Timeline for Retriangulating manifolds via triangulations of low local complexity
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 28, 2022 at 14:23 | comment | added | Sam Nead | I don't see how a triangulation of $M \times [0, 1]$ gives a sequence of Pachner moves. I tried to say this in the last (parenthetical) paragraph of my post. | |
| Jun 27, 2022 at 17:08 | comment | added | Andi Bauer | I asked a very similar question. I was wondering: If it was true that there is $L'$ such that any PL $n+1$-manifold can be triangulated with local complexity at most $L'$, would that imply the answer to your question? Since any triangulation of $M\times [0,1]$ triangulated by $T_1$ on $M\times 0$ and $T_2$ on $M\times 1$ yields a sequence of Pachner moves connecting $T_1$ and $T_2$ where all intermediate triangulations have complexity at most $L'$? | |
| Dec 27, 2020 at 18:53 | comment | added | Fedya | @MoisheKohan, I think this is orthogonal to the work of Nabutovsky and Weinberger. The smooth version of this question is: let's say we want to stay in the realm of Riemannian manifolds with sectional curvatures <1 and injectivity radius >1. From a given metric on S^n we can (by rescaling) walk to the standard one through such metrics, but (according to N-W) the intermediate metrics may have vastly larger volume. I suspect the same thing works in PL: we might have to vastly increase the number of vertices along the way, but we can retain bounded geometry. But it's no longer trivial. | |
| Dec 27, 2020 at 13:21 | comment | added | Sam Nead | The $(n+1, 1)$ and $(1, n+1)$ moves remove and add a single vertex, respectively. | |
| Dec 26, 2020 at 21:08 | comment | added | Moishe Kohan | You should also allow barycentric subdivision since Pachner moves proserve the number of vertices. I do not really understand the question, but it is likely related to the work of Nabutosvky and Weinberger on the “landscapes” of spaces of Riemannian metrics. | |
| Dec 26, 2020 at 16:36 | history | asked | Sam Nead | CC BY-SA 4.0 |