Timeline for Is there a version of the Poincaré–Hopf theorem for manifold with corners?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2022 at 19:46 | vote | accept | Ya He | ||
| Feb 7, 2022 at 16:59 | comment | added | Ya He | @MoisheKohan Very sorry, it's my first time to use the two websites. I have deleted the similar post on StackExchange. Thank you. | |
| Feb 7, 2022 at 1:15 | comment | added | Moishe Kohan | Cross-posted here. | |
| Feb 6, 2022 at 21:34 | comment | added | Ya He | @BenMcKay Thanks for the clarification. The motivation of my question is that: simplices and cubes frequently occur in economic theory such as general equilibrium and Nash equilibrium. Those economics papers apply PH index theorem (as shown in the above picture) to manifolds with corners (e.g., simplices and cubes). | |
| Feb 6, 2022 at 18:29 | history | became hot network question | |||
| Feb 6, 2022 at 16:04 | history | edited | LSpice | CC BY-SA 4.0 |
Poincare -> Poincaré; link to book
|
| Feb 6, 2022 at 14:42 | comment | added | Ben McKay | @YaHe: the simplex is a manifold with corners. In particular, the simplex you have written down is an equilateral triangle. Note that angles at corners less than 180 can be altered by change of variables, to any other angle less than 180. | |
| Feb 6, 2022 at 14:35 | answer | added | mlk | timeline score: 11 | |
| Feb 6, 2022 at 13:56 | history | edited | Ya He | CC BY-SA 4.0 |
added 395 characters in body
|
| Feb 6, 2022 at 13:31 | comment | added | mlk | @LoïcTeyssier I think quarter-singularity might be a misleading term here. Corners need not be right angle, so any number of them could join up to form a full singularity. But I guess the definition of degree as signed relative area of $S^{n-1}$ covered by the map would work safely for any angle. | |
| Feb 6, 2022 at 12:26 | comment | added | Ya He | @LoïcTeyssier Thanks for your comments. I have a question: Is the simplex (e.g., $S = \{ ({x_1},{x_2},{x_3}):\sum\limits_{i = 1}^3 {{x_i}} = 1,{\rm{ }}{x_i} \ge 0\}$) a manifold with corners? It has three angles. | |
| Feb 6, 2022 at 12:09 | comment | added | Loïc Teyssier | The obvious thing that discriminates both situations from the point of view of vector fields is whether or not a singularity sits in a corner. Then, you'll have to define what is the index of these "quarter"-singularities (which is OK if the boundary of the manifold is assumed invariant by the vector field). | |
| Feb 6, 2022 at 12:07 | comment | added | Loïc Teyssier | If what you seek exists, you should consider an object that encodes the $C^1$-difference between a closed disc and a closed square: at the topological level both objects coincide. Probably the number of corner / the collection of their angles is relevant. | |
| S Feb 6, 2022 at 10:26 | review | First questions | |||
| Feb 6, 2022 at 12:13 | |||||
| S Feb 6, 2022 at 10:26 | history | asked | Ya He | CC BY-SA 4.0 |