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.
Questions tagged [simplicial-complex]
559 questions
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Criterion for the inverse map to be simplicial
Let $K$ and $L$ be abstract simplicial complexes and let $V(K)$, $V(L)$ denote their vertex sets. Then a simplicial map $K \to L$ is a map $f\colon V(K)\to V(L)$ such that $\{v_0,\dots, v_n\}\in K$ implies $\{f(v_0),\dots,f(v_n)\}\in L$. Now an…
johnnytrain
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Linear embeddings are simplicial?
Let $K,L$ be finite simplicial complexes. Suppose there is a topological embedding $f: |K| \to |L|$ such that $f$ restricted to simplices of $K$ is linear (in particular $f(S)$ is completely inside a single simplex of $L$ for a simplex $S \in…
mathquest
- 298
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Star of a simplex
The star of a simplex $\sigma$ is defined to be the union of the interiors of the simplices that have $\sigma$ as a face. I need to show that the star of $\sigma$ is the intersection of all the star of its vertices, that…
LanaDR
- 503
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What does $\Delta^H$ mean in this context?
I am (still) reading Evasiveness of Graph Properties and Topological Fixed Point Theorems. On page 52 it states:
Suppose that $H$ is a group of order $p^m$. With $m \geq 1$, which acts on $\Delta$ in such a way that $\Delta^h$ is a subcomplex for…
Dair
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Relation between the number of facets and of free faces
First, give the definition.
A facet is any simplex in a complex that is not a face of any larger simplex. (aka maximal face)
A simplex $\tau$ is called a free face if it is the face of only one facet in a simplicial complex.
Here is an example.…
Sooner
- 59
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Characteristic function of simplicial complex
Is there some sort of explicit (ideally polynomial) characteristic function of simplicial complexes?
Let $C = \{\{1, 2, 3\}, \{1,2\}, \{2, 3\}, \{1,3\}, \{3, 4\}, \{1\}, \{2\}, \{3\}, \{4\}, \emptyset\}$. Here we have a filled triangle $(1,2,3)$…
Ordev Agens
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