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Questions tagged [triangulations]

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19
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0answers
486 views

Homeomorphisms of the sphere mapping a geodesic triangulation to another one

Consider the standard Riemannian 2-sphere $S$, equipped with a geodesic triangulation $T$. Let $L(S,T)$ be the space of homeomorphisms of $S$ which map $T$ to a geodesic triangulation. What is the ...
8
votes
0answers
482 views

Can any smooth triangulation of a smooth manifold be blurred?

For the purposes of this question, let's say that a blurring of a smooth triangulation $T$ of a smooth manifold $X$ is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id}...
5
votes
0answers
156 views

Are there non-cuspy triangulations of smooth manifolds?

In (as it turned out my misunderstanding of) the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a ...
4
votes
0answers
151 views

Most regular way to triangulate $\mathbb{R}^3$?

By "regular", I'm going by a property of Delaunay Triangulation, which is to maximize the minimum angle. In $\mathbb{R}^2$, tiling with equilateral triangles gives you a minimum angle of 60 degrees. ...
4
votes
0answers
219 views

(∞,n)-category of triangulated cobordisms

What is an accepted definition of a (∞,n)-category of triangulated cobordisms? Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
4
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0answers
203 views

Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold

Work by Tamura (extending results by Luo and Stong) shows the following. Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for ...
3
votes
0answers
205 views

regular triangulations of the product of two simplices

Is description of all regular triangluations of $\Delta^n\times \Delta^k$ known? (Regular triangulations are those which correspond to vertices of Gelfand--Kapranov--Zelevinsky secondary polytope, or, ...
2
votes
0answers
73 views

Existence of a “generic enough” lattice point interior to a lattice triangle

Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...
2
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0answers
79 views

Find a certain triangulation subordinate to a given covering of a manifold

Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...
2
votes
0answers
77 views

Does any smooth oriented closed orbifold have a fundamental class

This thread:triangulation of orbifolds has shown that any smooth closed orbifold has a triangulation. My further question is: if the difference of any two triangulations $P$ and $Q$ is a boundary of a ...
2
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0answers
69 views

Existence of triangulation of Lipschitz domains

Consider a bounded Lipschitz domain $\Omega \subset \mathbb R^n$. Q1: Can its closure $\overline\Omega$ be triangulated? Q2: If yes, can the triangulation be chosen as finite? Furthermore, how ...
2
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0answers
154 views

Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure

I am a beginner in Contact Geometry. To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes. ...
1
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0answers
74 views

Interior and boundary vertices of weighted graphs

Xu He's article Rigidity of Infinite Disk Patterns and I have a problem with a statement he makes on page 7. He considers weighted embedded planar graphs $G=(V, E)$ with weight function $\Theta: E \...
1
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0answers
20 views

Finding Equal-volume Triangulations in Homogenic Coordinates

Given the $n$-dimensional triangulation $\mathbb{T}^n$ of a finite set of points $\{p_1,\ ...\ ,\ p_{n+k}\} \subset \mathbb{R}^n$, is it always possible to find $n+k$ positive real weights $\{\omega_1,...
1
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0answers
176 views

dissections and vertices of non-convex polytopes

Let us call a finite union $P$ of $n$-dimensional compact convex polytopes in $\mathbb{R}^n$ a non-convex polytope. Recall that a dissection of $P$ is a finite collection $T$ of $n$-dimensional ...
0
votes
0answers
28 views

dual and intersection of a simplex

In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote ...
0
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0answers
32 views

Edgebreaker algorithm over 2-manifolds

Suppose I triangulate a 2-manifold and make a dual pseudograph over it. If I do the Edgebreaker compression algorithm over this graph to generate a spanning tree, can exist an edge with three or more ...