score: 2 | |
| Dec 8, 2020 at 16:32 | vote | accept | Simon Henry | ||
| Dec 8, 2020 at 16:31 | answer | added | Phil Tosteson | timeline score: 5 | |
| Dec 7, 2020 at 19:38 | comment | added | Simon Henry | Very roughly, I have some 'combinatorialy' generated CW complexes that I want to show are homotopy equivalent to such configuration spaces. In "dimension 2" I convinced myself that I could use discrete Morse theory (or something similar) to reduce them known CW decomposition of the classyfing space of Braid groups. At the end of the day that's probably not the right way to solve my problem but I was hopping that doing some concrete calculations on these complexes would give me a better idea of how to do it more generally, hence the question. | |
| Dec 7, 2020 at 18:50 | comment | added | Ryan Budney | When the number of points $n$ is small compared to the dimension $d$ there are some very cute (and explicit) CW-decompositions coming from "electrostatic potential functions", i.e. take the function that is the sum of the inverse of the pairwise distances between the points. You need to replace $\mathbb R^d$ with $\mathbb D^d$. These functions give you Bott-style Morse functions that decompose these configuration spaces into a union of disc bundles over submanifolds. What do you want to use these CW-decompositions for? | |
| Dec 7, 2020 at 18:44 | comment | added | Simon Henry | Thank you very much. You should write this as an answer ! | |
| Dec 7, 2020 at 18:32 | comment | added | Phil Tosteson | I don't know a canonical source, but I think that this paper: arxiv.org/abs/1110.4137 and the references in it could be a good place to start, in particular Ayala and Hepworth's paper. | |
| Dec 7, 2020 at 18:03 | comment | added | Simon Henry | @PhilTosteson : That seems to be exactly what I'm after. Do you have a recommendation for a reference on the topic ? For the ordered vs unordered, yes of course: that's why I said "using the Salvetti complex", and not that the Salvetti complex was such a presentation. | |
| Dec 7, 2020 at 16:55 | comment | added | Phil Tosteson | Technically the Salvetti complex is for ordered configuration spaces. I would recommend looking at the Fox Neuwirth Fuks cells: these aren't quite a CW decomposition of the configuration space, but they are a stratification into contractible pieces-- which is just as good for homotopical purposes. The combinatorics of the Fox-Neuwirth-Fuks stratification is essentially the same as the quotient of Salvetti by S_n. Also the stratification generalizes to configurations in R^d, and is closely related to Joyal's Theta_d category. | |
| Dec 7, 2020 at 16:13 | history | asked | Simon Henry | CC BY-SA 4.0 |