Questions tagged [triangulations]
The triangulations tag has no usage guidance.
122
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High-Dimensional Analogs of Polygon Spaces
[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.]
Background: Polygon spaces
Given a ...
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0
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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 ...
17
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1
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How can gauge theory techniques be useful to study when topological manifolds can be triangulated?
I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...
17
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2
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What are some triangulations of Grassmannians?
A while ago I heard that there was no known triangulation of the Grassmannian of 3-planes in 6-space.
To believe a statement like that, you have to be a little bit ungenerous about what you mean by "...
17
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1
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550
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Finite union of closed convex sets is triangulable?
I posted this question on math.stackexchange.com, but didn't get an answer.
Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that ...
16
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1
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Can one determine the dimension of a manifold given its 1-skeleton?
This may be an easy question, but I can't think of the answer at hand.
Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...
15
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1
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603
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Acute triangles in "obtuse" polygons?
Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute?
I conjecture ...
14
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1
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Comparing layered triangulations of 3-manifolds which fiber over the circle.
I am sorry but I am reposting this question because I wasn't logged in when I first asked it.
Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which ...
14
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0
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379
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Minimum number of distinct triangles for tesselating the sphere
Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
12
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3
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Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?
There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
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What is the number of equitriangulations of the n-cube?
I wonder if this question has been considered before and if anything is known. My search attempts have failed so far.
Let's consider the n-dimesnional cube, $[0,1]^n$, and let's call a simplex with ...
12
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3-manifolds with stacked links
Stacked spheres
A triangulation of a 2-dimensional sphere is called a stacked sphere if it is obtained inductively from the boundary of a 3-simplex by deleting a 2-face (triangle) $T$ adding a new ...
12
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Finite list of neighborhoods to approximate any finite simplicial complex
It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$...
11
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Critical dimensions D for "smooth manifolds iff triangulable manifolds"
I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
For ...
11
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2
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Triangulation with simplices of same volume
Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?
...
11
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Proving the Gauss-Bonnet theorem for embedded surfaces using triangulations
I heard this really neat elementary proof of the "Gauss-Bonnet Theorem" :
Let $S$ be a surface embedded in $\mathbb{R}^3$. Now take a triangulation of that surface, and approximate the surface with a ...
10
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2
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On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$
In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds.
$$\...
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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}...
9
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4
answers
410
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Minimum number of common edges of triangulations
Let $S$ and $T$ be two triangulations.
We define
$c(S,T)$ as the number of edges shared by $S$ and $T$.
With this, we can define
$f(n) = \min_{P} \min_{S,T} c(S,T)$.
Here the first minimum goes over ...
9
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3
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388
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Are there invariants of cell complexes similar to the Euler characteristic?
The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely
\begin{equation}
\chi=\...
9
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1
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441
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Refining a triangulation
I'm reading Thurston's article "Shapes of polyhedra and triangulations of the sphere." In the introduction he claims the following:
"${}^{(1)}$There are procedures to refine and modify ...
9
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1
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558
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Local behavior of smooth triangulations
If $M$ is a smooth $n$- manifold, a smooth triangulation is defined to be a homeomorphism from a simplicial complex $K$ to $M$ whose restriction to each simplex is a smooth embedding. It's a well-...
9
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1
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424
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Hamiltonian circuit
Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior.
Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...
9
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0
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203
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Is every compact smooth Riemannian manifold bilipschitz equivalent to a finite simplicial complex?
Let $M$ be a compact smooth Riemannian manifold. Then it admits a triangulation, i.e. a finite simplicial complex $K$ which is homeomorphic to $M$. Any such simplicial complex carries a natural metric ...
8
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2
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535
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Presentations of exotic 4-manifolds
TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed).
Are there known presentations of $4$-manifolds $M$ with exotic structures, ...
8
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1
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592
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When is a triangulation of sphere two-colorable?
Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors.
I ...
8
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2
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742
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Three-dimensional triangulations with fixed number of vertices
My question is the following:
Are there triangulations of $S^3$ which (a) are non-degenerate, (b)
have four vertices, and (c) have no edges of degree two?
A side question:
If one represents this ...
8
votes
1
answer
522
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coincidence between minimal triangulation numbers and chromatic numbers
A friend and I were reading up on the classification of compact surfaces when we realized that the minimal number of triangles in a triangulation of the sphere, torus, and projective plane are 4 (...
8
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3
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798
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Alternating Sum Involving Catalan Numbers
I was wondering if anyone knew how to obtain a simpler closed form of the following sum(or had any other insights regarding it):
$$\sum_{k=0}^n (-1)^k{n \choose k} C_{2n-2-k} $$
Here $C_n = \frac{1}{n+...
8
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3
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781
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Efficient topological triangulations of non-convex polyhedra
Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a topological triangulation with complexity $O(n)$?
Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ ...
8
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3
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248
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Colorings of triangulations as a generalization of the four-color problem
My question can be viewed as a generalization of the four-color problem. Instead of a planar graph, consider a triangulation of a d-sphere. One wants to color vertices with N colors so that no two ...
8
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1
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328
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Smoothing of piecewise Euclidean Riemannian metrics
Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...
8
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Is there a combinatorial representation of general topological manifolds similar to triangulations?
Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
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183
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Are triangulations of n-dimensional manifolds determined by lower-dimensional skeleta?
Suppose that $M$ is an $n$-dimensional manifold equipped with a triangulation $T$. Given $n\ge 1$, in order to recover $T$ (up to an isomorphism of simplicial complexes) one needs to know at least ...
7
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2
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The number of $2$-simplices and the number of $1$-simplices in a $4$-dimensional simplicial complex
Given a $4d$ simplicial complex (a triangulation of $4$-manifold), is there any relation between the number of $2$-simplices (triangles) and the number of $1$-simplices (edges)? Generically, is the ...
7
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1
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Bordism for oriented triangulable manifolds without smooth differentiable structures
We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$.
$$\...
7
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2
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Comparing different layered structures for fibered 3-manifolds: example request.
Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular ...
7
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1
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Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?
Given a planar triangulation of (say) a convex region,
imagine the following process to convert it to a triangulation with
no obtuse angles:
Pick an arbitrary obtuse angle at vertex $a$ of $\triangle ...
7
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1
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How many maximal triangulations of a rectangle?
I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow.
Let $L_{m,n}\subset\mathbb R^2$ be a rectangle given by $[0,m]×[0,n]$ with $m,...
6
votes
1
answer
671
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Triangulation of Surfaces without Jordan-Schoenflies
Does anyone know of a proof of the fact that any 2-manifold can be triangulated that does not use the Jordan-Curve Theorem or the Jordan-Schoenflies Theorem? Thanks for your help
6
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1
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Are triangulations of compact manifolds PL homeomorphic?
I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes ...
6
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1
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534
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Properties a triangulation must have in order to describe a manifold
I am mainly interested in the $3$-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every $3$-dimensional topological manifold (with ...
6
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2
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147
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Eberhard-type theorems for Fisk triangulations?
A triangulation of a surface is called a Fisk triangulation if the degree of all but two vertices is even, and these two exceptional vertices of odd degree are neighbors.
I would like to know what ...
6
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1
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349
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What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant)
This is my first question on mathoverflow! It relates to a project I'm undertaking with a student.
Work by Tamura (extending results by Luo and Stong) shows that for any closed 3-manifold $M$ and any ...
6
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1
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Combinatorial curvature of real projective plane
There is a notion of combinatorial curvature due to Forman, see here (published paper) or here (preprint). I checked for a couple of small triangulations of $\mathbb{RP}^2$ (6-vertex, 7-vertex, 9-...
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1
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Does this iterated Delaunay triangulation process always "explode"?
Let $P$ be a set of three noncollinear points in $\mathbb{R}^2$.
Iteratively form the
Delaunay triangulation
$\cal T$ of $P$, and then
augment $P$ by the circumcircle centers of all triangles in $\...
6
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Hamiltonicity for triangulations of the 3-sphere
A classical theorem of Whitney states that the 1-skeleton of every triangulation of the 2-sphere $\mathbb{S}^2$ has a Hamilton cycle as long as each of its 3-cycles bounds a triangle.
I'm wondering if ...
6
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Can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable, PL or DIFF manifold, if $X_4$ is a non-triangulable manifold? [duplicate]
Question: If $X_4$ is a non-triangulable topological (TOP) manifold,
can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable manifold?
can $X_4 \times S^1$, $X_4 \...
6
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If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ be a triangulable manifold?
If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ always be a triangulable manifold?
Let $X_d$ be a $d$-manifold which is NOT a ...
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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. ...