All Questions
Tagged with triangulations gt.geometric-topology
13 questions with no upvoted or accepted answers
20
votes
0
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540
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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 ...
12
votes
0
answers
229
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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 ...
- 24.7k
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
10
votes
0
answers
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
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
8
votes
0
answers
170
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 ...
- 3,001
8
votes
0
answers
184
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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 ...
- 12.3k
6
votes
0
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209
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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 ...
- 10.5k
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
5
votes
0
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226
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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
4
votes
0
answers
106
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Retriangulating manifolds via triangulations of low local complexity
Suppose that $M$ is closed, connected PL $n$-manifold. We say that a triangulation of $M$ has local complexity at most $L$ if every zero-cell of $T$ meets at most $L$ $n$-simplices. (An alternative ...
- 28.2k
3
votes
0
answers
93
views
Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
- 183
3
votes
0
answers
102
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Find a certain triangulation subordinate to a given covering of a manifold
Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...
- 2,789