Timeline for Morse theory for manifolds with boundary
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 13, 2023 at 17:03 | history | edited | Anton Petrunin | CC BY-SA 4.0 |
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| Aug 13, 2023 at 16:58 | comment | added | Anton Petrunin | @RyanBudney thank you, Vakhrameev contains everything I need (I did not check the rest of the papers). | |
| Aug 12, 2023 at 6:42 | comment | added | Ryan Budney | I wrote a paper using Morse theory on manifolds with cubical corners, so one step up from manifolds with boundary. As references I used Goresky-MacPherson, also Handron's "Generalized billiard paths and Morse theory on manifolds with corners" as well as Vakhrameev "Morse lemmas for smooth functions on manifolds with corners". I forget which reference had which results, but I think between Handron and Vakhrameev you should find everything you need. | |
| Aug 11, 2023 at 23:18 | comment | added | Anton Petrunin | @RyanBudney, still I need a formal reference. | |
| Aug 11, 2023 at 23:10 | history | became hot network question | |||
| Aug 11, 2023 at 21:24 | comment | added | Ryan Budney | The main issue with manifolds with boundary is you get some additional cells where the gradient of the function (restricted to the boundary) is zero. You only get cells if the gradient of the original function is pointing "out" of the manifold. Well, this is for the upwards flow, i.e. flow with gradient. If you take the downwards flow then you get cells when the gradient is pointing in. You can express Alexander duality quite nicely via this out/in duality. | |
| Aug 11, 2023 at 20:11 | comment | added | Tim Perutz | There's a highly illuminating account in Kronheimer-Mrowka, "Monopoles and 3-Manifolds", introductory chapter. | |
| Aug 11, 2023 at 18:47 | answer | added | Gael Meigniez | timeline score: 6 | |
| Aug 11, 2023 at 17:59 | comment | added | Liviu Nicolaescu | Have a look at the local model $f(x_1,\dotsc, x_n)=\pm x_1+\sum_{j=2}^n \pm x_j^2$ in the halfspace $\{x_1\geq 0\}$ | |
| Aug 11, 2023 at 16:04 | history | edited | Anton Petrunin | CC BY-SA 4.0 |
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| Aug 11, 2023 at 15:41 | comment | added | Andy Putman | There are probably more elementary sources, but I suspect whatever you need could be extracted from Goresky-MacPherson’s book “Stratified Morse Theory”. The point is that a manifold with boundary is a stratified space with two strata (the interior, and the boundary). | |
| Aug 11, 2023 at 15:09 | history | asked | Anton Petrunin | CC BY-SA 4.0 |