29
votes

### Can one determine the dimension of a manifold given its 1-skeleton?

You cannot hope to find the dimension exactly from the 1-skeleton alone. The complete graph on seven vertices is both the 1-skeleton of a triangulation of the two dimensional torus and of the five ...

- 83.2k

24
votes

Accepted

### Critical dimensions D for "smooth manifolds iff triangulable manifolds"

All smooth manifolds are triangulable, as you say. This follows from Morse theory, which dictates that you only need to know how to triangulate (PL) handle-attachments, which one can do by hand. The ...

- 8,850

24
votes

Accepted

### Acute triangles in "obtuse" polygons?

Take a very obtuse isosceles triangle and chop its acute angles.

- 41.2k

17
votes

Accepted

### Triangulation with simplices of same volume

I don't know a reference, but I think that we can use a theorem of Moser to get what you want.
Start with any triangulation $\mathcal{T}$ of $M$, whose cardinal is denoted by $k$ and denote by $\...

- 13.9k

14
votes

### Are there invariants of cell complexes similar to the Euler characteristic?

In a certain restricted setting the answer is that the Euler characteristic is the only such invariant. This is not a complete answer to the question since more general things are allowed. However, it ...

- 7,426

11
votes

### The number of $2$-simplices and the number of $1$-simplices in a $4$-dimensional simplicial complex

Perhaps the best results in this direction are given by Klee's Dehn-Somerville equations:
Klee, V., A combinatorial analogue of Poincar\'e's duality theorem, Can. J. Math. 16, 517-531 (1964). ZBL0134....

- 33.8k

11
votes

Accepted

### Are triangulations of compact manifolds PL homeomorphic?

Pachner‘s Theorem is about PL-homeomorphic manifolds, while the Hauptvermutung is asking for a PL-homeomorphism between manifolds which a priori are only homeomorphic.

- 10.1k

11
votes

Accepted

### When is a triangulation of sphere two-colorable?

As Fedor Petrov mentions in the comments, a necessary and sufficient condition is that each vertex has even degree. Here is a proof.
Let $T^*$ be the dual graph of the triangulation. That is, the ...

- 29.7k

8
votes

Accepted

### Is every triangulation of a Euclidean ball by convex tetrahedra shellable?

I haven't had time to check through the details of the construction, but the example B_3_9_18 found in the proof of Theorem 2 here by Frank Lutz appears to be embeddable in 3-space.
Frank specializes ...

- 15.1k

8
votes

### Presentations of exotic 4-manifolds

I guess that this is as explicit and low-tech as it gets: if $X$ is a K3 surface (i.e. a non-singular quartic hypersurface in $\mathbb{CP}^3$, with the complex orientation), then $X \# \overline{\...

- 9,439

7
votes

### Minimum weight triangulation of lattice points in a circle

I would like to propose a suggestion for finding some asymptotic bound, I think it should be $$\frac{1}{2}\cdot (2+\sqrt{2})\cdot \pi r^2.$$
Namely, the ratio of this number to the actual weight will ...

- 28.4k

7
votes

### Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?

$\newcommand{\RP}{\mathbb{RP}}\newcommand{\C}{\mathbb C}\newcommand{\cC}{\mathcal C}$Here's a TFT-style argument
for why it should be possible in principle to use an invariant of triangulations to ...

- 6,626

7
votes

Accepted

### Is every (not necessarily PL-) triangulation of a manifold pure, non-branching and strongly-connected?

Suppose that $M$ is a connected $d$-dimensional topological manifold without boundary. (We make the last assumption to simplify matters.) Let $\Delta$ be the given pseudo-triangulation. So the ...

- 22.4k

7
votes

Accepted

### Refining a triangulation

Suppose that $S$ is a closed, connected surface with negative Euler characteristic. Suppose that $T$ is a triangulation of $S$.
Define "refine" to mean "replace each triangle by four ...

- 22.4k

7
votes

### Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$

A nice class of examples are the (generalized) triangulations with only one edge. The manifolds obtained by removing an open neighborhood of the vertex have totally geodesic boundary (or a cusp in the ...

- 63.2k

6
votes

### Finite union of closed convex sets is triangulable?

This is not true even for unions of two compact convex sets.
To construct a suitable counterexample, consider the unit disk $$D=\{z\in\mathbb C:|z|\le 1\}$$in the complex plane $\mathbb C$. By $\...

- 37.2k

6
votes

Accepted

### Properties a triangulation must have in order to describe a manifold

From the comments:
Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In ...

6
votes

Accepted

### Triangulation of a simplex

You are looking for the edgewise subdivision:
Edelsbrunner, H.; Grayson, D. R., Edgewise subdivision of a simplex, Discrete Comput. Geom. 24, No. 4, 707-719 (2000). ZBL0968.51016.
The basic idea is to ...

- 924

5
votes

### Eberhard-type theorems for Fisk triangulations?

This is only a partial answer.
First, a sphere does not admit Fisk triangulations. Fisk has proved it by using the 4-color theorem, but there is a simpler proof using Fisk's method of coloring ...

- 6,137

5
votes

Accepted

### Minimum weight triangulation of lattice points in a circle

The examples shown below for $\ $ $r=4,\ 5,\ $ and $\ 6\ $ illustrate the idea described in Dmitri's answer. Some modifications in the interior plus the choices of edges near the boundary of the ...

- 7,030

5
votes

### The number of $2$-simplices and the number of $1$-simplices in a $4$-dimensional simplicial complex

The answer is yes for pure abstract simplicial complexes. Thus we consider a finite set $X$ and a system of subsets $S$ in $X$, such that $S$ is closed under taking subsets, every $s \in S$ has at ...

- 315

5
votes

Accepted

### Do random triangulation edge-flips maintain randomness?

It disturbs uniformity. What you have is a Markov chain and it converges to a distribution given by the Perron eigenvector of the transition matrix. To preserve uniformity you need that eigenvector ...

- 35.3k

4
votes

### What is the number of equitriangulations of the n-cube?

Finding the exact number of unimodular triangulations of a cube in higher dimension is, I would say, out of reach. In fact, I would say the number is doubly exponential in n (that is, exponential in ...

- 1,422

4
votes

### Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?

I'll convert my comment to an answer:
Yes, triangulations can distinguish two non-diffeomorphic smooth structures on any 4-dimensional manifold; in particular, given an exotic $RP^4$, there exists an ...

- 8,234

4
votes

Accepted

### Do combinatorially equivalent polytopes have the same triangulations?

Just to mark this question as answered, the comment by Tobias Fritz is spot on. In this answer to a Math Stack Exchange question, Francisco Santos completely resolves your questions. On the one hand, ...

Community wiki

4
votes

### Distance between two points using triangulation

It seems to me that in the general setting of a metric space, what one learns from the sampling data will be precisely the bounds provided by the instances of the triangle inequality that must be ...

- 206k

4
votes

### Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$

In addition to the answers provided above, you might consider looking at
Heard, Damian; Hodgson, Craig; Martelli, Bruno; Petronio, Carlo, Hyperbolic graphs of small complexity, Exp. Math. 19, No. 2, ...

- 5,171

4
votes

### Proof of Lemma 37.5 in Pak's Lectures on Discrete and Polyhedral Geometry

Yes you are right, each $\gamma_{ij}$ is a shortest path that (as it is stated at the start of the proof). And yes, it is implicitly assumed the geodesics between $v_1$, $v_2$, $v_3$, and $v_4$ do not ...

- 41.2k

3
votes

### Complexity of Random Delaunay Triangulation in 3D

Yes, at least for periodic boundary conditions so we don't have to worry about what happens near the boundary. See
Dwyer, Rex A.
Higher-dimensional Voronoĭ diagrams in linear expected time.
Discrete ...

- 18.3k

3
votes

### Geometric realization of an abstract triangulation of the plane

Yes. For finite complexes this is completely standard, for infinite graphs, this follows by a compactness argument (in particular, there is an infinite circle packing, though it is not unique in ...

- 94.5k

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