Timeline for Deeper meanings of barycentric subdivision
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 25, 2014 at 15:23 | comment | added | Jean Van Schaftingen | The oldest reference that I am aware of generalizing the subdivision of an equilateral triangle into four equilateral triangle is Freudenthal (1942) and the decomposition into four triangles which share a common vertex is the book of Whitney (1957) in Appendix 1, §1. | |
| Jul 9, 2010 at 11:21 | comment | added | Tom Goodwillie | Or maybe it can distort them a lot. But somehow, if you do it $m$ times, the diameter of a little simplex is bounded by $\frac{D}{2^m}$ for a constant $D$ even though you don't get this by working out a bound for what happens when you do it one time. | |
| Jul 9, 2010 at 11:11 | comment | added | Tom Goodwillie | By arbitrarily close to $1$ I mean: you can have a $3$-simplex in $\mathbb R^3$ with diameter $1$ such that when you subdivide it into $2^3$ simplices in this way then some $1$-simplex of the subdivision has diameter $>1-\epsilon$. This $1$-simplex connects the midpoints of two opposite edges. This does not contradict the D/k bound. Iterated barycentric subdivision creates some very distorted simplices; iterated edgewise subdivision does not. | |
| Jul 9, 2010 at 7:25 | comment | added | Kerry | I don't really understand how it could be arbitrarily close to 1, but the $\frac{D}{k}$ bound makes sense. This is second time I got answers from the book authors. I'm glad Hatcher is still alive. | |
| Jul 8, 2010 at 22:23 | comment | added | Tom Goodwillie | Yet it's not hard to see that in the $k$-fold edgewise subdivision of an $n$-simplex every little simplex has diameter bounded by $\frac{D}{k}$ where $D$ is the diameter of a cube containing the big simplex. And edgewise subdivision repeated $m$ times is $2^m$-fold edgewise subdivision. | |
| Jul 8, 2010 at 22:21 | comment | added | Tom Goodwillie | Thinking it over, I see that these edgewise subdivisions do make things smaller in the sort of way one needs. But the story is rather different from the barycentric one. In his book Allen Hatcher shows that the diameter of an $n$-simplex shrinks by at worst some definite amount depending only on $n$ (in fact $\frac{n}{n+1}$), no matter what metric (inherited from an ambient Euclidean space) is used. This fails for at least one kind of edgewise subdivision of a $3$-simplex (where new vertices are placed at midpoints of edges); the ratio of diameters can be arbitrarily close to $1$. | |
| Jul 8, 2010 at 22:10 | comment | added | Dev Sinha | I would say that other subdivisions are not so rare, in the sense that while individually each subdivision might not appear in many different places, needing an alternate subdivision of some sort is not unusual. Immediately I can think of a generalization of edgewise subdivision used by Bokstedt, Hsiang and Madsen (I now see that Goodwillie mentions it in his answer) and a decomposition of a simplex into products of lower-dimensional simplices used by McClure-Smith (first case: a 2-simplex as the union of a product of 1-simplices (that is, a square) and two 2-simplices). | |
| Jul 8, 2010 at 17:59 | comment | added | Tom Goodwillie | There are two kinds of edgewise subdivision. See my edited answer to this question. I think that Segal's is the second one that I mention there. | |
| Jul 8, 2010 at 17:10 | comment | added | Dan Ramras | I'm pretty sure this discussion is about what's usually called (Segal's) edge-wise subdivision, and it seems to come up quite a bit actually. A google search turns up a bunch of references. | |
| Jul 8, 2010 at 16:24 | comment | added | Michael Hutchings | That's one of the ways in which singular homology is easier with cubes than with simplices: there is a natural way to subdivide a cube and it is obvious that sufficiently many iterations produce arbitrarily small cubes. | |
| Jul 8, 2010 at 15:26 | history | answered | Allen Hatcher | CC BY-SA 2.5 |