All Questions
Tagged with simplicial-stuff simplex
20 questions
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Reference for function space definition of geometric realization of abstract simplicial complex
The following is described on Wikipedia as one way to define the geometric realization of an abstract simplicial complex $K$:
The construction goes as follows. First, define $|K|$ as a subset of $[0,...
- 2,827
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41
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Triangulation of N datapoints in $\mathbb{R}^d$
Given a set of $N$ data points in $\mathbb{R}^d$, is a triangulation always composed of the same number of "triangles", i.e. $d$-simplices?
This seems to be the case when $d=2$. The number ...
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111
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Triangulating Product of Simplicial Complexes
I am currently working on a problem for which I believe the following result is crucial.
The result of this problem was discussed in this post.
Product of simplicial complexes?
However it is not ...
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3
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2
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274
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Geometric realization of simplicial sets via nondegenerate simplices
I have just started studying simplicial sets, and I was given the definition of geometric realization of a simplicial set X as $$|X| = (\coprod_{n \in \mathbb{N}} X_n \times \Delta_n) / \sim$$ with $(...
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Let K be a simplicial complex and σ a simplex of K. Show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty.
I wanna know if I choose the right way to do my proof? it's correct?
To show that the link $\textrm{Lk}(\sigma)$ is a subcomplex of $\mathcal{K}$, if $\textrm{Lk}(\sigma)$ is not empty, we need to ...
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Understanding the Beck-Chevalley condition (II)
The older question here on the site asks about the intuitive meaning of the Beck-Chevalley condition. Accidentally one of the answers has caught my eye.
It made me wonder if it is possible to describe ...
- 1,062
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3
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517
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Do we distinguish two singular simplices if they have different vertex orders?
We define a $\textbf{singular $n$-simplex}$ in $X$ to be a continuous map $\sigma:\Delta^n\to X$ where $\Delta^n$ is the standard $n$-simplex. Now, as an example, Let $X$ be a singleton $\{p\}$. Then ...
- 1,573
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Barycentric subdivision of a simplex is a geometric simplicial complex
My question is part of a question which has been asked a few years ago here: Barycentric subdivisions of simplices yield a simplicial complex, but has not been answered to my satisfaction:
The ...
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570
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Understanding the Eilenberg-Zilber map on singular chains
I am having trouble to understand how the image of an element $a\otimes b\in C_p(X)\otimes C_q(Y)$ under the Eilenberg-Zilber map on singular chains is an element of $C_{p+q}(X\times Y)$.
I am using ...
- 6,443
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the difference between chains and cochains
I have heard the following definitions, for concreteness I will refer a simplicial complex $K$ and a ring $R$, though the definitions can be extended:
A chain is a formal linear combination of ...
- 1,229
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191
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How to divide a unit space into many simplices?
I'm sorry, it may be simple and stupid but I didn't find any relative solutions on the Internet.
Given the unit hypercube $C$ in the Euclidean space $R^n$, how to divide (or we can say "partition", I ...
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2
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2
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434
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What are the vertices of an abstract simplicial complex?
So a simplex is the $k$-dimensional convex hull of $k+1$ vertices. The convex hulls of a subset of it's vertices are it's faces. A simplicial complex is a collection of simplices such that each face ...
- 2,685
1
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90
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Lemma for Hurewicz Theorem (Bredon)
I am trying to understand the following lemma:
If $f,g:I\rightarrow X$ are paths s.t. $f(1)=g(0)$ then the 1-chain $f*g-f-g$ is a boudary.
Proof: On the standard 2-complex (should it say simplex?) $...
- 415
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$2$-skeleton of k-th horn of $\Delta^n$
I'm trying to show that $Sk_2(\Lambda_k^n)=Sk_2(\partial \Delta^n)=Sk_2(\Delta^n)$ for $n \geq 4$
is it true for $n=3$? any solution or reference is very much appreciated.
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If $lk(v;K)$ is collapsible, then $K$ collapses to $del(v;K)$.
I'm doing exercises about simplicial complexes and I'm stuck with one for which I'll first give some definitions.
Let $K$ a simplicial complex and $v\in K^0$ a $0$-simplex (vertex).
The star of $v$ ...
- 6,443
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131
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Homotopy type of simplicial complexes
I am confused about the following question:
Let $K_{\bullet}$ be a simplicial complex. Let $( \sigma, \tau )$ be a pair of its simplices with $\sigma$ the only codimension $1$ coface of $\tau$.
...
- 1,691
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What does it mean that interior of a simplex is a vector space?
Let the $n-1$ simplex be denoted as:
$$S_{n-1} = \{x = (x_1, \ldots, x_n) | \sum\limits_{i = 1}^nx_i = 1, 0\leq x_i \leq 1\}$$
Then the interior of a simplex is simply:
$$S_{n-1}^\circ = \{x = (x_1,...
- 8,374
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Homology of the simplicial complex obtained from an octahedron by removing 4 faces.
Suppose you have the surface of an octahedron and you remove 4 of the eight faces as follows:
If you remove one face then you don't remove all the adyacent faces and so on.
You can look at this as ...
- 897
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Why is the second example not a Simplicial Complex?
This is my first encounter of simplex and simplicial complex, hence I am not very sure about the concepts.
In the definition of Simplicial Complex
, I am not sure why is the second picture not a ...
- 1,165
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223
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Degeneracies of simplex $y$ which appears as any face of some simplex $x$
Let $K$ be simplicial set and $d_i:K_n\rightarrow K_{n-1}$, $s_i: K_n\rightarrow K_{n+1}$ ($i = 0,...,n$) face and degeneracy maps respectively.
Suppose we have some $x\in K_n$ with $d_0x = ... = ...
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