All Questions
Tagged with triangulations simplicial-complexes
14 questions
9
votes
0
answers
186
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What is an intuitive explanation for a manifold to have no triangulation?
It is known that some topological manifolds, even compact and simply-connected ones, do not have admit a triangulation. One example is the E8 manifold in a dimension as low as $4$.
I am trying to ...
- 5,327
5
votes
0
answers
190
views
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
6
votes
2
answers
370
views
Does every triangulable manifold have a vertex-transitive triangulation?
Does every triangulable manifold have a vertex-transitive triangulation?
When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically ...
- 345
4
votes
1
answer
304
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Do combinatorially equivalent polytopes have the same triangulations?
A triangulation of a convex polytope $P\subset\Bbb R^n$ is a partition of $P$ into $n$-simplices $\{\Delta_1,...,\Delta_m\}$ each of which has all its vertices among the vertices of $P$. A polytope ...
- 13.6k
1
vote
1
answer
135
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Annulus theorem for pseudomanifolds
Lets say I take an arbitrary closed and smooth $d$-manifolds $\mathcal{M}$. Now, it is a well-known fact that whenever I take two (sufficiently nice embedded) closed $d$-balls $B_{1}$ and $B_{2}$ in $\...
- 1,429
6
votes
0
answers
123
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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 ...
- 1,936
9
votes
1
answer
793
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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 ...
- 1,429
8
votes
0
answers
170
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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 ...
- 3,001
6
votes
1
answer
466
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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 ...
- 165
12
votes
0
answers
133
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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 $> 3$...
- 1,842
16
votes
1
answer
906
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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 ...
- 6,290
9
votes
3
answers
403
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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=\...
- 3,001
7
votes
2
answers
187
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 ...
- 109
17
votes
1
answer
582
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 ...
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