All Questions
Tagged with gt.geometric-topology simplicial-complexes
24 questions with no upvoted or accepted answers
12
votes
0
answers
133
views
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
12
votes
0
answers
359
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Consequences of Zeeman's conjecture
Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.
Zeeman showed that this implies the Poincaré conjecture in ...
- 2,104
9
votes
0
answers
186
views
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
123
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Is every simplicial $d$-sphere linearly embeddable in $\Bbb R^{d+1}$?
A simplicial $d$-sphere is a simplicial complex homeomorphic to the $d$-sphere. It is known that not every such complex can be embedded into $\Bbb R^{d+1}$ as the boundary complex of a convex ...
- 13.6k
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
149
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Small flag triangulations
In geometric group theory and low-dimensional topology, asking for a triangulation of a specific space to be flag can often be somewhat more cumbersome than just turning a CW-complex into a simplicial ...
- 101
7
votes
0
answers
148
views
Cycles in Tits building
Tits building for an $n$-dimensional vector space $V$ is defined to be the simplicial complex corresponding to the poset of proper and non-zero subspaces of $V$. It is denoted by $T(V)$. This is known ...
- 5,901
6
votes
0
answers
132
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Is there any work in topological data analysis on something like "Voronoi complexes"?
Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated ...
- 15.4k
5
votes
0
answers
188
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
5
votes
0
answers
189
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Gaussian curvature/Euler characteristic of Facebook clusters
If I look at a connected subgraph on a small collection of actors (such as a small cluster) in the Facebook social network, and I find that
1) The Euler characteristic of the clique complex built on ...
- 640
4
votes
0
answers
263
views
Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?
Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$.
Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...
- 13.6k
4
votes
0
answers
183
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In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?
I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...
- 13.6k
4
votes
0
answers
273
views
Are triangulations with common refinements PL-homeomorphic?
Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...
- 165
3
votes
0
answers
88
views
Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?
Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
- 13.6k
3
votes
0
answers
191
views
Skeleton of $\mathcal{G}$-simplicial complex
I'm trying to prove a result with simplicial complex endowed with an action of a groupoid. Let start with a formal definition :
$\textbf{Definition.}$ [$\mathcal{G}$-simplicial complex] Let $\mathcal{...
- 53
3
votes
0
answers
152
views
On a generalization of the Borsuk-Ulam theorem / Tucker's lemma for a map from simplex to its boundary
The Borsuk-Ulam theorem is equivalent to $S^{n-1}$ not being a retract of $B^n$. <totally wrong! or else 2+2=4 is equivalent to the Poincare conjecture/thm>
How shall i prove the following ...
2
votes
0
answers
78
views
Stiefel manifolds and "simplicial complex chromated Sitefel manifolds"
Let $K$ be a simplicial complex whose vertices are labelled by $1,2,\cdots,k$. I want to define a variant concept of the open Stiefel manifolds
$$
V_K(\mathbb{R}^n):=\{(v_1,v_2,\cdots,v_k)\in\prod_k\...
- 543
2
votes
0
answers
68
views
Any results on rayless simplicial complexes?
We define a closed ray in a topological space $X$ to be its closed subset homeomorphic to the real half-line $[0,\infty)\subseteq \mathbb{R}$. Call a topological space $X$ rayless if it does not ...
- 2,104
1
vote
0
answers
69
views
"Bad" valid edge contractions
In this paper, an edge contraction of a simplicial complex $\Gamma$ is defined as the operation of removing the neighborhood $N_e\Gamma$ of the edge $e=\{0,1\}$ and identifying $N_0\partial N_e\Gamma$ ...
- 11
1
vote
0
answers
68
views
Complex of groups of an orbifold
I have just started to look a bit into orbifolds. The English Wikipedia page mentions that there is a complex of groups associated to an (effective) orbifold.
However I couldn't manage to find the ...
- 19
1
vote
0
answers
160
views
Simplicial sets and oriented simplicial complexes
$\DeclareMathOperator\Sing{Sing}$I'm writing a paper about simplicial sets and how they may “replace” simplicial complexes in some known results. To do this, we need to check that they induces the ...
- 11
1
vote
0
answers
49
views
Smooth subdivision which is not rectilinear
What is the simplest example of a smooth subdivision of the standard simplex $\Delta^n$ which can not be realized as a rectilinear subdivision?
That is I want a simplicial complex $K$ for which there ...
- 313
1
vote
0
answers
125
views
Do bistellar flips in a simplicial sphere preserve facet connectivity?
Define two facets of a simplicial sphere as k-connected if there are at least k disjoint edge-vertex paths between them in their facet-ridge graphs.
Do bistellar flips preserve facet k-connectivity?
...
0
votes
0
answers
260
views
Another definition of singular homology
The singular homology is defined via standard simplex. Now if I propose another definition of singular homology groups, based on arbitrary simplex, as follows:
Let $X$ be a topological space. A $n$-...
- 781