Timeline for Local topology of Whitney stratified spaces
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 29, 2021 at 19:13 | vote | accept | Jesse Wolfson | ||
| Mar 20, 2021 at 17:06 | comment | added | Benoît Kloeckner | @JesseWolfson beware that "distance" is not to take literally, you have functions with some compatibility conditions and the euclidean distance in your example does not satisfy them (the $\ell_\infty$ one would work). However there is a subtelty left, as you may define something discontinuous if you apply what I had in mind bluntly. | |
| Mar 19, 2021 at 18:59 | vote | accept | Jesse Wolfson | ||
| Oct 29, 2021 at 19:12 | |||||
| S Mar 19, 2021 at 18:26 | history | suggested | Jesse Wolfson | CC BY-SA 4.0 |
added the reference David provided.
|
| Mar 19, 2021 at 18:17 | review | Suggested edits | |||
| S Mar 19, 2021 at 18:26 | |||||
| Mar 19, 2021 at 18:06 | comment | added | Jesse Wolfson | @Benoit: the kind of example that I'm getting turned around on with min distance args is the linear subspace arrangement I included in my question. Taking n=1, this is just the union of the axes in $\mathbb{R}^2$. If I naively try to implement the min distance to strata, I never get a point to retract to the origin (since one axis or another is always closer, by inspection.) I can fix this in this setting as I explained. But I haven't figured out how to generalize this to an arbitrary example. Is there something I'm missing that you have in mind? | |
| Mar 19, 2021 at 18:03 | comment | added | Jesse Wolfson | @David, thanks! I'll check this out! | |
| Mar 19, 2021 at 17:57 | comment | added | David C | Sorry Jesse I was lazy, in fact Thom-Mather stratifications, are part of a larger class of stratified spaces defined by F. Quinn and called "Homotopically stratified spaces" and these stratified spaces have such neighourhoods for a recent treatment : "Strongly stratified homotopy theory" by D. A. Miller can be useful in particular proposition 2.7 (due to F. Quinn). Anyway, as Benoit suggested I think there is a direct proof without using this stuff. | |
| Mar 19, 2021 at 17:52 | comment | added | Jesse Wolfson | @Benoit, thanks for the reference. The \min distance to strata idea is one I've been trying to implement but getting turned around with. I'll check out your notes! | |
| Mar 19, 2021 at 17:47 | comment | added | Benoît Kloeckner | @JesseWolfson : Mather/Thom stratification come by definition with tubular neighborhoods; there is something left to prove to go the closure of a stratum, which should involve taking the argmin of the "distance to strata" functions over all adherent strata to choose to which one project. You might find useful an expository paper of mine (in French, sorry!): perso.math.u-pem.fr/kloeckner.benoit/papiers/… especially definition 3.1 and Theorem 3.2 | |
| Mar 19, 2021 at 17:29 | comment | added | Jesse Wolfson | Hi David, thanks for the answer. How does admitting a structure of an abstract stratified set imply that for any union of strata there is an open neighborhood of the union which deformation retracts onto it? | |
| Mar 19, 2021 at 16:57 | history | answered | David C | CC BY-SA 4.0 |