Questions tagged [simplicial-complex]
A finite simplicial complex can be defined as a finite collection $K$ of simplices in $\mathbb{R}^N$ that satisfies the following conditions : (1) Any face of a simplex from $K$ is also in $K$ and (2) The intersection of any two simplices in $K$ is a face of both simplices.
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How is a Coxeter complex a simplicial complex?
In e.g. the first definition on Wikipedia, a Coxeter complex is given as a quotient of the Tits cone (with 0 removed) by the positive reals. This makes perfect sense as a topological space, but then ...
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Triangulations of manifolds are non-branching
Let $X$ be an $n$-manifold and let $K$ be a triangulation of that manifold. I am looking for a proof of the fact that $K$ is non-branching, which means:
There is no simplex $S \in K$ of dimension $n-1$...
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Prove continuity of the affine extension mapping between geometric simplicial complexes
Let $\Delta_1$ and $\Delta_2$ be geometric simplicial complexes. Let $K_1$ and $K_2$ be their associated abstract simplicial complexes. Let $f: V(K_1) \to V(K_2)$ be a simplicial mapping.
We define ...
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Relation between local homeomorphism and homological dimension
For a given topological space $X$ (one can assume a simplicial complex if required), define it's homological dimension $\operatorname{hdim}(X)$ as the largest integer $n$ such that $H_n(X,A)\ne 0$ for ...
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Free abelian group functor preserves Kan fibration
Suppose, $\mathbb{Z}$ is the free abelian group functor from simplicial sets to simplicial abelian groups. Then does $\mathbb{Z}$ preserves Kan fibrations i.e. if $X \to Y$ is a Kan fibration between ...
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Spaces with free $\pi_1$ but not homotopy equivalent to a wedge sum of spheres?
Background
The Rips complex of a metric space $X$ at scale parameter $r \in [0, \infty)$, denoted $\mathcal R(X)_r$, is the simplicial complex with simplices all finite subsets of $X$ with diameter at ...
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minimal cuts and pastes to make a t-shirt?
(I have not found an exercise about this in do Carmo or Struik, and I think it could go a long way towards building intuition about curvature and develop skills in computing special 2D areas, as well ...
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Is every (finite) simplicial complex the nerve of some covering?
I need to prove that for every finite simplicial complex $\Delta$ exists a Hausdorff paracompact space $X$ and a good covering $\mathfrak{U}$ of $X$ such that the nerve of the complex is $\Delta$. I ...
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Homeomorphism between geometric realization of abstract simplicial complex and triangulation of set.
I would like to understand the relation between two different definitions of triangulations. The first one is defined directly within the triangulated space, and the second one is defined via a ...
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How to set up this problem geometrically? (Hatcher AT Page 131 Problem 3)
I'm attempting to go through all of Hatcher's problems on homology. I was able to do 1, 2, 4, and 5 so far, but I don't know what he's asking for geometrically in 3. I see this thread (Hatcher ...
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Orientation of boundary $\partial_2$ on torus (and other CW complexes)?
This is a simple and stupid question, but I still can't figure out how the map $\partial_2$ gets oriented. For the torus $T$:
Hatcher and Wikipedia give the boundary as:
$$ \partial_2 U = a + b - c = ...
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Explanation of "boundary of a boundary is 0" in an unoriented simplicial complex
I have already known that in algebraic topology $\partial^2=0$ holds, or $\partial_{n-1}\partial_n=0$ to be more specifically. If one chooses all $n$-simplicies as the basis of the $n$-chain vector ...
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If $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$, Prove that $G/H \cong \mathbb{Z}$
I am trying to prove that if $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$ with both of them being free abelian groups, then $G/H \cong \...
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How to understand May's proof that counit map is a weak equivalence?
A similar question was asked about 4 years ago here, but received no answers, so I hope it is appropriate to post a new question. I am trying to read the singular homology section in May's Concise ...
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Definition for orders corresponding to directed acyclic graphs (DAG)
My question
What is the name for a binary relation $R$ on $V$ that corresponds to a graph $G = (V,E)$ that is a directed acyclic (simple) graph?
Background
There is a bijection between simple directed ...
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