Questions tagged [triangulations]

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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 ...
Gil Kalai's user avatar
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20 votes
0 answers
534 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 ...
François Laudenbach's user avatar
17 votes
1 answer
1k views

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 ...
Yuji Tachikawa's user avatar
17 votes
2 answers
1k views

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 "...
David Treumann's user avatar
17 votes
1 answer
550 views

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 ...
kleenstar's user avatar
  • 271
16 votes
1 answer
887 views

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 ...
Simon Rose's user avatar
  • 6,240
15 votes
1 answer
603 views

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 ...
Scattering State's user avatar
14 votes
1 answer
885 views

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 ...
leone slavich's user avatar
14 votes
0 answers
379 views

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 ...
Arthur B's user avatar
  • 1,882
12 votes
3 answers
833 views

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 $...
Joe's user avatar
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12 votes
4 answers
1k views

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 ...
Gjergji Zaimi's user avatar
12 votes
0 answers
222 views

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 ...
Gil Kalai's user avatar
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12 votes
0 answers
133 views

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 $> 3$...
Lev Soukhanov's user avatar
11 votes
1 answer
839 views

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 ...
wonderich's user avatar
  • 10.3k
11 votes
2 answers
304 views

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 ? ...
Thibault Lefeuvre's user avatar
11 votes
2 answers
2k views

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 ...
Jean-Philippe Burelle's user avatar
10 votes
2 answers
735 views

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. $$\...
S.Lim's user avatar
  • 449
10 votes
0 answers
732 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}...
Dmitri Pavlov's user avatar
9 votes
4 answers
410 views

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 ...
Till's user avatar
  • 469
9 votes
3 answers
388 views

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=\...
Andi Bauer's user avatar
  • 2,901
9 votes
1 answer
441 views

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 ...
Joaquin Lema's user avatar
9 votes
1 answer
558 views

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-...
Adam Levine's user avatar
9 votes
1 answer
424 views

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 ...
Anonymous's user avatar
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9 votes
0 answers
203 views

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 ...
Omalley's user avatar
  • 91
8 votes
2 answers
535 views

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, ...
Student's user avatar
  • 5,008
8 votes
1 answer
592 views

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 ...
Hailong Dao's user avatar
  • 30.3k
8 votes
2 answers
742 views

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 ...
Kregnach's user avatar
8 votes
1 answer
522 views

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 (...
Yan X Zhang's user avatar
8 votes
3 answers
798 views

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+...
interstice's user avatar
8 votes
3 answers
781 views

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 $n$ ...
JeffE's user avatar
  • 336
8 votes
3 answers
248 views

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 ...
Anton Kapustin's user avatar
8 votes
1 answer
328 views

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 ...
Benoît Kloeckner's user avatar
8 votes
0 answers
163 views

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 ...
Andi Bauer's user avatar
  • 2,901
8 votes
0 answers
183 views

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 ...
Moishe Kohan's user avatar
  • 9,664
7 votes
2 answers
175 views

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 ...
color's user avatar
  • 99
7 votes
1 answer
246 views

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 $M$. $$\...
wonderich's user avatar
  • 10.3k
7 votes
2 answers
329 views

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 ...
leone slavich's user avatar
7 votes
1 answer
130 views

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 ...
Joseph O'Rourke's user avatar
7 votes
1 answer
182 views

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,...
Suzanne's user avatar
  • 71
6 votes
1 answer
671 views

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
John's user avatar
  • 63
6 votes
1 answer
440 views

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 ...
user136604's user avatar
6 votes
1 answer
534 views

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 ...
G. Blaickner's user avatar
  • 1,137
6 votes
2 answers
147 views

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 ...
domotorp's user avatar
  • 18.3k
6 votes
1 answer
349 views

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 ...
Aaron Trout's user avatar
6 votes
1 answer
152 views

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-...
Matthias Wendt's user avatar
6 votes
1 answer
220 views

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 $\...
Joseph O'Rourke's user avatar
6 votes
0 answers
121 views

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 ...
Agelos's user avatar
  • 1,844
6 votes
0 answers
162 views

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 \...
wonderich's user avatar
  • 10.3k
6 votes
0 answers
207 views

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 ...
wonderich's user avatar
  • 10.3k
6 votes
0 answers
212 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. ...
user137794's user avatar