# Tag Info

### 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
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
Accepted

### Acute triangles in "obtuse" polygons?

Take a very obtuse isosceles triangle and chop its acute angles.
• 41.2k
Accepted

• 9,439

### 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

### 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
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
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

### 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

• 41.2k