Timeline for Can any smooth triangulation of a smooth manifold be blurred?
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19 events
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| Apr 2, 2016 at 14:31 | comment | added | Dmitri Pavlov | @JesseC.McKeown: With respect to your last proposal, it is unclear how to construct “normal scalings around each facet”. In private correspondence with somebody else it was suggested to me that the existence of such scalings can be included in the definition of a smooth triangulation (the term “conical triangulation” was used to describe such objects). However, this simply moves the problem to a different place: we still have to prove that any smooth manifold admits a conical smooth triangulation. | |
| Apr 1, 2016 at 20:58 | comment | added | Jesse C. McKeown | Still unfinished-thought: for a simplex $s$ with facets $a$ partially-ordered by inclusion, find an open cover of $s$ such that $a \subset \cup_{b\leq a} U_b$, $\overline U_a \cap \partial a = \emptyset$, and $U_a \cap U_b = \emptyset$ whenever $a$ and $b$ are incomparable; and choose a smooth partition of unity subordinate to this covering; a sum of normal scalings around each facet weighted by the partition should, within $U_a$, stabilize all planes incident to $a$. | |
| Apr 1, 2016 at 8:10 | comment | added | Dmitri Pavlov | @JesseC.McKeown: “stabilizing the simplices intersecting with O_1” is really difficult in my opinion, at least I don't see any obvious way to do it. | |
| Apr 1, 2016 at 7:57 | comment | added | Dmitri Pavlov | @JesseC.McKeown: Your second to the last comment is right on: the preimage of a generic point on a codimension 1 simplex looks like two rays emanating from that point, and there is no reason for the tangent vectors to match. | |
| Apr 1, 2016 at 3:43 | comment | added | Jesse C. McKeown | New Idea: instead of inflating a balloon inside one simplex at a time, fix neighborhoods $O_1 \supset C_1 \supset O_2 \supset \sigma $, $O_j$ open and $C_1$ closed; find a partial bluring $h_t$, acting as the identity outside $O_1$, such that $h_1 O_2 = \sigma $, and stablizing the simplices intersecting with $O_1$. | |
| Apr 1, 2016 at 3:34 | comment | added | Jesse C. McKeown | Well, this has been fascinating! The closest I'm yet to understanding what you're telling me (thank you, too, for your patience) is that the generic singular fibers of $h_1$ need not match up across the boundary at all. | |
| Mar 31, 2016 at 18:27 | comment | added | Dmitri Pavlov | @JesseC.McKeown: Not really, in the counterexample above we can choose the ambient manifold to be R^n and the simplices to be affine (i.e., each n-simplex is the convex hull of its vertices). So it's as smooth as one could possibly wish for. | |
| Mar 31, 2016 at 18:21 | comment | added | Jesse C. McKeown | OK... that makes it sound like the smoothness of a triangulation is a very strange sort of smoothness indeed. I'd have worried about codimension-2 things, but ... well, golly. | |
| Mar 31, 2016 at 18:00 | comment | added | Dmitri Pavlov | @JesseC.McKeown: The entire family of approaches cannot work: no matter which blurrings Δ^n→Δ^n you choose (functorial in n or not, S_n-equivariant or not), the resulting blurring of the entire triangulation will be smooth away from simplices of codimension 1, but only continuous (and not smooth) on simplices of codimension 1. Even more strongly: for any fixed n≥2 and for any smooth map f: Δ^n→Δ^n one can construct a smooth triangulation T with two adjacent n-simplices a and b such that the map induced by f on T is not smooth on the intersection of a and b. | |
| Mar 31, 2016 at 17:18 | comment | added | Jesse C. McKeown | @dmitri-pavlov, I haven't presented or pretended to present a finished solution --- in fact, I went out of my way to express incomplete knowledge of my first guess; and I mention "convex interpolation" only to equivariantize, not to construct a homotopy. Was there some other similarity of phrasing that you were refering to? | |
| Mar 31, 2016 at 17:05 | comment | added | Dmitri Pavlov | @JesseC.McKeown: What you're suggesting is the first approach discussed in the main post (see the paragraph that starts with “For example, one can construct a family of smooth maps…”). It doesn't work because no matter what the maps b_n are (S_n-equivariant or not), the map h_1 will not be smooth on simplices of codimension 1 and higher (if the ambient manifold has dimension 2 or higher). | |
| Mar 31, 2016 at 16:59 | comment | added | Jesse C. McKeown | Thinking about it another minute, you want an equivariant blurring of a single $n$-simplex, w.r.t. $S_n$, so the "composite" blurrings won't work right away.... but if one did have ANY blurring of one $n$-simplex, you could amplify it into an equivariant blurring, either by barycentric subdivision or by convex interpolation. | |
| Mar 31, 2016 at 16:51 | comment | added | Jesse C. McKeown | It should be enough to argue such a blurring for single $n$-simplices with lots of stationarity at facets; and I'd want to guess that you could amplify a blurring of the real line at $0$ into such a thing by composing blurings along facet normals, but there's more chain rule in that guess than I want to calculate right away. | |
| Mar 31, 2016 at 15:50 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Mar 31, 2016 at 15:13 | comment | added | Dmitri Pavlov | @BenMcKay: I added an example in the main post. | |
| Mar 31, 2016 at 15:13 | history | edited | Dmitri Pavlov | CC BY-SA 3.0 |
Added an example
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| Mar 31, 2016 at 7:44 | comment | added | Dmitri Pavlov | @BenMcKay: The problem lies with the last sentence (“But then“). What's happening here is that the map h_1 collapses a neighborhood of each coordinate plane to that plane. In your example, h_t can be constructed explicitly: choose a cutoff function c that vanishes in a small neighborhood of 0 and is identity outside of a larger neighborhood. Apply c to all three coordinates of a point in the sphere and then project back to the sphere using a radial projection. This gives you h_1, and h_t can be constructed by linearly interpolating between c and id. | |
| Mar 30, 2016 at 18:46 | comment | added | Ben McKay | I am trying to picture this. Cut up a tennis ball into octants, through the coordinate planes through the centre of the ball. Use those octants as my triangulation. This $h_t$ takes each octant to itself, I think, for each $t$. But then so does $h_1$. So $h_1$ doesn't map any point outside a given octant into that octant, because that point belongs to the other octants. But then $h_1$ doesn't map a neighborhood of our given octant to that octant. Where am I lost? | |
| Mar 29, 2016 at 16:38 | history | asked | Dmitri Pavlov | CC BY-SA 3.0 |