All Questions
Tagged with triangulations smooth-manifolds
11 questions
5
votes
1
answer
453
views
Every manifold can be cut into cubes
I saw the following statement in my advanced calculus text, which was presented without proof:
If $\bar{D}$ is a compact domain in the plane (that is, closure of an open, connected and bounded subset ...
- 151
5
votes
0
answers
190
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Triangulating piecewise-linear manifolds
Question 1: Is this the mainstream definition of a PL-manifold?
Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
- 396
0
votes
1
answer
101
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A question on relation of different triangulations of a triangulable space
Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, i.e. the restriction on the boundary is itself a triangulation, is it these ...
- 781
8
votes
2
answers
630
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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, ...
- 5,230
12
votes
1
answer
738
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-...
- 779
4
votes
0
answers
161
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(Non-)Orientability of non-triangulable manifolds
We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least ...
- 2,147
3
votes
0
answers
221
views
Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]
Please let me denote the following
(TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold
(PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF
(PL) ...
- 2,147
13
votes
1
answer
899
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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 ...
- 10.5k
5
votes
0
answers
226
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 ...
- 8,723
11
votes
2
answers
326
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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 ?
...
10
votes
0
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742
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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}...
- 37.8k